Distinction of Classes in the Feature Space

Even if we take the best features available, there may be classes that can- not be separated. In such a case, it is always worth reminding us that separating the objects in well-defi ned classes is only a model of reality. Often, the transition from one class to another may not be abrupt but rather gradual. For example, anomalies in a cell may be present to a vary-

520                                                                                                    20 Classifi cation

 

Figure 20.6: Illustration of the recognition of letters with very similar shape such as the large ‘O’ and the fi gure ‘0’, or the letters ‘I’ and ‘l’ and the fi gure ‘1’.

 

ing degree, there not being two distinct classes, “normal” and “patho- logical”, but rather a continuous transition between the two. Thus, we cannot expect to fi nd well-separated classes in the feature space in every case. We can draw two conclusions. First, it is not guaranteed that we will fi nd well-separated classes in the feature space, even if optimal fea- tures have been selected. Second, this situation may force us to recon- sider the object classifi cation. Two object classes may either in reality be only one class or the visualization techniques to separate them may be inadequate.

In another important application, optical character recognition, or OCR, we do have distinct classes. Each character is a well-defi ned class. While it is easy to distinguish most letters, some, e. g., the large ‘O’ and the fi gure ‘0’, or the letters ‘I’ and ‘l’ and the fi gure ‘1’, are very similar,

i. e., lie close to each other in the feature space (Fig. 20.6). Such well- defi ned classes that hardly diff er in their features, pose serious problems for the classifi cation task.

How can we then distinguish the large letter ‘O’ from the fi gure ‘0’ or an ‘l’ from a capital ‘I’? We can give two answers to this question. First, the fonts can be redesigned to make letters better distinguishable from each other. Indeed, special font sets have been designed for automated character recognition.

Second, additional information can be brought into the classifi cation process. This requires, however, that the classifi cation does not stop at the level of individual letters; it must be advanced to the word level. Then, it is easy to establish rules for better recognition. One simple rule which helps to distinguish the letter ‘O’ from the fi gure ‘0’ is that letters and fi gures are not mixed in a word. As a counterexample to this rule, take British or Canadian zip codes which contain a blend of letters and fi gures. Anybody who is not trained to read this unusual mix has serious problems in reading and memorizing them. As another example, the capital ‘I’ can be distinguished from the lowercase ‘l’ by the rule that capital letters occur only as the fi rst letter in a word or in an all-capital-letter word.

We close this section with the comment that asking whether a clas- sifi cation is at all possible for a given problem either by its nature or by the type of possible features is at least as important, if not more so, than the proper selection of a classifi cation method.

20.2 Feature Space                                                                       521

 

Principal Axes Transform

The discussion in the previous section suggested that we must choose the object features very carefully. Each feature should bring in new in- formation which is orthogonal to what we already know about the ob- ject classes; i. e., object classes with a similar distribution in one feature should diff er in another feature. In other words, the features should be uncorrelated. The correlation of features can be studied with the statis- tical methods discussed in Section 3.3, provided that the distribution of the features for the diff erent classes is known (supervised classifi cation). The important quantity is the cross-covariance of two features m_p and m_q from the P-dimensional feature vector for one object class, which

is defi ned as                                                                       

C_pq =.m_p − m_pΣ  .m_q − m_q)Σ.                                      (20.1)

If the cross-covariance C_pq is zero, the features are said to be uncorre- lated or orthogonal. The variance

2

C_pp =.m_p − m_pΣ

 

(20.2)

 

is a measure for the variance of the feature. A good feature for a certain object class should show a small variance indicating a narrow extension of the cluster in the corresponding direction of the feature space. With P features, we can form a symmetric matrix with the coeffi cients C_pq, the covariance matrix



C = 

C11  C12  ...    C1, P

C12  C22  ...    C2, P

 

                      



 .                        (20.3)

.     .  ...    .

                               

 C1, P C2, P... CP, P 

The diagonal elements of the covariance matrix contain the variances of the P features, while the off -diagonal elements constitute the cross- covariances. Like every symmetric matrix, the covariance matrix can be diagonalized (compare the discussion on the tensor representation of neighborhoods in Section 13.3). This procedure is called the principal- axes transform. The covariance matrix in the principal-axes coordinate system reads

   C₁'1       0  ··· 0 

=



(20.4)

.

C' 

0  C₂'2

   

...    .

..

 .

.    0



   0 ··· 0  C_p' p   

The diagonalization shows that we can fi nd a new coordinate system

in which all features are uncorrelated. Those new features are linear

522                                                                                                    20 Classifi cation

 

Figure 20.7: Illustration of correlated features and the principal-axes transform.

 

 

combinations of the old features and are the eigenvectors of the covari- ance matrix. The corresponding eigenvalues are the variances of the transformed features. The best features show the lowest variance; fea- tures with large variances are not of much help since they are spread out in the feature space and, thus, do not contribute much to separating diff erent object classes. Thus, they can be omitted without making the classifi cation signifi cantly worse.

A trivial but illustrative example is the case when two features are nearly identical, as illustrated in Fig. 20.7. In this example, the features m₁ and m₂ for an object class are almost identical, since all points in the feature space are close to the main diagonal and both features show a

=  −

large variance. In the principal-axes coordinate system m₂'                               m₁  m₂ is

a good feature, as it shows a narrow distribution, while m₁' is as useless as m₁ and m₂ alone. Thus we can reduce the feature space from two dimensions to one without any disadvantages.

In this way, we can use the principal-axes transform to reduce the dimension of the feature space and fi nd a smaller set of features which does nearly as good a job. This requires an analysis of the covariance matrix for all object classes. Only those features can be omitted where the analysis for all classes gives the same results. To avoid misunder- standings, the principal-axes transform cannot improve the separation quality. If a set of features cannot separate two classes, the same feature set transformed to the principal-axes coordinate system will not do so either. Given a set of features, we can only fi nd an optimal subset and, thus, reduce the computational costs of classifi cation.

20.3 Simple Classifi cation Techniques                                            523

 

20.2.6 Supervised and Unsupervised Classifi cation

We can regard the classifi cation problem as an analysis of the struc- ture of the feature space. One object is thought of as a pattern in the feature space. Generally, we can distinguish between supervised clas- sifi cation and unsupervised classifi cation procedures. Supervision of a classifi cation procedure means determining the clusters in the feature space with known objects beforehand using teaching areas for identify- ing the clusters. Then, we know the number of classes and their location and extension in the feature space.

With unsupervised classifi cation, no knowledge is presumed about the objects to be classifi ed. We compute the patterns in the feature space from the objects we want to classify and then perform an analysis of the clusters in the feature space. In this case, we do not even know the number of classes beforehand. They result from the number of well- separated clusters in the feature space. Obviously, this method is more objective, but it may result in a less favorable separation.

Finally, we speak of learning methods if the feature space is updated by each new object that is classifi ed. Learning methods can compensate any temporal trends in the object features. Such trends may be due to simple reasons such as changes in the illumination, which could easily occur in an industrial environment because of changes in daylight, aging, or dirtying of the illumination system.

 

20.3 Simple Classifi cation Techniques

In this section, we will discuss diff erent classifi cation techniques. They can be used for both unsupervised and supervised classifi cation. The techniques diff er only by the method used to associate classes with clus- ters in the feature space (Section 20.2.6).

Once the clusters are identifi ed by either method, the further classi- fi cation process is identical for both of them. A new object delivers a feature vector that is associated with one of the classes or rejected as an unknown class. The diff erent classifi cation techniques diff er only by the manner in which the clusters are modeled in the feature space.

Common to all classifi ers is a many to one mapping from the fea- ture space M to the decision space D. The decision space contains Q elements, each corresponding to a class including a possible rejection class for unidentifi able objects. In the case of a deterministic decision, the elements in the decision space are binary numbers. Then only one of the elemets can be one; all others must be zero. If the classifi ers gener- ates a probabilistic decision, the elements in the decision space are real numbers. Then the sum of all elements in the decision space must be one.

524                                                                                                    20 Classifi cation

 

20.3.1 Look-up Classifi cation

This is the simplest classifi cation technique but in some cases also the best, since it does not perform any modeling of the clusters for the dif- ferent object classes, which can never be perfect. The basic approach of look-up classifi cation is very simple. Take the feature space as it is and mark in every cell to which class it belongs. Normally, a signifi cant amount of cells do not belong to any class and thus are marked with 0. In case the clusters from two classes overlap, we have two choices.

First, we can take that class which shows the higher probability at this cell. Second, we could argue that an error-free classifi cation is not pos- sible with this feature vector and mark the cell with zero. After this initialization of the feature space, the classifi cation reduces to a simple look-up operation (Section 10.2.2). A feature vector m is taken and is looked up in the multidimensional look-up table to see which class, if any, it belongs to.

× ×

Without doubt, this is a fast classifi cation technique which requires a minimum number of computations. The downside of the method — as with many other fast techniques — is that it requires huge amounts of memory for the look-up tables. An example: a three-dimensional fea- ture space with only 64 bins per feature requires 64 64 64 = 1/4 MB of memory — if no more than 255 classes are required so that one byte is suffi cient to hold all class indices. We can conclude that the look-up table technique is only feasible for low-dimensional feature spaces. This suggests that it is worthwhile to reduce the number of features. Alterna- tively, features with a narrow distribution of feature values for all classes are useful, since then a rather small range of values and, thus, a small number of bins per feature suffi ciently reduce the memory requirements.

 

20.3.2 Box Classifi cation

The box classifi er provides a simple modeling of the clusters in the fea- ture space. A cluster of one class is modeled by a bounding box tightly surrounding the area covered by the cluster (Fig. 20.8). It is obvious that the box method is a rather crude modeling. If we assume that the clus- ters are multidimensional normal distributions, then the clusters have an elliptic shape. These ellipses fi t rather well into the boxes when the axes of the ellipse are parallel to the axes of the feature space. In a two- dimensional feature space, for example, an ellipse with half-axes a and b has an area of π ab, the surrounding box an area of 4ab. This is not too bad.

When the features are correlated with each other the clusters become long and narrow objects along diagonals in the feature space. Then the boxes contain a lot of void space and they tend much more easily to overlap, making classifi cation impossible in the overlapping areas. How-

20.3 Simple Classifi cation Techniques                                            525

0.8

0.6

0.4

0.2

0

                  

 

                                                                                                                                                                                                                                                                                                        

 

                                                                                                                                                                                                                                                                                             

 

                                                                                                                                                                                                                                                                                                        

 

                       

0       200     400    600     800    1000  1200

 

Figure 20.8: Illustration of the box classifi er for the classifi cation of diff erent seeds from Fig. 20.2 into peppercorns, lentils, and sunfl ower seeds using the two features area and eccentricity.

 

Table 20.1: Parameters and results of the simple box classifi cation for the seeds shown in Fig. 20.2. The corresponding feature space is shown in Fig. 20.8.

 

	Area	Eccentricity	Number
total	—	—	122
peppercorns	100–300	0.0–0.22	21
lentils	320–770	0.0–0.18	67
sunfl ower seeds	530–850	0.25–0.65	15
rejected			19

 

 

ever, correlated features can be avoided by applying of the principal-axes transform (Section 20.2.5).

The computations required for the box classifi er are still modest. For each class and for each dimension of the feature space, two compari- son operations must be performed to decide whether a feature vector belongs to a class or not. Thus, the maximum number of comparison operations for Q classes and a P-dimensional feature space is 2PQ. In contrast, the look-up classifi er required only P address calculations; the number of operations did not depend on the number of classes.

To conclude this section we discuss a realistic classifi cation prob- lem. Figure 20.2 showed an image with three diff erent seeds, namely sunfl ower seeds, lentils, and peppercorns. This simple example shows many properties which are typical for a classifi cation problem. Although the three classes are well defi ned, a careful consideration of the features to be used for classifi cation is necessary since it is not immediately evi-

526                                                                                                    20 Classifi cation

 

A                                                                    b

C                                                                    d

Figure 20.9: Masked classifi ed objects from image Fig. 20.2 showing the classifi ed

a peppercorns, b lentils, c sunfl ower seeds, and d rejected objects.

