Interactive Gray Value Evaluation

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Homogeneous point operators implemented via look-up tables are a very useful tool for inspecting images. As the look-up table operations work in real-time, images can be manipulated interactively. If only the output look-up table is changed, the original image content remains unchanged. Here, we demonstrate typical tasks.

Evaluating and Optimizing Illumination. With the naked eye, we can hardly estimate the homogeneity of an illuminated area as demonstrated

250 10 Pixel Processing

P(q)

Figure 10.3: Illustration of a nonlinear look-up table with mapping of multiple values onto one and missing output value leading to uneven steps.

in Fig. 10.4a, b. A histogram reveals the gray scale distribution but not its spatial variation (Fig. 10.4c, d). Therefore, a histogram is not of much help for optimizing the illumination interactively. We need to mark gray scales such that absolute gray levels become perceivable for the human eye. If the radiance distribution is continuous, it is suﬃ cient to use equidensities. This technique uses a staircase type of homogeneous point operation by mapping a certain range of gray scales onto one. This point operation is achieved by zeroing the p least signiﬁ cant bits with a logical and operation:

q' = P (q) = q ∧ (2p − 1), (10.9)

−

∧

where denotes the logical (bitwise) and and overlining denotes nega- tion. This point operation limits the resolution to Q p bits and, thus, 2Q− p quantization levels. Now, the jump between the remaining quan- tization levels is large enough to be perceived by the eye and we see contour lines of equal absolute gray scale in the image (Fig. 10.4). We can now try to homogenize the illumination by making the distance be- tween the contour lines as large as possible.

Another way to mark absolute gray values is the so-called pseudo- color image that has already been discussed in Section 10.2.2. With this technique, a gray level q is mapped onto an RGB triple for display. As color is much better recognized by the eye, it helps reveal absolute gray levels.

Detection of Underﬂ ow and Overﬂ ow. Under- and overﬂ ows of the gray values of a digitized image often go unnoticed and cause a serious bias in further processing, for instance for mean gray values of objects or the center of gravity of an object. In most cases, such areas cannot

10.2 Homogeneous Point Operations 251

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C d

Figure 10.4: a The irradiance is gradually decreasing from the top to the bot- tom, which is almost not recognized by the eye. The gray scale of this ﬂ oating- point image computed by averaging over 100 images ranges from 160 to 200. b Histogram of a; c and d (contrast enhanced, gray scale 184–200): Edges ar- tiﬁ cially produced by a staircase LUT with a step height of 1.0 and 2.0 make contours of constant irradiance easily visible.

be detected directly. They may only become apparent in textured ar- eas when the texture is bleached out. Over and underﬂ ow are detected easily in histograms by strong peaks at the minimum and/or maximum gray values (Fig. 10.5). With pseudocolor mapping, the few lowest and highest gray values could be displayed, for example, in blue and red, re- spectively. Then, gray values dangerously close to the limits immediately pop out of the image and can be avoided by correcting the illumination lens aperture or gain of the video input circuit of the frame grabber.

Contrast enhancement. Because of poor illumination conditions, it of- ten happens that images are underexposed. Then, the image is too dark and of low contrast (Fig. 10.6a). The histogram (Fig. 10.6b) shows that the image contains only a small range of gray values at low gray values. The appearance of the image improves considerably if we apply a point operation which maps a small grayscale range to the full contrast range

252 10 Pixel Processing

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Figure 10.5: Detection of underﬂ ow and overﬂ ow in digitized images by his- tograms; a image with underﬂ ow and b its histogram; c image with overﬂ ow and d its histogram.

≥

= =

(for example with this operation: q' 4q for q < 64, and q' 255 for q 64) (Fig. 10.6c). We only improve the appearance of the image but not the image quality itself. The histogram shows that the gray value resolution is still the same (Fig. 10.6d).

The image quality can be improved. The best way is to increase the object irradiance by using a more powerful light source or a better design of the illumination setup. If this is not possible, we can still increase the gain of the analog video ampliﬁ er. All modern image processing boards include an ampliﬁ er whose gain and oﬀ set can be set by software (see Figs. 10.1 and 10.2). By increasing the gain, the brightness and resolution of the image improve, but at the expense of an increased noise level.

Contrast Stretching. It is often of interest to analyze faint irradiance diﬀ erences which are beyond the resolution of the human visual system or the display equipment used. This is especially the case if images are printed. In order to observe faint diﬀ erences, we stretch a small gray scale range of interest to the full range available. All gray values outside this range are set to the minimum or maximum value. This operation

10.2 Homogeneous Point Operations 253

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Figure 10.6: Contrast enhancement; a underexposed image and b its histogram;

c interactively contrast enhanced image and d its histogram.

requires that the gray values of the object of interest fall into the range selected for contrast stretching. An example of contrast stretching is shown in Fig. 10.7a, b. The wedge at the bottom of the images, rang- ing from 0 to 255, directly shows which part of the gray scale range is contrast enhanced.