 

 

dent which parameters can be successfully used to distinguish between the three classes. Furthermore, the shape of the seeds, especially the sunfl ower seeds, shows considerable fl uctuations. The feature selection for this example was already discussed in Section 20.2.3.

Figure 20.8 illustrates the box classifi cation using the two features area and eccentricity. The shaded rectangles mark the boxes used for the diff erent classes. The conditions for the three boxes are summa- rized in Table 20.1. As the fi nal result of the classifi cation, Fig. 20.9 shows four images. In each of the images, only objects belonging to one of the subtotals from Table 20.1 are masked out. From a total of 122 ob- jects, 103 objects were recognized. Thus 19 objects were rejected. They could not be assigned to any of the three classes for one of the following reasons:

•

Two or more objects were so close to each other that they merged into one object. Then the values of the area and/or the eccentricity are too high.

20.3 Simple Classifi cation Techniques                                           527

0.8

0.6

0.4

0.2

0

                       

 

                                                                                                                                                                                                                                                                                                        

 

                                                                                                                                                                                                                                                                                                        

 

                                                                                                                                                                                                                                                                                                        

 

                                                                                                                                                                                                                                                                                                      

 

                                                                                                                                                                                                                                                                                                        

 

                                                                                                                                                                                                                                                                                                        

 

                                                                                                                                                                                                                                                                                                        

 

                                                                                                                                                                                                                                                                                                        

 

                                                                                                                                                                                                                                                                                                      

                                

0       200     400    600     800    1000  1200

 

Figure 20.10: Illustration of the minimum distance classifi er with the classifi - cation of diff erent seeds from Fig. 20.2 into peppercorns, lentils, and sunfl ower seeds using the two features area and eccentricity. A feature vector belongs to the cluster to which it has the minimal distance to the cluster center.

 

•

The object was located at the edge of the image and thus was only partly visible. This leads to objects with relatively small area but high eccentricity.

•

Three large sunfl ower seeds were rejected because of too large an area. If we increased the area for the sunfl ower seed class then also merged lentils would be recognized as sunfl ower seeds. Thus this classifi cation error can only be avoided if we avoid the merging of objects with a more advanced segmentation technique.

 

20.3.3 Minimum Distance Classifi cation

The minimum distance classifi er is another simple way to model the clusters. Each cluster is simply represented by its center of mass m_q. Based on this model, a simple partition of the feature space is given by searching for the minimum distance from the feature vector to each of the classes. To perform this operation, we simply compute the distance of the feature vector m to each cluster center m_q:

P

q

d2 = |m − m_q|2 =. (m_p − m_qp)2.                                 (20.5)

p=1

The feature is then assigned to the class to which it has the shortest distance.

Geometrically, this approach partitions the feature space as illus- trated in Fig. 20.10. The boundaries between the clusters are hyper-

528                                                                                                    20 Classifi cation

 

planes perpendicular to the vectors connecting the cluster centers at a distance halfway between them.

The minimum distance classifi er, like the box classifi er, requires a number of computations that is proportional to the dimension of the feature space and the number of clusters. It is a fl exible technique that can be modifi ed in various ways.

The size of the cluster could be taken into account by introducing a scaling factor into the distance computation Eq. (20.5). In this way, a feature must be closer to a narrow cluster to be associated with it. Sec- ondly, we can defi ne a maximum distance for each class. If the distance of a feature is larger than the maximum distance for all clusters, the object is rejected as not belonging to any of the identifi ed classes.

 

20.3.4 Maximum Likelihood Classifi cation

The maximum likelihood classifi er models the clusters as statistical prob- ability density functions. In the simplest case, P-dimensional normal distributions are taken. Given this model, we compute for each feature vector the probability that it belongs to any of the P classes. We can then associate the feature vector with the class for which it has the maximum likelihood. The new aspect with this technique is that probabilistic deci- sions are possible. It is not required that we decide to put an object into a certain class. We can simply give the object probabilities for member- ship in the diff erent classes.

 

20.4 Further Readings‡

 

From the vast amount of literature about classifi cation, we mention only three monographs here. The book of Schü rmann [166] shows in a unique the common concepts of classifi cation techniques based on classical statistical techniques and on neural networks. The application of neural networks for classifi cation is detailed by Bishop [9]. One of the most recent advances in classifi cation, the so-called support vector machines, are very readably introduced by Christianini and Shawe-Taylor [20].

 

Part V

Reference Part

 

 

 

A Reference Material

R1

Selection of CCD imaging sensors (Section 1.7.1)

(C: saturation capacity in 1000 electrons [ke], eNIR: enhanced NIR sensi- tivity, FR: frame rate in s− 1, ID: image diagonal in mm, QE: peak quantum effi ciency)

 

Chip                            Format

H × V

 

FR ID Pixel size

H × V, µm

 

Comments

 

Interlaced EIA video

Sony1 ICX258AL        768 × 494   30 6.09 6.35 × 7.4      1/3", eNIR

Sony1 ICX248AL        768 × 494   30 8.07 8.4 × 9.8        1/2", eNIR

Sony1 ICX082AL        768 × 494   30 11.1 11.6 × 13.5 2/3"

Interlaced CCIR video

Sony1 ICX259AL        752 × 582   25 6.09 6.5 × 6.25      1/3", eNIR

Sony1 ICX249AL        752 × 582   25 8.07 8.6 × 8.3        1/2", eNIR

Sony1 ICX083AL        752 × 582   25 10.9 11.6 × 11.2 2/3"

Progressive scanning interline

Sony1 ICX098AL        659 × 494   30 4.61 5.6 × 5.6        1/4"

Sony1 ICX084AL        659 × 494   30 6.09 7.4 × 7.4        1/3"

Sony1 ICX204AL       1024 × 768 15 5.95 4.65 × 4.65  1/3"

Kodak2 KAI-0311M     648 × 484   30 7.28 9.0 × 9.0        1/2", QE 0.37 @ 500 nm

×                                ×

Sony1 ICX074AL        659  494   40 8.15  9.9  9.9       1/2", C 32 ke,

QE 0.43 @ 340 nm

Sony1 ICX075AL        782 × 582   30 8.09 8.3 × 8.3        1/2"

Sony1 ICX205AL       1360 × 1024 9.5 7.72 4.65 × 4.65 1/2", C 12 ke

×                                 ×

Sony1 ICX285AL       1360 1024 10 11.0 6.45 6.45 2/3", eNIR, C 18 ke,

QE 0.65 @ 500 nm

×                               ×

Sony1 ICX085AL       1300 1030 12.5 11.1  6.7  6.7            2/3", C 20 ke,

QE 0.54 @ 380 nm

Kodak2 KAI-1020M 1000 × 1000 49 10.5 7.4 × 7.4                   QE 0.45 @ 490 nm

Kodak2 KAI-1010M 1008 × 1018 30 12.9 9.0 × 9.0                   QE 0.37 @ 500 nm

Kodak2 KAI-2000M 1600 × 1200 30 14.8 7.4 × 7.4                   QE 0.36 @ 490 nm

Kodak2 KAI-4000M 2048 × 2048 15 21.4 7.4 × 7.4                   QE 0.36 @ 490 nm

1 http: //www.sony.co.jp/en/Products/SC-HP/Product_List_E/index.html

2 http: //www.kodak.com/go/ccd

 

531

B. Jä hne, Digital Image Processing                                                                                                       Copyright © 2002 by Springer-Verlag

ISBN 3–540–67754–2                                                                                                    All rights of reproduction in any form reserved.

532                                                                                            A Reference Material

 

R2

Selection of CMOS imaging sensors (Section 1.7.1)

(C: saturation capacity in 1000 electrons [ke], FR: frame rate in s− 1, PC: pixel clock in MHz, QE: peak quantum effi ciency)

 

Chip                              Format

H × V

 

FR PC Pixel size

H × V, µm

 

Comments

 

Linear response

PhotonFocus1                          640 × 480     30 10 10.5 × 10.5 32% fi ll factor

×                                   ×

Kodak2 KAC-0311         640  480     60 20  7.8  7.8     C 45ke,

QE 0.22 @ 500 nm

Fillfactory3 LUPA1300 1280 × 1024 450 40 12.0 × 12.0 16 parallel ports Photobit4 PB-MV40 2352 × 1728 240 80 7.0 × 7.0 16 parallel 10-bit ports

Kodak2 KAC-1310        1280 × 1024  15      20 6.0 × 6.0     C 40 ke

Fast frame rate linear response

 

Photobit4 PB-MV13      1280 × 1024 600     80 12.0 × 12.0 10 parallel 10-bit ports

Photobit4 PB-MV02       512 × 512    4000 80 16.0 × 16.0 16 parallel 10-bit ports

Logarithmic response

IMS HDRC VGA 5                640 × 480     25  8 12 × 12

×                                    ×

PhotonFocus1                        1024  1024  28     28 10.6 10.6 linear response at low

light levels with ad- justable transition to logarithmic response

1 http: //www.photonfocus.com

2 http: //www.kodak.com/go/ccd

3 http: //www.fillfactory.com

4  http: //www.photobit.com

5 http: //www.ims-chips.de

533

 

R3

Imaging sensors for the infrared (IR, Section 1.7.1)

(C: full well capacity in millions electrons [Me], IT: integration time, NETD: Rausch" aquivalente Temperaturdiff erenz, QE: peak quantum effi ciency)

 

Chip                             Format

H × V

 

FR PC Pixel size

H × V, µm

 

Comments

 

Near infrared (NIR)

Indigo1 InGaAs             320 × 256  345           30 × 30     0.9–1.68 µm, C 3.5Me

 

 

Mid wave infrared (MWIR) AIM2 PtSi 640 × 486 50 12 24 × 24 3.0–5.0 µm, NETD < 75 mK @ 33 ms IT Indigo1 InSb Indigo1 InSb 320 × 256 345 100   30 × 30 2.0–5.0 µm, C 18 Me 2.0–5.0 µm, C 11 Me AIM2 HgCdTe 384 × 288 120 20 24 × 24 3.0–5.0 µm, NETD < 20 mK @ 2 ms IT AIM2/IaF FhG3 QWIP 640 × 512 30 18 24 × 24 3.0–5.0 µm, NETD < 15 mK @ 20 ms IT Long wave infrared (LWIR) AIM2 HgCdTe 256 × 256 200 16 40 × 40 8–10 µm, NETD < 20 mK @ 0.35ms IT Indigo1 QWIP 320 × 256 345   30 × 30 8.0–9.2 µm, C 18 Me, NETD < 30 mK AIM2/IaF FhG3 QWIP 256 × 256 200 16 40 × 40 8.0–9.2 µm, NETD < 8 mK @ 20 ms IT AIM2/IaF FhG3 QWIP 640 × 512 30 18 24 × 24 8.0–9.2 µm, NETD < 10 mK @ 30 ms IT Uncooled sensors           Indigo1 Microbolo- meter 320 × 240 60   30 × 30 7.0–14.0 µm, NETD < 120 mK

640 × 512                    25 × 25

 

 

1 http: //www.indogosystems.com

2 http: //www.aim-ir.de

3 http: //www.iaf.fhg.de/ir/qwip/index.html

534                                                                                            A Reference Material

 

R4

Properties of the W -dimensional Fourier transform (Section 2.3.5)

g(x) ◦ • gˆ (k) and h(x) ◦      • hˆ (k) are Fourier transform pairs: RW ⊆ → C:

− ∞

∞

gˆ (k) =  ∫   g(x) exp.− 2π ik^TxΣ  d^W x =.exp.2π ik^TxΣ  .g(x).

 

s is a real, nonzero number, a and b are complex constants; A and U are W× W matrices, R is an orthogonal rotation matrix (R− 1 = RT, det R = 1)

Property                                Spatial domain                          Fourier domain

Linearity                                 ag(x) + bh(x)                           agˆ (k) + bhˆ (k)

Similarity                                      g(sx)

gˆ (k/s)/|s|

Generalized similarity

g(Ax)

gˆ.(A− 1)T kΣ  / det A

Rotation                                        g(Rx)

.

W

gˆ (Rk)

.

W

Separability

g_w(x_w)

w=1

gˆ _w(k_w)

w=1

 

0

Shift in x space                            g(x − x₀)                        exp(− 2π ik^T x₀)gˆ (k) Finite diff erence  g(x + x₀/2) − g(x − x₀/2)                                         2i sin(π xT k)gˆ (k)

0

Shift in k space                    exp(2π ik^T x)g(x)

T

gˆ (k − k₀)

Modulation                           cos(2π k₀ x)g(x)              .gˆ (k − k₀) + gˆ (k + k₀)Σ   2

 

Diff erentiation in x space

∂ g(x)

∂ x_p

 

2π ik_pgˆ (k)

 

Diff erentiation in k space

∂ g ˆ (k)

− 2π ix_pg(x)                                         ∂ kp

∞

Defi nite integral, mean

∫ g(x')dW x'

∞

.0

− ∞

gˆ (0)

 

Moments

∫ xm

 

xng(x)dW x

. i

Σ m+n . ∂ m+ngˆ (k) Σ.

 

p  q                                                 2π

 

∂ km∂ kn

 

 

Convolution

 

 

Spatial correlation

∫ h(x')g(x − x')dW x'

− ∞

∫ h(x')g(x' + x)dW x'

− ∞

hˆ (k)gˆ (k)

 

gˆ ∗ (k) hˆ (k)

∞

Multiplication                            h(x)g(x)

∞

∫   hˆ (k^')gˆ (k − k^')dW k'

− ∞ ∞

Inner product

∫ g∗ (x) h(x)dW x

− ∞

∫   gˆ ∗ (k)hˆ (k)dW k

− ∞

535

 

R5

Elementary transform pairs for the continuous Fourier transform

2- Dand 3-Dfunctions are marked by † and ‡, respectively.                                       

Space domain                                               Fourier domain

Delta, δ (x)                                                   const., 1

const., 1                                                       Delta, δ (k)

2

cos(k₀x)                                                       1 (δ (k − k₀) + δ (k + k₀))

2

sin(k₀x)                                                         i (δ (k − k₀) − δ (k + k₀))

.=

sgn(x)      1  x ≥ 0

− 1 x < 0

0 |x|≥ 1/2

Box, Π (x) =. 1 |x| < 1/2

− i

π k

 

=

sinc(k)   sin(π k)

π k

 

Disk,

† 1

π r 2

Π .|x| Σ

Bessel, J₁(2π r|k|)

π r|k|

2r

Ball,  ‡ Π  .|x| Σ                                              sin ( |k| ) − |k| cos ( |k| )

2

Bessel, J 1 (2π x)

x

|k|3/(4π )

2

2(1 − k)1/2Π . k Σ

 

exp(− |x|), exp(− |x|)†                                                                2     ,          2π           †

1 + (2π k)2 (1 + (2π |k|)2)3/2

 

R6

Functions invariant under the Fourier transform

Space domain                                               Fourier domain

Gaussian, exp.− π xTxΣ                                   Gaussian, exp.− π k^TkΣ

x_p exp.− π xTxΣ                                             − ik_p exp.− π k^TkΣ

 

=

sech(π x)                  1               

exp(π x) + exp(− π x)

sech(π k)                  1             

=

exp(π k) + exp(− π k)

 

Hyperbola, |x|− W/2                                                               |k|− W/2

∞                                                                  ∞

n=− ∞

v=− ∞

1-D δ comb, III(x) =  . δ (x − n)                  III(k) =. δ (k − v)

536                                                                                            A Reference Material

 

R7

Properties of the 2-D DFT (Section 2.3.5)

×

G and H are complex-valued M N matrices, Gˆ  and Hˆ  their Fourier trans- forms,

1 M− 1N− 1

gˆ u, v  =

MN.

 

. g_{m, n}w^{− mu}w^{− nv}, w_N = exp (2π i/N)

M         N

m=0n=0

M− 1N− 1

gm, n =

.  . gˆ u, v wmuwnv,

M     N

u=0 v=0

∈                 ×

and a and b complex-valued constants. Stretching and replication by factors K, L N yields KM LN matrices. For proofs see Cooley and Tukey [22], Poularikas [141].

Property                           Space domain                        Wave-number domain

..

1 M− 1N− 1

Mean

MN          Gmn

m=0n=0

gˆ 0, 0

Linearity                            aG + bH                                  aGˆ  + bHˆ

 

Spatial stretching (up- sampling)

Replication (frequency stretching)

gKm, Ln

 

gm, n (gkM+m, lN+n = gm, n)

gˆ _uv/(KL)

(gˆ kM+u, lN+v = gˆ u, v )

gˆ Ku, Lv

 

M

N

Shifting                              gm− m', n− n'                                                   w− m'uw− n'v gˆ uv

M

N

Modulation                      w^u'mw^v'ng_{m, n}

gˆ u− u', v− v'

Finite diff erences              (g_{m+1, n} − g_{m− 1, n})/2

(gm, n+1 − gm, n− 1)/2

M− 1 N− 1

i sin(2π u/M)gˆ _uv i sin(2π v/N)gˆ _uv

Convolution

.  . hm'n' gm− m', n− n'  MNhˆ uv gˆ uv

m'=0n'=0 M− 1 N− 1

Spatial correlation

.  . hm'n' gm+m', n+n'  MNhˆ uv gˆ u∗ v

m'=0n'=0

M− 1 N− 1

Multiplication                   g_mnh_mn

.. hu'v' gu− u', v− v'

 

Inner product Norm

M− 1N− 1

..

gm∗  nhmn

..

m=0n=0 M− 1N− 1

|g_mn|2

m=0n=0

u'=0v'=0 M− 1N− 1

..

gˆ u∗ v hˆ uv

..

u=0 v=0 M− 1N− 1

|gˆ _uv|2

u=0 v=0

537

 

R8

Properties of the continuous 1-D Hartley transform (Section 2.4.2)

g(x) ◦ • gˆ (k) and h(x) ◦      • hˆ (k) are Hartley transform pairs: R ⊆ → R,

 

hgˆ (k) =

 

with

∫ g(x) cas(2π kx)dx ◦     • g(x) =

∞

− ∞

∫   hgˆ (k) cas(2π kx)dk

∞

− ∞

cas 2π kx = cos(2π kx) + sin(2π kx). s is a real, nonzero number, a and b are real constants.

Property          Spatial domain                   Fourier domain

Linearity            ag(x) + bh(x)                    agˆ (k) + bhˆ (k)

Similarity         g(sx)

gˆ (k/s)/|s|

Shift

in x space

g(x − x₀)                            cos(2π kx₀)gˆ (k)− sin(2π kx₀)gˆ (− k)

Modulation        cos(2π k₀x)g(x)                 .gˆ (k − k₀) + gˆ (k + k₀)Σ   2

Diff erentiation  ∂ g(x)                                       − 2π k_pgˆ (− k)

in x space

 

Defi nite integral, mean

∂ x_p

∞

∫ g(x')dx'

− ∞

 

∞

 

 

gˆ (0)

Convolution

∫   h(x')g(x − x')dx'                [gˆ (k)hˆ (k) + gˆ (k)hˆ (− k)

− ∞                                                             +gˆ (− k)hˆ (k) − gˆ (− k)hˆ (− k)]/2

Multiplication  h(x)g(x)                                  [gˆ (k) ∗ hˆ (k) + gˆ (k) ∗ hˆ (− k)

+gˆ (− k) ∗ hˆ (k) − gˆ (− k) ∗ hˆ (− k)]/2

∞

Autocorrelation∫   g(x')g(x' + x)dx'                               [gˆ 2(k) + gˆ 2(− k)]/2

                                       − ∞                                                                                                                                                                     

 

1. Fourier transform expressed in terms of the Hartley transform

1                                        i

gˆ (k) = 2 .hgˆ (k) +h gˆ (− k)Σ  − 2 .hgˆ (k) − h gˆ (− k)Σ

2. Hartley transform expressed in terms of the Fourier transform

h                                                                                      1                                  i

gˆ (k) = ≡ [gˆ (k)] − ¥ [gˆ (k)] = 2 .gˆ (k) + gˆ ^∗ (k)Σ  + 2 .gˆ (k) − gˆ ^∗ (k)Σ

538                                                                                            A Reference Material

 

R9

Probability density functions (PDFs, Section 3.4).

Name Defi nition Mean Variance Discrete PDFs fn       Poisson P(µ) µn exp(− µ)   , n ≥ 0 µ µ

Defi nition, mean, and variance of some PDFs

 

 

n!

−              ≤                                          −

Binomial B(Q, p)            Q!    pn(1  p)Q− n, 0   n< Q  Qp           Qp(1  p)

n! (Q − n)!

Continuous PDFs f (x)

Uniform U(a, b)      1  

b − a

a + b

2

(b − a)2

12

 

Normal N(µ, σ )

1

√ 2π σ

exp.−

(x − µ)2                                           µ           σ 2

Σ

2σ 2

Rayleigh R(σ )          x  exp.  x2  Σ  ,  x > 0                         σ , π /2  σ 2(4 − π )/2

 

σ 2             − 2σ 2

2 Σ

xQ/2− 1                              x  

Chi-square

χ 2(Q, σ )

 

2Q/2

 

Γ (Q/2)σ

Q exp.−

, x > 0   Qσ 2           2Qσ 4

2σ

 

 

Addition theorems for independent random variables g₁ and g₂

 

PDF	g1	g2	g1 + g2
Binomial	B(Q1, p)	B(Q2, p)	B(Q1 + Q2, p)
Poisson	P(µ1)	P(µ2)	P(µ1 + µ2)

Normal                      N(µ₁, σ ₁)            N(µ₂, σ ₂)            N(µ₁ +µ₂, (σ 2 + σ 2)1/2)

1       2

Chi-square                 χ 2(Q₁, σ )            χ 2(Q₂, σ )            χ 2(Q₁ + Q₂, σ )

PDFs of functions of independent random variables g_n

1

2

PDF of variable                        Function                            PDF of function

g_n: N(0, σ )                               (g2 + g2)1/2                                           R(σ )

g_n: N(0, σ )                              arctan(g2/g2)                       U(0, 2π )

 

g_n: N(0, σ )

2   1

Q

n

. g2

 

χ 2(Q, σ )

n=1

539

 

R10

Error propagation (Sections 3.2.3, 3.3.3, and 4.2.8)

=

=

f_g is the PDF of the random variable (RV) g, a, and b constants, g'  p(g) a diff erentiable monotonic function with the derivative dp/dg and the inverse function g p− 1(g').

×

Let g be a vector with P RVs with the covariance matrix cov(g), g' a vector with Q RVs and with the covariance matrix cov(g'), M  a Q                                                                                                  P

matrix, and a a column vector with Q elements.

 

1. PDF, mean, and variance of a linear function g' = ag + b

f '(g') = fg((g' − a)/b),            µ ' = aµ

 

+ b, σ 2

= a2σ 2

g                                       |a|

g                g                       g'                   g

2. PDF of monotonous diff erentiable nonlinear function g' = p(g)

'           f_g(p− 1(g'))

f_g'(g ) =

dp(p− 1

,

.

(g'))/dg.

3. Mean and variance of diff erentiable nonlinear function g' = p(g)

2

.      .

dg2

g

dg

σ ² d2p(µ )           2          dp(µ )        2

µ_g' ≈ p(µ_g) + 2

g,   σ _g' ≈          ^g    σ

.

.