Range Compression. In comparison to the human visual system, a digital image has a considerably smaller dynamical range. If a minimum resolution of 10 % is demanded, the gray values must not be lower than

≈

10. Therefore, the maximum dynamical range in an 8-bit image is only 255/10 25. The low contrast range of digital images makes them appear of low quality when high-contrast scenes are encountered. Either the bright parts are bleached or no details can be recognized in the dark parts. The dynamical range can be increased by a transform that was introduced in Section 2.2.6 as the gamma transform. This nonlinear homogeneous point operation has the form

q = q.

' 255 γ (10.10)

255γ

254 10 Pixel Processing

A b

C d

Figure 10.7: b – d Contrast stretching of the image shown in a. The stretched range can be read from the transformation of the gray scale wedge at the bottom of the image.

The factors in Eq. (10.10) are chosen such that a range of [0, 255] is mapped onto itself. This transformation allows a larger dynamic range to be recognized at the cost of resolution in the bright parts of the image. The dark parts become brighter and show more details. This contrast transformation is better adapted to the logarithmic characteristics of the human visual system. An image presented with diﬀ erent gamma factors is shown in Fig. 10.8.

Noise Variance Equalization. From Section 3.4.5, we know that the variance of the noise generally depends on the image intensity according to

σ 2(g) = σ 2 + α g. (10.11)

A statistical analysis of images and image operations is, however, much easier if the noise is independent of the gray value. Only then all the error propagation techniques discussed in Section 3.3.3 are valid.

Thus we need to apply a nonlinear gray value transform h(g) in such a way that the noise variance becomes constant. In ﬁ rst order, the vari-

10.2 Homogeneous Point Operations 255

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Figure 10.8: Presentation of an image with diﬀ erent gamma values: a 0.5, b 0.7,

c 1.0, and d 2.0.

ance of h(g) is σ 2 ≈ . dh Σ

σ 2(g) (10.12)

according to Eq. (3.35) [47]. If we set σ 2 to be constant, we obtain

dh σ h dg.

 σ 2(g)

Integration yields

h(g) = σ _h∫

dg'

+ C. (10.13)

0  σ 2(g')

The two free parameters σ _h and C can, for instance, be used to ﬁ t the values of h into a suitable interval. With the linear variance function Eq. (10.11), the integral in Eq. (10.13) yields

 h

2σ h(g) = √ α

σ 2 + α g + C. (10.14)

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A b

Figure 10.9: Noise reduction by image averaging: a single thermal image of small temperature ﬂ uctuations on the water surface cooled by evaporation; b same, averaged over 16 images; the full gray value range corresponds to a tem- perature range of 1.1 K.

The nonlinear transform becomes particularly simple for an ideal imag- ing sensor with σ ₀ 0. Then a square root transform must be applied to obtain an intensity independent noise variance:

2σ _h

h(g) = √ α , g. (10.15)

10.3 Inhomogeneous Point Operations†

Homogeneous point operations are only a subclass of point operators. In general, a point operation depends also on the position of the pixel in the image. Such an operation is called an inhomogeneous point opera- tion. Inhomogeneous point operations are mostly related to calibration procedures. Generally, the computation of an inhomogeneous point op- eration is much more time consuming than the computation of a homo- geneous point operation. We cannot use look-up tables since the point operation depends on the pixel position and we are forced to calculate the function for each pixel.

The subtraction of a background image without objects or illumina- tion is a simple example of an inhomogeneous point operation which is written as:

G_m' n = P_mn(G_mn) = G_mn − B_mn, (10.16)

where B_mn is the background image.

Image Averaging

One of the simplest inhomogeneous point operations is image averag- ing. There are a number of imaging sensors available which show a

10.3 Inhomogeneous Point Operations† 257

considerable noise level. Prominent examples include thermal imaging (Section 6.4.1) and all sensor types such as slow-scan CCD imagers or image ampliﬁ ers where only a limited number of photons are collected.

Figure 10.9a shows the temperature diﬀ erences at the water surface of a wind-wave facility cooled at 1.8 m/s wind speed by evaporation. Be- cause of a substantial noise level, the small temperature ﬂ uctuations can hardly be detected. Taking the mean over several images signiﬁ cantly reduces the noise level (Fig. 10.9b).

The error of the mean (Section 3.3.3) taken from N samples is given

− 1

σ 2 ≈ (N

σ 2 1

) N(N

− 1

). (g − g)2. (10.17)

= √

If we take the average of N images, the noise level is reduced by 1/ N

compared to a single image. Taking the mean over 16 images thus re- duces the noise level by a factor of four. Equation (10.17) is only valid, however, if the standard deviation σ _g is signiﬁ cantly larger than the standard deviation related to the quantization (Section 9.4).

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