4. Covariance matrix of a linear combination of RVs, g' = Mg + a

cov(g') = M cov(g)MT

5. Covariance matrix of a nonlinear combination of RVs, g' = p(g)

∂ gp

cov(g') ≈ J cov(g)JT with the Jacobian J,               j_{q, p} = ∂ p q .

6. Homogeneous stochastic fi eld: convolution of a random vector by the fi lter h g' = h ∗ g (Section 4.2.8)

(a) With the autocovariance vector c

2

c' = c > (h > h)      ◦ • c ˆ ^'(k) = c ˆ (k).h ˆ (k).  .

(b) With the autocovariance vector c = σ 2δ _n (uncor.relate.d elements)

2

c' = σ 2(h > h)      ◦ • c ˆ ^'(k) = σ 2 .h ˆ (k).  .

. .

540                                                                                            A Reference Material

 

R11

1-D LSI fi lters (Sections 4.2.6, 11.2, and 12.2)

1. Transfer function of a 1-D fi lter with an odd number of coeffi cients (2R + 1, [h_{− R},..., h_{− 1}, h₀, h₁,..., h_R])

(a) General

hˆ (k˜ ) =

R v'.=− R

h_v' exp(− π iv'k˜ )

(b) Even symmetry (h_{− v} = h_v)

.

R

hˆ v = h₀ + 2          h_v' cos(π v'k˜ )

v'=1

(c) Odd symmetry (h_{− v} = − h_v)

.

R

hˆ v = − 2i        h_v' sin(π v'k˜ )

v'=1

2. Transfer function of a 1-Dfi lter with an even number of coeffi cients (2R, [h_{− R},..., h_{− 1}, h₁,..., h_R], convolution results put on intermedi- ate grid)

(a) Even symmetry (h_{− v} = h_v)

.

R

hˆ v = 2       h_v' cos(π (v' − 1/2)k˜ )

v'=1

(b) Odd symmetry (h_{− v} = − h_v)

.

R

hˆ v = − 2i        h_v' sin(π (v' − 1/2)k˜ )

v'=1

541

 

R12

1-D recursive fi lters (Section 4.3).

1. General fi lter equation

 

gn'

 

S

.

=−    an'' gn' − n'' +

n''=1

R n'.=− R

 

hn' gn− n'

2. General transfer function

 

 

R

S

.  h_n' exp(− π in'k˜ )

hˆ (k˜ ) = n'=− R

.

a_n'' exp(− π in''k˜ )

n''=0

3. Factorization of the transfer function using the z transform and the fundamental law of algebra

 

 

4. Relaxation fi lter

 

2R

hˆ (z) =       S− R  n'=1h                  z− R

. (1 − c_n' z− 1)

S

.

(1 − d_n'' z− 1)

n''=1

(a) Filter equation (|α | < 1)

gn'

(b) Point spread function

 

±n =

±r

 

= α g_n' ∓ 1 + (1 − α )g_n

 

.(1 − α )α n  n ≥ 0

 

 

(c) Transfer function of symmetric fi lter (running fi lter successively in positive and negative direction)

rˆ (k˜ ) =              1            , .rˆ (0) = 1, rˆ (1) =     1           Σ

 

 

with

1 + β − β cos π k˜

1 + 2β

β = 2α   , α = 1 + β  −  , 1 + 2β, β ∈ ] − 1/2, ∞ [

(1 − α )2                                           β

542                                                                                            A Reference Material

 

5. Resonance fi lter with unit response at resonance wave number k˜ 0 in the limit of low damping 1 − r, 1

(a) Filter equation (damping coeffi cient r ∈ [0, 1[, resonance wave number k˜ 0 ∈ [0, 1])

gn'

= (1 − r 2) sin(π k˜ 0)g_n + 2r cos(π k˜ 0)g_n' ∓ 1 − r 2g_n' ∓ 2

(b) Point spread function

h±n =   0                                                          n< 0



  (1 − r 2)rn sin[(n + 1)π k˜ 0]  n ≥ 0

(c) Transfer function of symmetric fi lter (running fi lter successively in positive and negative direction)

˜                                             sin²(π k˜ 0)(1 − r 2)2

sˆ (k) =

˜ ˜           2

˜ ˜           2

.1 − 2r cos[π (k − k₀)] + r  Σ  .1 − 2r cos[π (k + k₀)] + r  Σ

(d) For low damping, the transfer function can be approximated by

sˆ (k) ≈

0

4r 2π 2

˜                     1                1 + (k˜  − k˜   )2  (1− r 2)2

for 1 − r, 1

(e) Halfwidth ∆ k, defi ned by sˆ (k˜ 0 + ∆ k) = 1/2

∆ k ≈ (1 − r)/π

 

R13

Gaussian and Laplacian pyramids (Section 5.3)

1. Construction of the Gaussian pyramid G(0), G(1),..., G(Q− 1) with Q planes by iterative smoothing and subsampling by a factor of two in all directions

G(0) = G,    G(q+1) = B↓ 2G(q)

2. Condition for smoothing fi lter to avoid aliasing

2

Bˆ (k˜ ) = 0  ∀ k˜ p ≥ 1

3. Construction of the Laplacian pyramid L(0), L(1),..., L(Q− 1) with Q

planes from the Gaussian pyramid

L(q) = G(q)− ↑ 2 G(q+1),          L(Q− 1) = G(Q− 1)

The last plane of the Laplacian pyramid is the last plane of the Gaussian pyramid.

 

4. Interpolation fi lters for upsampling operation ↑ ₂ (± R22)

543

5. Iterative reconstruction of the original image from the Laplacian pyra- mid. Compute

G(q− 1) = L(q− 1)+ ↑ 2 Gq

= −

starting with the highest plane (q Q 1). When the same upsam- pling operator is used as for the construction of the Laplacian pyra- mid, the reconstruction is perfect except for rounding errors.

6. Directio-pyramidal decomposition in two directional components

G(q+1) = ↓ 2 BxBy G(q)

L(q)            = G(q)− ↑ 2 G(q+1)

L

(q) x

L

(q) y

= 1/2(L(q) − (B_x

= 1/2(L(q) + (B_x

− B_y)G(q))

− B_y)G(q))

 

 

R14

Basic properties of electromagnetic waves (Section 6.2)

1. The frequency ν (cycles per unit time) and wavelength λ (length of a period) are related by the phase speed c (in vacuum speed of light c = 2.9979 × 108 m s− 1):

λ ν = c

2. Classifi cation of the ultraviolet, visible and infrared part of the elec- tromagnetic spectrum (see also Fig. 6.2)

 

Name	Wavelength range	Comment
VUV (vacuum UV)	30–180 nm	Strongly absorbed by air; re-
		quires evacuated equipment
UV-C	100–280 nm	CIE standard defi nition
UV-B	280–315nm	CIE standard defi nition
UV-A	315–400 nm	CIE standard defi nition
Visible (light)	400–700 nm	Visible by the human eye
VNIR (very near IR)	0.7–1.0 µm	IR wavelength range to which standard silicon image sen-
		sors respond
NIR (near IR)	0.7–3.0 µm
TIR (thermal IR)	3.0–14.0 µm	Range of largest emission at environmental temperatures
MIR (middle IR)	3–100 µm
FIR (far IR)	100–1000 µm

544                                                                                            A Reference Material

 

3. Energy and momentum of particulate radiation such as β radiation (electrons), α radiation (helium nuclei), neutrons, and photons (elec- tromagnetic radiation):

 

ν =  E/h  Bohr frequency condition,

λ =  h/p  de Broglie wavelength relation.

 

 

R15

Radiometric and photometric terms (Section 6.3)

dA₀ is an element of area in the surface, θ the angle of incidence, Ω the solid angle. For energy-, photon-, and photometry-related terms, often the indices e, p, and ν, respectively, are used.

Term                     Energy-related      Photon-related  Photometric quantity

Energy                  Radiant energy Q

[Ws]

Photon number [1]

Luminous energy [lm s]

Energy fl ux (power)

 

Incident energy fl ux density

 

Excitant energy fl ux density

 

 

Energy fl ux per solid angle

Radiant fl ux

=

Φ dQ [W]

dt

Irradiance

=

E d Φ  [W m− 2]

dA₀

Radiant excitance (emittance)

=

M d Φ  [W m− 2]

dA₀

Radiant intensity

=

I d Φ  [Wsr− 1]

dΩ

Photon fl ux [s− 1]

 

Photon irradi- ance [m− 2s− 1]

 

Photon fl ux density [m− 2s− 1]

 

 

Photon intensity [s− 1sr− 1]

Luminous fl ux [lumen (lm)]

 

Illuminance [lm/m2 = lux [(lx)]

 

Luminous excitance [lm/m2]

 

 

Luminous intensity [lm/sr = candela (cd)]

Energy fl ux density per solid angle

Radiance

L

d2Φ

= dΩ dA₀ cos θ

Photon radiance [m− 2s− 1sr− 1]

Luminance [cd m− 2]

[W m− 2 sr− 1]

Energy/area         Energy density [Ws m2]

 

Photon density [m− 2]

 

Exposure

[lms m− 2 = lx s]

 

Computation of luminous quantities from the corresponding radiomet- ric quantity by the spectral luminous effi cacy V (λ ) for daylight (photopic) vision:

∫

780 nm

W

Q_v = 683 lm

Q(λ )V(λ ) dλ

380 nm

545

 

Table with the 1980 CIE values of the spectral luminous effi cacy V (λ )

for photopic vision
λ [µm]	V (λ )	λ [µm]	V (λ )	λ [µm]	V (λ )
380	0.00004	520	0.710	660	0.061
390	0.00012	530	0.862	670	0.032
400	0.0004	540	0.954	680	0.017
410	0.0012	550	0.995	690	0.0082
420	0.0040	560	0.995	700	0.0041
430	0.0116	570	0.952	710	0.0021
440	0.023	580	0.870	720	0.00105
450	0.038	590	0.757	730	0.00052
460	0.060	600	0.631	740	0.00025
470	0.091	610	0.503	750	0.00012
480	0.139	620	0.381	760	0.00006
490      0.208                  630      0.265                  770      0.00003
500	0.323	640	0.175	780	0.000015
510	0.503	650	0.107

 

R16

Color systems (Section 6.3.4)

1. Human color vision based on three types of cones with maximal sen- sitivities at 445 nm, 535 nm, and 575 nm (Fig. 6.5b).

2. RGB color system; additive color system with the three primary col- ors red, green, and blue. This could either be monochromatic colors with wavelengths 700 nm, 646.1 nm, and 435.8 nm or red, green, and blue phosphor as used in RGB monitors (e. g., according to the Euro- pean EBU norm). Not all colors can be represented by the RGB color system (see Fig. 6.6a).

3. Chromaticity diagram: reduction of the 3-Dcolor space to a 2-Dcolor plane normalized by the intensity:

r =    R    , g =    G    , b =    B   .

R + G + B              R + G + B             R + G + B

It is suffi cient to use the two components r and g: b = 1 − r − g.

4. XY Z color system (Fig. 6.6c): additive color system with three vir- tual primaries X, Y, and Z that can represent all possible colors and is given by the following linear transform from the EBU RGB color

546                                                                                            A Reference Material

 

system:

 X   0.490 0.310 0.200   R 

 

     = 0.177  0.812  0.011            G 

                                  

.

 Z   0.000 0.010 0.990   B 

5. Color diff erence or YUV system: color system with an origin at the white point (Fig. 6.6b).

6. Hue-saturation (HSI) color system: color system using polar coordi- nates in a color diff erence system. The saturation is given by the radius and the hue by the angle.

 

R17

Thermal emission (Section 6.4.1)

1. Spectral emittance (law of Planck)

 

with

M_e(λ, T ) =

2π hc2

λ 5

1

kBTλ

exp. hc Σ − 1

h = 6.6262 × 10− 34 J s               Planck constant,

k_B = 1.3806 × 10− 23 J K− 1 Boltzmann constant, and

c = 2.9979 × 108 m s− 1                      speed of light in vacuum.

2.

B

Total emittance (law of Stefan and Boltzmann)

M_e =

2 k4 π 5

 

T 4 = σ T 4 with σ ≈ 5.67 · 10^{− 8}W m^{− 2}K^{− 4}

15 c2h3

3. Wavelength of maximum emittance (Wien’s law)

T

λ m ≈ 2898K µm

 

 

R18

Interaction of radiation with matter (Section 6.4)

1. Snell’s law of refraction at the boundary of two optical media with the indices of refraction n₁ and n₂

sin θ ₁     n₂

sin θ ₂ = n₁

θ ₁ and θ ₂ are the angles of incidence and refraction, respectively.

2. Refl ectivity ρ: ratio of the refl ected radiant fl ux to the incident fl ux at the surface. Fresnel’s equations give the refl ectivity for parallel polarized light

ρ     tan2(θ ₁ − θ ₂)

⊗ = tan2(θ ₁ + θ ₂),

547

 

for perpendicular polarized light

 

ρ

 

and for unpolarized light

sin²(θ      θ )

− 1          2

⊥ = sin²(θ ₁ + θ ₂),

 

=

ρ   ρ _⊗ + ρ _⊥ .

2

3. Refl ectivity at normal incidence (θ ₁ = 0) for all polarization states

ρ   (n₁ − n₂)2               (n − 1)2

= (n1 + n2)2 = (n + 1)2 with n = n₁/n₂

4. Total refl ection. When a ray enters into a medium with lower refrac- tive index, beyond the critical angle θ _c all light is refl ected and none enters the optically thinner medium:

=c

θ    arcsin n 1 n₂

 

with n₁ < n₂

 

 

R19

Optical imaging

1. Perspective projection with pinhole camera model

x1           d'X₁

 

d'X₂

2

=− X3,     x =− X3

 

Pinhole located at origin of world coordinate system [X₁, X₂, X₃]^T, d' is distance of image plane to projection center, X₃ axis aligned perpendicular to image plane.

2. Image equation (Newtonian and Gaussian form)

dd' = f 2  or       1    + 1  = 1

d' + f    d + f    f

d and d' are the distances of the object and image to the front and back focal points of the optical system, respectively (see Fig. 7.7).

3. Lateral magnifi cation

ml = x₁     f   d'

 

 

4. Axial magnifi cation

X₁ = d = f

m    d'   f 2       d'2              2

a ≈ d = d2 = f 2 = ml

548                                                                                            A Reference Material

 

5. The f -number n_f of an optical system is the ratio of the focal length and diameter of lens aperture

=f

n     f

2r

6. Depth of focus (image space)

f

∆ x₃ = 2n_f.1 + d' Σ  H = 2n_f (1 + m_l)H

 

7. Depth of fi eld (object space)

m2

Distant objects (∆ X₃, d)                     ∆ X₃ ≈ 2n_f · 1 + m l H

 

dmin

 

for range including infi nity d_min ≈

f 2

 

4n_f H

Microscopy (m_l

$ 1)                        ∆ X₃

2n_f H

≈ ml

8. Resolution with a diff raction-limited optical systems: angular reso- lution

r

Angular resolution                                  ∆ θ ₀ = 0.61 λ

n'a

Lateral resolution at image plane          ∆ x  = 0.61 λ

=

Lateral resolution at object plane  ∆ X                    0.61 λ

n_a

The resolution is given by the Rayleigh criterion (see Fig. 7.15b); n_a and n_a' are the object-sided and image sided numerical aperture of the light cone entering the optical system:

n_a = n sin θ ₀ = 2n = nr ;

n_f    f

n is the index of refraction.

9. Relation of the irradiance at image plane E' to the object radiance L

(see Fig. 7.10)

 

E' =

 

2

.     Σ r

tπ  

f + d'

 

cos⁴ θ L ≈

tπ cos4 θ L for  d          f

n

2

$

f

549

 

R20

Homogeneous point operation (Section 10.2)

Point operation that is independent of the position of the pixel

G_m'  n = P (G_mn)

1. Negative

 

P_N(q) = Q − 1 − q

2.

  

Detection of underfl ow and overfl ow by a pseudocolor [r, g, b] map- ping



[0, 0, Q − 1] (blue) q = 0

 

P_uo(q) = [q, q, q]                (grey) q ∈ [1, Q − 2] [Q − 1, 0, 0]  (red)          q = Q − 1

3.

  0                              q< q1

Contrast stretching of range [q₁, q₂]

 

 

Puo

(q) =    (q − q₁)(Q − 1)

q ∈ [q₁, q₂]

 

q  − q2 1

Calibration procedures

  Q − 1                       q> q₂

 

1.

R21

Noise equalization (Section 10.2.3)

If the variance of the noise depends on the image intensity, it can be equalized by a nonlinear grayscale transformation

g

g' = h(g)σ _h∫       dg'    + C

0  σ 2(g')

with the two free parameters σ _h and C. With a linear variance func- tion (Section 3.4.5)

σ 2(g) = σ 2 + α g

g

the transformation becomes

 h

2σ h(g) = √ α

0

 

σ 2 + α g + C.

2.

0

Linear photometric two-point calibration (Section 10.3.3)

Two calibration images are taken, a dark image B without any illumi- nation and a reference image R with an object of constant radiance. A normalized image corrected for both the fi xed pattern noise and inhomogeneous sensitivity is given by

=

G' c G − B .

R − B

550                                                                                            A Reference Material

 

R22

Interpolation (Section 10.6)

1. Interpolation of continuous function from sampled points at dis- tances ∆ x_w is an convolution operation:

g_r (x) =.n  g(x _n)h(x − x _n).

Reproduction of the grid points results in the interpolation condition

 

.=n

.

h(x  )        1    n = 0

0    otherwise

 

2. Ideal interpolation function

 

h(x) =

 

W

.

sinc(x_w/∆ x_w)        ◦ •

w=1

hˆ (k) =

 

W

.

Π (k˜ w /2)

w=1

3.

Discrete 1-D interpolation fi lters for interpolation of intermediate grid points halfway between the existing points

Type                Mask                                   Transfer function

Linear                   1  1    /2                          cos(π k˜ /2)

8

˜                     ˜

Cubic                − 1 9 9 − 1  /16 9 cos(π k/2) − cos(3π k/2)

˜                     ˜

Cubic B-spline  1 23 23 1  /48

23 cos(π k/2) + cos(3π k/2)

√

√

3,

 3 −            − 2  /2^†

3

16 + 8 cos(π k˜ )

 

†Recursive fi lter applied in forward and backward direction, see Section 10.6.5

551

 

R23

Averaging convolution fi lters (Chapter 11)

1.

Summary of general constraints for averaging convolution fi lters

Property                            Space domain        Wave-number domain

Preservation of mean          .h_n = 1

hˆ (0) = 1

 

Zero shift, even symmetry

h_{− n} = h_n                     ¥ .hˆ (k)Σ  = 0

Monotonic decrease        — from one to zero

hˆ (k˜ 2)  ≤  hˆ (k˜ 1) if k˜ 2  >  k˜ 1, hˆ (k) ∈ [0, 1]

Isotropy                              h(x) = h(|x|)           hˆ (k) = hˆ (|k|)

 

2.

1-Dsmoothing box fi lters

Mask                                           Transfer function          Noise suppression†

3                                                                               1 2        ˜                  1

R = [1 1 1]/3

3 + 3 cos(π k)

√ 3 ≈ 0.577

 

4R = [1 1 1 1]/4                             cos(π k˜ ) cos(π k˜ /2)            1/2 = 0.5

 

2R+1R = [1  ...  1]/(2R + 1)               sin(π (2R + 1)k˜ /2)             √  1

                    

 

2RR = [1... 1]/(2R)

(2R + 1) sin(π k˜ /2)

sin(π Rk˜ ) 2R sin(π k˜ /2)

2R + 1

1

√ 2R

 

†For white noise

 

3.

1-Dsmoothing binomial fi lters

Mask                         TF                 Noise suppression†

8

B2 = [1 2 1]/4              cos2(π k˜ /2)     ,  3 ≈ 0.612

 

, =

B4  [1464 1]/16  cos4(π k˜ /2)                      35

128

≈ 0.523

1/2

1/4

2R                                                      2R ˜

. Γ (R + 1/2) Σ

 

. 1 Σ .

1 Σ

B

†For white noise

cos

(π k/2)

√ π Γ (R + 1)

≈ Rπ

1 − 16R

552                                                                                            A Reference Material

 

R24

First-order derivative convolution fi lters (Chapter 12)

1.

Summary of general constraints for a fi rst-order derivative fi lter into the direction x_w

Property                             Space domain         Wave-number domain

Zero mean                           .h_n = 0

hˆ (k ˜ )

.k˜ w

=0 = 0

Zero shift, odd symmetry

h_{− n} = − h_n                     ≡ .Hˆ (k)Σ  = 0

First-order derivative            .n_wh_n

∂ hˆ (k ˜ )

= 1                  ˜ .

= π i

ˆ ˜         ˜ ˆ ˜

n                                                     ∂ k_w

.k˜ w =0

 

Isotropy

h(k) = π ik_wb(.k.)  with

ˆ                  ˆ ˜

b(0) = 1, ∇ _kb(.k.) = 0

..

2. First-order discrete diff erence fi lters

Name                                      Mask                               Transfer function

D_x                                           Σ   1  − 1  Σ                         2i sin(π k˜ x/2)

Σ                   Σ

Symmetric diff erence, D_2x             1  0  − 1     /2        i sin(π k˜ x)

sin(π k˜   )

Cubic B-spline D_2x ±R            Σ 1 0  − 1  Σ  /2,              i                  ^x ˜

Σ 3 −

√ 3,

√ 3 − 2 Σ †

2/3 + 1/3 cos(π k_x)

 

†Recursive fi lter applied in forward and backward direction, see Section 10.6.5

553

 

3.

Regularized fi rst-order discrete diff erence fi lters

Name                    Mask                         Transfer function

 

2 × 2, D B

1  1 − 1 

 

2i sin(π k˜   /2) cos(π k˜

 

/2)

x  y                 2  1  − 1                                x                           y

1  1 0 –1 

y

                 

Sobel, D_2xB2

8 2 0 –2

 1 0 –1 

i sin(π k˜ x) cos2(π k˜ y /2)

1   3  0   –3  

                     

Optimized Sobel

y

D_2x(3B2 + I)/4         32



10 0 –10

3  0   –3 

i sin(π k˜ x)(3 cos2(π k˜ y /2) + 1)/4

 

 

4.

Performance characteristics of edge detectors: angle error, magni- tude error, and noise suppression for white noise. The three values in the two error columns give the errors for a wave number range of 0–0.25, 0.25–0.5, and 0.5–0.75, respectively.

Name                  Angle error [°]      Magnitude error       Noise factor

D_x                                                                                                 √ 2 ≈ 1.414

D_2x                       1.36 4.90 12.66 0.026 0.151 0.398 1/√ 2 ≈ 0.707

D_2x ±R                 0.02 0.33 2.26     0.001 0.023 0.220  , 3 ln 3/π ≈ 1.024

D_xB_y                    0.67 2.27 5.10      0.013 0.079 0.221 1

y

D2xB2                                              0.67 2.27 5.10 0.012 0.053 0.070   D2x(3B2 + I)/4 0.15 0.32 0.72 0.003 0.005 0.047 √ 59/16 ≈ 0.480

√ 3/4 ≈ 0.433

y

554                                                                                            A Reference Material

 

R25

Second-order derivative convolution fi lters (Chapter 12)

1.

Summary of general constraints for a second-order derivative fi lter into the direction x_w

Property                            Space domain    Wave-number domain

Zero mean                          .h_n = 0

hˆ (k ˜ )

.k˜ w

 

=0 = 0

Zero slope                          .n_wh_n

∂ hˆ (k ˜ )

= 0             ˜ .    = 0

n                                              ∂ k_w

.k˜ w =0

Zero shift, even symmetry

h_{− n} = h_n                  ≡ .Hˆ (k)Σ  = 0

Second-order derivative  .n2 h

2       ∂ h(k).

2π 2

2 ˆ ˜

w

=

n  w  n                                 ∂ k˜ 2

=−

h(                          ..  ..k) = − (π k  )  b(  k  )  withw

.k˜ w =0

 

Isotropy

ˆ ˜             ˜ 2 ˆ ˜

ˆ                  ˆ ˜

b(0) = 1, ∇ _kb(.k.) = 0

..

2. Second-order discrete diff erence fi lters

Name                    Mask                           Transfer function

x

1-D Laplace D2

Σ   1  − 2  1  Σ               − 4 sin²(π k˜ x/2)

1 − 4 1

˜

 0   1 0 

2-D Laplace L

    0    1  0   

                        

1   1      2 1 

− 4 sin (π k_x/2)− 4 sin (π k_y/2)

 

 

2-D Laplace L'

4 2 − 12 2

1     2  1

4 cos2(π k_x/2) cos2(π k_y/2) − 4

 

 

 

B Notation

Because of the multidisciplinary nature of digital image processing, a consistent and generally accepted terminology — as in other areas — does not exist. Two basic problems must be addressed.

•

Confl icting terminology. Diff erent communities use diff erent symbols (and even names) for the same terms.

•

Ambiguous symbols. Because of the many terms used in image process- ing and the areas it is related to, one and the same symbol is used for multiple terms.

There exists no trivial solution to this awkward situation. Otherwise it would be available. Thus confl icting arguments must be balanced. In this textbook, the following guidelines are used:

•

Stick to common standards. As a fi rst guide, the symbols recom- mended by international organizations (such as the International Or- ganization for Standardization, ISO) were consulted and several ma- jor reference works were compared [40, 111, 116, 141]. Additionally cross checks were made with several standard textbooks from dif- ferent areas [11, 54, 133, 143]. Only in a few confl icting situations deviations from commonly accepted symbols are used.

•

Use most compact notation. When there was a choice of diff erent notations, the most compact and comprehensive notation was used. In rare cases, it appeared useful to use more than one notation for the same term. It is, for example, sometimes more convenient to use indexed vector components (x = [x₁, x₂]T ), and sometimes to use x = [x, y]T .

•

Allow ambiguous symbols. One and the same symbol can have diff er- ent meanings. This is not so bad as it appears at fi rst glance because from the context the meaning of the symbol becomes unambiguous. Thus care was taken that ambiguous symbols were only used when they can clearly be distinguished by the context.

In order to familiarize readers coming from diff erent backgrounds to the notation used in this textbook, we will give here some comments on deviating notations.

 

555

B. Jä hne, Digital Image Processing                                                                                                       Copyright © 2002 by Springer-Verlag

ISBN 3–540–67754–2                                                                                                    All rights of reproduction in any form reserved.

556                                                                                                                 B Notation

 

=

Wave number. Unfortunately, diff erent defi nitions for the term wave number exist:

=

k'  2π

λ

and  k   1 .                                 (B.1)

λ

=      =

=

Physicists usually include the factor 2π in the defi nition of the wave number: k' 2π /λ, by analogy to the defi nition of the circular fre- quency ω 2π /T 2π ν. In optics and spectroscopy, however, it is defi ned as the inverse of the wavelength without the factor 2π (i. e., number of wavelengths per unit length) and denoted by ν ˜ = λ − 1.

Imaginary unit. The imaginary unit is denoted here by i. In electrical engineering and related areas, the symbol j is commonly used.

Time series, image matrices. The standard notation for time series [133], x[n], is too cumbersome to be used with multidimensional signals: g[k][m][n]. Therefore the more compact notation x_n and g_{k, m, n} is chosen.

Partial derivatives. In cases were it does not lead to confusion, partial derivates are abbreviated by indexing: ∂ g/∂ x = ∂ _xg = g_x

Typeface          Description

e, i, d, w           Upright symbols have a special √ meaning; examples: e for the

base of natural logarithm, i =

dg, w = e2π i

a, b, …             Italic (not bold): scalar

− 1, symbol for derivatives:

g, k, u, x, … Lowercase italic bold: vector, i. e., a coordinate vector, a time series, row of an image, …

G, H, J, … Uppercase italic bold: matrix, tensor, i. e., a discrete image, a 2-Dconvolution mask, a structure tensor; also used for signals with more than two dimensions

B R F

, , , …   Caligraphic letters indicate a representation-independent op- erator

N, Z, R, C Blackboard bold letters denote sets of numbers or other quan- tities

 

Accents            Description

k ¯ , n ¯ , …           A bar indicates a unit vector

k˜, k ˜ , x ˜ , …       A  tilde  indicates  a  dimensionless  normalized  quantity  (of  a quantity with a dimension)

G ˆ , gˆ (k), …       A hat indicates a quantity in the Fourier domain

557

Subscript         Description

g_n                     Element n of the vector g

g_mn                   Element m, n of the matrix G

g_p                     Compact notation for fi rst-order partial derivative of the con- tinuous function g into the direction p: ∂ g(x)/∂ x_p

g_pq                   Compact notation for second-order partial derivative of the continuous function g(x) into the directions p and q:

∂ 2g(x)/(∂ x_p∂ x_q)

 

Superscript      Description

A− 1, A− g                   Inverse of a square matrix A; generalized inverse of a (non- square) matrix A

AT                                 Transpose of a matrix

a>                                   Conjugate complex

A>                                  Conjugate complex and transpose of a matrix

 

Indexing          Description

K, L, M, N        Extension of discrete images in t, z, y, and x directions

k, l, m, n          Indices of discrete images in t, z, y, and x directions

r, s, u, v          Indices of discrete images in Fourier domain in t, z, y, and x

directions

P Number of components in a multichannel image; dimension of a feature space

Q Number of quantization levels or number of object classes

R Size of masks for neighborhood operators

W                     Dimension of an image or feature space

p, q, w              Indices of a component in a multichannel image, dimension in an image, quantization level or feature

558                                                                                                                 B Notation

Function          Description

cos(x)              Cosine function

exp(x)              Exponential function

ld(x)                Logarithmic function to base 2

ln(x)                Logarithmic function to base e log(x)    Logarithmic function to base 10 sin(x)               Sine function

sinc(x)             Sinc function: sinc(x) = sin(π x)/(π x)

det(G)              Determinant of a square matrix

diag(G)            Vector with diagonal elements of a square matrix trace(G)      Trace of a square matrix

cov(g)              Covariance matrix of a random vector

E(g), var(G)     Expectation (mean value) and variance

 

Image operators Description

· Pointwise multiplication of two images

∗                           Convolution

>                            Correlation

∅, ⊕                       Morphological erosion and dilation operators

◦ , •                        Morphological opening and closing operators

⊗                           Morphological hit-miss operator

∨, ∧                       Boolean or and and operators

∪, ∩                       Union and intersection of sets

⊂, ⊆                       Set is subset, subset or equal

C                          Shift operator

↓

s                                         Sample or reduction operator: take only every sth pixel, row, etc.

↑

s                                         Expansion or interpolation operator: increase resolution in every coordinate direction by a factor of s, the new points are interpolated from the available points

559

 

Symbol   Defi nition, [Units]             Meaning

Greek symbols

α              [m− 1]                                Absorption coeffi cient

β              [m− 1]                                Scattering coeffi cient

δ (x), δ _n                                                 Continuous, discrete δ distribution

W ∂ 2

∆         . ∂ x2

Laplacian operator

w=1      w

H            [1]                                     Specifi c emissivity

H            [m]                                   Radius of blur disk

κ              [m− 1]                                Extinction coeffi cient, sum of absorp- tion and scattering coeffi cient

T

∂ x1

∂ xW

∇              ∂  ,..., ∂                  Gradient operator

λ              [m]                                   Wavelength

ν              [s− 1], [Hz] (hertz)               Frequency

∇ ×                                                         Rotation operator

η              n + iξ, [1]                          Complex index of refraction

η              [1]                                    Quantum effi ciency

φ              [rad], [°]                               Phase shift, phase diff erence

φ _e            [rad], [°]                               Azimuth angle

Φ        [J/s], [W], [s− 1], [lm]          Radiant or luminous fl ux

Φ _e, Φ _p    [W], [s− 1], [lm]                  Energy-based radiant, photon-based

radiant, and luminous fl ux

ρ, ρ _⊗ , ρ _⊥ [1]                                         Refl ectivity for unpolarized, parallel polarized, and perpendicularly polar- ized light

ρ              [kg/m3]                             Density

σ _x                                                          Standard deviation of the random variable x

σ              5.6696 · 10− 8Wm− 2K− 4 Stefan-Boltzmann constant

σ _s            [m2]                                  Scattering cross-section

τ              [1]                                    Optical depth (thickness)

τ              [1]                                    Transmissivity

τ              [s]                                     Time constant

θ              [rad], [°]                               Angle of incidence

θ _b            [rad], [°]                               Brewster angle (polarizing angle)

θ _c            [rad], [°]                               Critical angle (for total refl ection)

θ _e            [rad], [°]                               Polar angle

θ _i            [rad], [°]                               Angle of incidence

continued on next page

560                                                                                                                 B Notation

Symbol   Defi nition, [Units]             Meaning

continued from previous page

Ω         [sr] (steradian)                  Solid angle

ω             ω = 2π ν, [s− 1], [Hz]           Circular frequency

Roman symbols

A [m2]                                  Area

a, a          a = x_tt = u_t, [m/s2]               Acceleration

bˆ (k ˜ )                                                     Transfer function of binomial mask

B [Vs/m2]                             Magnetic fi eld

B                                                          Binomial fi lter mask

B Binomial convolution operator

c 2.9979 · 108 ms− 1                        speed of light

C set of complex numbers

d [m]                                   Diameter (aperture) of optics, dis- tance

d'                     [m]                                   Distance in image space

dˆ (k ˜ )                                                     Transfer function of D

D [m2/s]                               Diff usion coeffi cient

D                                                          First-order diff erence fi lter mask

D First-order diff erence operator

e 1.6022 · 10− 19 As               Elementary electric charge e 2.718281...     Base for natural logarithm

E [W/m2], [lm/m2], [lx]         Radiant (irradiance) or luminous (illu-

minance) incident energy fl ux density

E [V/m]                                Electric fi eld

e ¯             [1]                                     Unit eigenvector of a matrix

f, f_e         [m]                                   (Eff ective) focal length of an optical system

f_b, f_f       [m]                                   Back and front focal length

f Optical fl ow

f                                                           Feature vector

F [N] (newton)                     Force

G Image matrix

H General fi lter mask

h             6.6262 · 10− 34 Js                Planck’s constant (action quantum)

h h/(2π ) [Js]

i √ − 1                                   Imaginary unit

I             [W/sr], [lm/sr]                   Radiant or luminous intensity

I             [A]                                    Electric current

continued on next page

561

Symbol   Defi nition, [Units]             Meaning

continued from previous page

I                                                           Identity matrix

I Identity operator

J Structure tensor, inertia tensor

k_B           1.3806 · 10− 23 J/K              Boltzmann constant

k             1/λ, [m− 1]                          Magnitude of wave number

k             [m− 1]                                Wave number (number of wave- lengths per unit length)

k˜              k∆ x/π                               Wave number normalized to the max- imum wave number that can be sam- pled (Nyquist wave number)

K_q           [l/mol]                              Quenching constant

K_r           Φ _ν /Φ _e, [lm/W]                   Radiation luminous effi ciency

K_s           Φ _ν /P [lm/W]                     Lighting system luminous effi ciency

K_I           [1]                                    Indicator equilibrium constant

L            [W/(m2sr)], [1/(m2sr)], [lm/(m2sr)], [cd/m2]

Radiant (radiance) or luminous (lumi- nance) fl ux density per solid angle

L                                                          Laplacian fi lter mask

L Laplacian operator

m            [kg]                                   Mass

m            [1]                                    Magnifi cation of an optical system

m Feature vector

M [W/m2], [1/(s m2)]             Excitant radiant energy fl ux density

(excitance, emittance)

M_e          [W/m2]                              Energy-based excitance

M_p          [1/(s m2)]                          Photon-based excitance

M Feature space

n [1]                                    Index of refraction

n_a            [1]                                    Numerical aperture of an optical sys- tem

n_f            f/d, [1]                              Aperture of an optical system

n ¯             [1]                                     Unit vector normal to a surface

N Set of natural numbers: {0, 1, 2,...}

p             [kg m/s], [W m]                 Momentum

p             [N/m2]                              Pressure

pH          [1]                                    pH value, negative logarithm of pro- ton concentration

Q [Ws] (joule), [lm s]             Radiant or luminous energy number of photons

Q_s           [1]                                    Scattering effi ciency factor

continued on next page

562                                                                                                                 B Notation

Symbol   Defi nition, [Units]             Meaning

continued from previous page

r [m]                                   Radius

T

r_{m, n}          r_{m, n} = Σ m∆ x, n∆ yΣ  T

Translation vector on grid

r ˆ _{p, q}          r ˆ _{p, q} = Σ p/∆ x, q/∆ yΣ           Translation vector on reciprocal grid

R Φ /s, [A/W]                       Responsivity of a radiation detector

R                                                          Box fi lter mask

R                                                          Set of real numbers

s [A]                                    Sensor signal

T [K]                                    Absolute temperature

t [s]                                     Time

t [1]                                    Transmittance

u [m/s]                                 Velocity

u             [m/s]                                 Velocity vector

U [V]                                    Voltage, electric potential

V [m3]                                  Volume

V (λ )        [lm/W]                              Spectral luminous effi cacy for pho- topic human vision

V '(λ )       [lm/W]                              Spectral luminous effi cacy for sco- topic human vision

w e2π i

w_N          exp(2π i/N)

T                     T

x Σ x, yΣ , [x₁, x₂]

Image coordinates in the spatial do- main

X            [X, Y, Z]^T, [X₁, X₂, X₃]^T World coordinates

Z, Z+                                                                                 Set of integers, positive integers

 

 

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Index

Symbols

3- Dimaging 205

4- neighborhood 33

6-neighborhood  33

8-neighborhood  33

 

A

absorption coeffi cient 170

accurate 77

acoustic imaging 153

acoustic wave 152

longitudinal 152

transversal 152

action quantum 150

action-perception cycle 16

active contour 442

active vision 16, 18

adder circuit 297

adiabatic compressibility 152

aerial image 514

AI 515

aliasing 233

alpha radiation 151

AltiVec 25

amplitude 56

amplitude of Fourier component 56 anaglyph method 210

analog-digital converter 247

analytic function 361

analytic signal  361

and operation  481

aperture problem 210, 379, 384,

385, 391, 394, 401, 450, 464

aperture stop 189

area 508

ARMA 116

artifi cial intelligence 18, 515

associativity 110, 484

astronomy 3, 18

autocorrelation function 94

autocovariance function 94 autoregressive-moving average

process  116 averaging

recursive 303

axial magnifi cation 187

 

B

B-splines 276

back focal length 186 band sampling 156

band-limited 236

bandwidth-duration product          55

bandpass decomposition 135, 139

bandpass fi lter 121, 128 base

orthonormal 39

basis image 39, 107

BCCE 386, 391

bed-of-nails function 236

Bessel function 199

beta radiation 151

bidirectional refl ectance distribution function        170

bimodal distribution 428

binary convolution 481

binary image 36, 427

binary noise 296

binomial distribution 89, 291

binomial fi lter 392

bioluminescence 173

bit reversal 68, 69

blackbody 163, 166

block matching  392

Bouger’s law 170

bounding box  499

box fi lter 286, 392

box function 196

BRDF 170

Brewster angle 169

 

 

575

576                                                                                                                           Index

 

 

brightness change constraint		colorimetry 160
equation   386		commutativity 109, 484
butterfl y operation 70		complex exponential 114, 117
		complex number 41
C		complex plane 43
calibration error 77		complex polynomial 117
camera coordinates 178		complex-valued vector 43
Camera link 24		computational complexity 65
Canny edge detector 333		computer graphics 17
Cartesian coordinates 90		computer science 17
Cartesian Fourier descriptor	504	computer vision 18
cartography 207		confocal laser scanning microscopy
Cauchy–Schwarz inequality	407	215
causal fi lter 115, 116		connected region 32
CCD 21		connectivity 433
CD-ROM 24		constant neighborhood 308
center of mass 500		continuity equation 386
central limit theorem 90		continuous-wave modulation 217
central moment 80, 500		controlled smoothness 452
centroid 504		convolution 52, 86, 95, 195, 235,
chain code 495, 498		350
characteristic value 114		binary 481
characteristic vector 114		cyclic 105
charge coupled device 21		discrete 102
chemiluminescence 173		normalized 309
chess board distance 34		convolution mask 52
chi density 91		convolution theorem 52, 108, 114
chi-square density 91, 92		Cooley-Tukey algorithm 72
child node 497		coordinates
circular aperture 201		camera 178
circularity 510		Cartesian 90
circularly polarized 150		homogeneous 183
city block distance 34		polar 90
classifi cation 16, 516		world 177
object-based 516		correlation 112
pixel-based 516		cyclic 95
supervised 523		correlation coeffi cient 83
unsupervised 523		correspondence
classifi er 523		physical 381
closing operation 486		visual 381
cluster 517		correspondence problem 379
CMOS image sensor 22		cosine transform 62, 63
co-spectrum 97		covariance 83, 94
coherence 150		covariance matrix 83, 111, 465, 521
coherence function 97		cross section 172
coherency measure 349		cross-correlation coeffi cient 407

coherency radar 8, 218

coherent 150

color diff erence system 161 color image 283

cross-correlation function 95

cross-correlation spectrum 97

cross-covariance 521

cross-covariance function 95

Index                                                                                            577

 

cyclic 343

cyclic convolution 105, 297

cyclic correlation 95

 

D

data space 463

data vector 461, 468

decimation-in-frequency FFT 73

decimation-in-time FFT 68

decision space 523

deconvolution 113, 476

defocusing 474

deformation energy 450 degree of freedom 465 delta function, discrete 115 depth from

multiple projections 208

phase 207

time-of-fl ight  207

triangulation  207

depth from paradigms 207 depth imaging 205, 206

depth map 6, 213, 441

depth of fi eld 188, 212, 477 depth of focus 187

depth range 208

depth resolution 208

depth-fi rst traversal 497

derivation theorem 53 derivative

directional 358

partial 316

derivative fi lter 350

design matrix 461, 468

DFT  43

DHT  63

diff erence of Gaussian 335, 358 diff erential cross section 172 diff erential geometry 404

diff erential scale space 135, 139

diff erentiation 315

diff raction-limited optics 200

diff usion coeffi cient 129

diff usion equation 472

diff usion tensor 459

diff usion-reaction system 455

digital object 32

digital signal processing 77 digital video disk 24 digitization 15, 177, 233

 

dilation operator 482

direction 342

directional derivative 358 directiopyramidal decomposition

141, 411, 420

discrete convolution 102 discrete delta function 115 discrete diff erence 315

discrete Fourier transform 43, 116 discrete Hartley transform 63 discrete inverse problem 442 discrete scale space 136

disparity 209

dispersion 149

displacement vector 379, 385, 450

displacement vector fi eld 386, 442,

450

distance transform 493 distortion

geometric 190

distribution function 79

distributivity 110, 485

divide and conquer 65, 72

DoG 335, 358

Doppler eff ect 174

dual base 242

dual operators 486

duality 486

DV 379, 385

DVD 24

DVD+RW 24

DVF 386

dyadic point operator 264, 320

dynamic range 208

 

E

eccentricity 502

edge 308

in tree 435

edge detection 315, 323, 339 edge detector

regularized 331

edge strength 315

edge-based segmentation 431 eff ective focal length 186 eff ective inverse OTF 478 effi ciency factor 172

eigenimage 114

eigenvalue 114, 459

eigenvalue analysis 400

578                                                                                                                           Index

 

eigenvalue problem 346

eigenvector 114, 399

elastic membrane 450

elastic plate  451

elastic wave 152

elasticity constant 450

electric fi eld 147

electrical engineering 17

electromagnetic wave 147

electron 151

electron microscope 152

ellipse 502

elliptically polarized 150

emission 163

emissivity 165, 166

emittance 154

energy 58

ensemble average 93

ergodic 95

erosion operator 482 error

calibration 77

statistical 77

systematic 77

error functional 446

error propagation 465

error vector 461

Ethernet 24

Euclidian distance 34

Euler-Lagrange equation 445, 455

excitance 154

expansion operator 139

expectation value 80

exponential, complex 114

exposure time 87

extinction coeffi cient 171

 

F

fan-beam projection 224

Faraday eff ect 173

fast Fourier transform 66 father node 435

feature 99

feature image 15, 99, 339

feature space 517

feature vector 517

FFT 66

decimation-in-frequency 73

decimation-in-time 68

multidimensional 74

radix-2 decimation-in-time 66

radix-4 decimation-in-time 72 fi eld

electric 147

magnetic 147

fi ll operation 500

fi lter 52, 99

binomial 290

causal 115

diff erence of Gaussian 358 fi nite impulse response 116 Gabor 364, 396, 411

infi nite impulse response 116 mask 109

median 124, 307

nonlinear 124

polar separable 368

quadrature 396

rank value 123, 482

recursive  115

separable  110

stable 116

transfer function 109

fi ltered back-projection 228, 229 fi nite impulse response fi lter 116 FIR fi lter 116

Firewire 24

fi rst-order statistics 78

fi x point 308

fl uid dynamics 386

fl uorescence 173 focal plane array 533 focus series 477

forward mapping 265

four-point mapping 267

Fourier descriptor 495

Cartesian 504

polar 505

Fourier domain 556

Fourier ring 48

Fourier series 45, 503 Fourier slice theorem 227 Fourier torus 48

Fourier transform 29, 40, 42, 45, 95,

195

discrete 43

infi nite discrete 45

multidimensional 45

one-dimensional 42

windowed 127

Index                                                                                            579

 

Fourier transform pair 43 FPA 533

Fraunhofer diff raction 200

frequency 147

frequency doubling 149

Fresnel’s equations 168 front focal length 186 FS 45

 

G

Gabor fi lter 364, 396, 411

gamma transform 253

gamma value 38

Gaussian noise 296

Gaussian probability density 89 Gaussian pyramid 126, 137, 138 generalized image coordinates 183 generalized inverse 465

geodesy 207

geometric distortion 190

geometric operation 245 geometry of imaging 177 global optimization 441

gradient space 218

gradient vector 316

gray value corner 405, 406

gray value extreme 405, 406

grid vector 34

group velocity  365

 

H

Haar transform  64

Hadamard transform 64

Hamilton’s principle 445

Hankel transform 199

Hartley transform 63

Hesse matrix 317, 404

hierarchical processing 15 hierarchical texture organization

413

Hilbert fi lter 360, 411, 420

Hilbert operator 360

Hilbert space 62

Hilbert transform 359, 360

histogram 79, 517

hit-miss operator 487, 488

homogeneous 79, 107

homogeneous coordinates 183, 267 homogeneous point operation 246 homogeneous random fi eld 94

 

Hough transform 437, 463

HT 63

hue 161

human visual system 18, 158

hyperplane 463

 

I

IA-64 25

idempotent operation 486

IDFT 45

IEEE 1394 24

IIR fi lter  116

illumination slicing 208

illumination, uneven 257

image analysis 427

image averaging 256

image coordinates 181

generalized 183

image cube 381

image data compression 63 image equation 186

image fl ow 385

image formation 236

image preprocessing 15

image processing 17

image reconstruction 16

image restoration 16

image sensor 22

image sequence 8

image vector 467

impulse 308

impulse noise 296

impulse response 108, 115

incoherent 150

independent random variables 83 index of refraction 149

inertia tensor 356, 502

infi nite discrete Fourier transform 45

infi nite impulse response fi lter 116 infrared 23, 165

inhomogeneous background 283 inhomogeneous point operation 256 inner product 39, 42, 60, 356

input LUT 247

integrating sphere 259

intensity 161

interferometry 207

interpolation 239, 242, 269

interpolation condition  270

580                                                                                                                           Index

 

inverse fi ltering 113, 442, 476

inverse Fourier transform 42, 46

inverse mapping 265, 266 inverse problem

overdetermined 461

irradiance 29, 154 isotropic edge detector 319 isotropy 288

 

J

Jacobian matrix 86, 336

joint probability density function 83 JPEG 63

 

K

Kerr eff ect 173

 

L

Lagrange function 445

Lambert-Beer’s law 170

Lambertian radiator 164 Laplace of Gaussian 334 Laplace transform 118

Laplacian equation 449

Laplacian operator 129, 135, 317,

328

Laplacian pyramid 126, 137, 139,

411

lateral magnifi cation 186

leaf node 497 leaf of tree 435 learning 523

least squares 447

lens aberration 474

line sampling 156

linear discrete inverse problem 461 linear interpolation 272

linear shift-invariant operator 107 linear shift-invariant system 123,

194, 474

linear symmetry 341

linear time-invariant 107

linearly polarized 149

local amplitude 362

local orientation 363, 368

local phase 362, 363

local variance 417

local wave number 358, 368, 373,

420

LoG 334

log-polar coordinates 59 logarithmic scale space 135 lognormal 369, 373 longitudinal acoustic wave 152 look-up table 247, 320

look-up table operation 247

low-level image processing 99, 427

LSI 123, 194

LSI operator 107

LTI 107

luminance 161

luminescence 173

LUT 247

 

M

m-rotational symmetry 505

machine vision  18

magnetic fi eld  147

magnetic resonance 225 magnetic resonance imaging 8 magnifi cation

axial  187

lateral 186

marginal probability density function 83

Marr-Hildreth operator 334

mask 100

mathematics 17

matrix 556

maximization problem 346

maximum fi lter 124

maximum operator 482

mean 80, 416

measurement space 517

median fi lter 124, 307, 314

medical imaging 18

membrane, elastic 450

memory cache 71 metameric color stimuli 159 metrology 18

MFLOP 65

microscopy 189

microwave 165

Mie scattering 172

minimum fi lter 124

minimum operator 482

minimum-maximum principle 133

MMX 25

model 442

model matrix 461

Index                                                                                            581

 

model space 437, 463

model vector 461

model-based segmentation 427, 436 model-based spectral sampling 156 Moiré eff ect 233, 237

molar absorption coeffi cient 171 moment 495, 500

central 500

scale-invariant 501

moment tensor 502

monogenic signal 363

monotony 485

morphological operator 483

motility assay 9

motion 15

motion as orientation 383 motion fi eld 385, 386

moving average 133

MR 8, 225

multigrid representation 126, 137 Multimedia Instruction Set Extension

25

multiplier circuit 297

multiscale representation 126 multiscale texture analysis 414 multispectral image 283

multiwavelength interferometry 218

 

N

neighborhood 4- 33

6-  33

8-  33

neighborhood operation 99

neighborhood relation 32

network model  469

neural networks  18

neutron 151

node 69

node, in tree 435 noise 283

binary 296

spectrum 112

white 308

zero-mean 94, 95

noise suppression 295, 307

non-closed boundaries  505

non-uniform illumination 283

nonlinear fi lter 124

nonlinear optical phenomenon 149

 

norm 61, 178, 461

normal density 462

normal distribution 90 normal probability density 89 normal velocity 401, 411

normalized convolution 309

null space 346

numerical aperture 202

 

O

object-based classifi cation 516

occlusion 182

OCR 12, 513, 520

octree 498

OFC 386, 391

opening operation 486

operator 556

operator notation 101

operator, Laplacian 317

operator, morphological 483

optical activity 173

optical axis 178, 186

optical character recognition 12, 513, 520

optical depth 171

optical engineering 17

optical fl ow 385

optical fl ow constraint 386 optical illusions 19

optical signature 515

optical thickness  171

optical transfer function 197, 474

or operation 481

orientation 342, 343, 383, 416, 502

local 450

orientation invariant 372

orientation vector 348

orthonormal 178

orthonormal base 39

orthonormality relation 40

OTF 197, 474, 478

outer product 46

output LUT 247

oxygen 173

 

P

parallax 209

parameter vector 461, 468

partial derivative 316

particle physics 3

582                                                                                                                           Index

 

particulate radiation 151

Pascal’s triangle 292

pattern recognition 18, 513

PBA 174

PDF 79

pel 29

perimeter 509

periodicity 47, 48

DFT 47

perspective projection 181, 182, 184

phase 56, 358, 410

phase angle 41

phase of Fourier component 56 phosphorescence 173

photogrammetry 3, 18

photography 3

photometric stereo 220, 441

photometry 156

photon 150

photonics 17

photopic vision 158

photorealistic 17, 388

physical correspondence 381

physics 17

pinhole camera 181

pixel 29, 78

pixel-based classifi cation 516

pixel-based segmentation 427

Planck 163

Planck’s constant 150

plane polarized 149

plate, elastic 451

point operation 78, 99, 245, 350

homogeneous 246

inhomogeneous 256

point operator 81

point spread function 108, 112, 115,

194, 428, 474

Poisson distribution 151

Poisson process 88

polar coordinates 90

polar Fourier descriptor 505 polar separable 311, 368 polarization

circular 150

elliptical 150

linear  149

potential  450

power spectrum 57, 96, 112

precise 77

primary colors 160

principal axes 394

principal plane  186

principal point  186

principal ray 189

principal-axes transform 521

principle of superposition 107, 484 probability density function 79 process

homogeneous 79

projection operator  226

projection theorem  227

proton 151

pseudo-color image 248, 250

pseudo-noise modulation 217

PSF 108, 194, 478

pulse modulation 217

pyramid 21

pyramid linking 433 pyrene butyric acid 174

 

Q

quad-spectrum 97

quadrant 497

quadratic scale space 135 quadrature fi lter 359, 364, 396 quadrature fi lter pair 420 quadtree 495, 496

quantization 35, 79, 177, 243

quantum effi ciency 22, 92

quantum mechanics 62

quenching 173

 

R

radiant energy 153

radiant fl ux 153

radiant intensity 154 radiometric calibration

nonlinear  260

two-point  259

radiometry 153 radiometry of imaging 177 radiosity 388

radius 505

radix-2 FFT algorithm 66 radix-4 FFT algorithm 72 Radon transform 226

RAIDarray 24

random fi eld 78, 93

ergodic 95

Index                                                                                            583

 

homogeneous 94

random variable 79, 151

independent  83

uncorrelated  83

rank 346

rank-value fi lter 123, 307, 482

ratio imaging 220

Rayleigh criterion 201

Rayleigh density 90

Rayleigh theorem 58

reciprocal base 242

reciprocal grid 236

reciprocal lattice 241

reconstruction 16, 100, 441

rectangular grid 32, 33

recursive averaging 303

recursive fi lter 115, 116

refl ectivity 168

refraction 167

region of support 100

region-based segmentation 432

regions 283

regularized edge detector 331 relaxation fi lter 118, 119

remote sensing 18

rendering equation 388 representation-independent notation

101

resonance fi lter 118

responsivity 157

restoration 100, 441, 447, 474

Riesz transform 363

robustness  351

root 308, 497 root of tree  435

rotation 35, 178, 184, 266

run-length code 495

RV 79

 

S

sample variance 91, 93

sampling 236

standard 239

sampling theorem 137, 234, 236,

237

satellite image 514

saturation 161

scalar 556

scalar product 39, 356

scale 128, 416

 

scale invariance 132, 133

scale invariant 501

scale mismatch 125

scale space 126, 128, 456

scaler circuit 296

scaling 34, 184, 266

scaling theorem 199

scotopic vision 158

searching 65

segmentation 15, 427, 442

edge-based 431

model-based 436

pixel-based 427

region-based 432

semi-group property 133

sensor element 78 separability

FT 51

separable fi lter 110, 118

shape 481

shape from refraction 221

shape from shading 9, 207, 218, 441

shearing 266

shift invariant 94, 107, 484

shift operator 107, 484

shift theorem 52, 57, 128, 506

shift-register stage 297

SIMD 25

similarity constraint 441

simple neighborhood 341

sine transform 62, 63

single instruction multiple data 25 singular value decomposition 463 skewness 80

smoothing fi lter 350

smoothness 448

smoothness constraint 441

snake 442

Snell’s law 167

Sobel operator 351

software engineering 17

solid angle 154

son node 435

space-time image 381

spatiotemporal energy 396

spatiotemporal image 381

specifi c rotation 173

spectroradiometry 155

spectroscopic imaging 156

specular surface 168

584                                                                                                                           Index

 

speech processing 18

speech recognition 513 speed of light 147 speed of sound 152 spline 276

standard deviation 85

standard sampling 239

statistical error 77

steerable fi lter 310

Stefan-Boltzmann law 165

step edge 433

stereo image 441

stereo system 209

stereoscopic basis 209

Stern–Vollmer equation 174

stochastic process 78, 93

stretching 266

structure element 100, 483

structure tensor 439

subsampling 137

subtractor circuit 297

subtree 435

superposition principle 107, 484

supervised classifi cation 523 support vector machine 528 symmetry 505

DFT 48

system, linear shift-invariant 123 systematic error 77

 

T

target function  327

telecentric 5

telecentric illumination system 221 temperature distribution 165

tensor 556

terminal node 497

test image 289

text recognition  513

texture 15, 339, 413

theoretical mechanics 445

thermal emission 163

thermal imaging 257

thermography 165, 167

three-point mapping 267

TIFF 496

time series 58, 107, 556

tomography 16, 100, 208, 224, 441 total least squares 399

total refl ection 169

tracing algorithm 432

transfer function 108, 109, 474

recursive fi lter 117

translation 34, 178, 184, 266

translation invariance 499

translation invariant 107

transmission tomography 225

transmissivity  171

transmittance  171

transport equation 472 transversal acoustic wave 152 tree 435, 497

triangular grid 33

triangulation  207

tristimulus 160

 

U

ultrasonic microscopy 152

ultrasound 152

ultraviolet 23

uncertainty relation 55, 128, 139,

355

uncorrelated random variable 83 uneven illumination 257

uniform density 90

uniform distribution 82

unit circle 43

unit vector 556

unitary transform 29, 60

unsupervised classifi cation 523

upsampling 51

 

V

Van Cittert iteration 479 variance 80, 83, 93, 416, 465

variance operator 213, 417

variation calculus 444

vector 556

vector space 44

vector, complex-valued 43 vectorial feature image 283 vertex, in tree 435 vignetting 192

VIS 25

visual computing 17

visual correspondence 381

visual inspection 5 visual instruction set 25 visual perception 18

volume element 32

Index                                                                                            585

volumetric image     6

volumetric imaging 205, 206

voxel 32, 381

 

W

Waldsterben   514

wave

acoustic 152

elastic 152

electromagnetic        147

wave number 41, 155, 556

wavelength 41, 147, 155, 195

weighted averaging      309

white noise 97, 308

white point   161

white-light interferometry 8, 218

Wien’s law 165

window 100

window function 238, 261 windowed Fourier transform      127

windowing  261

world coordinates     177

 

X

x-ray 23

x86-64 25

XYZ color system   161

 

Z

z-transform 48, 118

zero crossing 328, 453

zero-mean noise    94

zero-phase fi lter 118, 284

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