Image Formation, Sampling, Windowing

Our starting point is an infi nite, continuous image g( x ), which we want to map onto a matrix G. In this procedure we will include the image formation process, which we discussed in Section 7.6. We can then dis- tinguish three separate steps: image formation, sampling, and the limi- tation to a fi nite image matrix.

 

Image Formation

Digitization cannot be treated without the image formation process. The optical system, including the sensor, infl uences the image signal so that we should include this process.

Digitization means that we sample the image at certain points of a discrete grid, r _{m, n} (Section 2.2.3). If we restrict our considerations to rectangular grids, these points can be written according to Eq. (2.2):

r _{m, n} = [m ∆ x₁, n ∆ x₂]^T with m, n ∈ Z.                                    (9.1)

Generally, we do not collect the illumination intensity exactly at these points, but in a certain area around them. As an example, we take an ideal CCD camera, which consists of a matrix of photodiodes without any light-insensitive strips in between. We further assume that the pho- todiodes are equally sensitive over the whole area. Then the signal at the grid points is the integral over the area of the individual photodiodes:

(m+1/2)∆ x₁ (n+1/2)∆ x₂

g( r _{m, n}) =       ∫          ∫   g'( x ) dx₁ dx₂.                    (9.2)

(m− 1/2)∆ x₁ (n− 1/2)∆ x₂

This operation includes convolution with a rectangular box function and sampling at the points of the grid. These two steps can be separated. We can perform fi rst the continuous convolution and then the sampling. In this way we can generalize the image formation process and separate it from the sampling process.

Because convolution is an associative operation, we can combine the averaging process of the CCDsensor with the PSF of the optical system (Section 7.6.1) in a single convolution process. Therefore, we can de- scribe the image formation process in the spatial and Fourier domain by the following operation:

∞

g( x ) =

∫ g'( x ')h( x − x ')d²x'  ◦       •

− ∞

gˆ ( k ) = gˆ ^'( k )hˆ ( k ),            (9.3)

where h( x ) and hˆ ( k ) are the resulting PSF and OTF, respectively, and

g'( x ) can be considered as the gray value image that would be obtained

236                                                        9 Digitization, Sampling, Quantization

 

by a perfect sensor, i. e., an optical system (including the sensor) whose OTF is identically 1 and whose PSF is a δ -function.

Generally, the image formation process results in a blurring of the image; fi ne details are lost. In Fourier space this leads to an attenuation of high wave numbers. The resulting gray value image is said to be band- limited.

 

Sampling

Now we perform the sampling. Sampling means that all information is lost except at the grid points. Mathematically, this constitutes a multipli- cation of the continuous function with a function that is zero everywhere except for the grid points. This operation can be performed by multiply- ing the image function g( x ) with the sum of δ functions located at the grid points r _{m, n} Eq. (9.1). This function is called the two-dimensional δ comb, or “bed-of-nails function”. Then sampling can be expressed as

 

g_s( x ) = g( x ). δ ( x − r _{m, n})          ◦ •

gˆ _s( k )         gˆ ( k   r ˆ _{p, q}),      (9.4)

.=     −

p, q

 

wobei

r ˆ p, q = Σ   p□ k₁

Σ with p, q ∈ Z and □ k_w = 1

 

 

(9.5)

 

q□ k2

∆ xw

are the points of the so-called reciprocal grid, which plays a signifi cant role in solid state physics and crystallography. According to the convo- lution theorem (Theorem 4, p. 52), multiplication of the image with the 2-D δ comb corresponds to a convolution of the Fourier transform of the image, the image spectrum, with another 2-D δ comb, whose grid con- stants are reciprocal to the grid constants in x space (see Eqs. (9.1) and (9.5)). A dense sampling in x space yields a wide mesh in the k space, and vice versa. Consequently, sampling results in a reproduction of the image spectrum at each grid point r ˆ _{p, q} in the Fourier space.

 

Sampling Theorem

Now we can formulate the condition where we get no distortion of the signal by sampling, known as the sampling theorem. If the image spec- trum is so extended that parts of it overlap with the periodically repeated copies, then the overlapping parts are alternated. We cannot distinguish whether the spectral amplitudes come from the original spectrum at the center or from one of the copies. In order to obtain no distortions, we must avoid overlapping.

A safe condition to avoid overlapping is as follows: the spectrum must be restricted to the area that extends around the central grid point

9.2 Image Formation, Sampling, Windowing                                   237

 

Figure 9.3: Explanation of the Moiré eff ect with a periodic structure that does not meet the sampling condition.

 

up to the lines parting the area between the central grid point and all other grid points. In solid state physics this zone is called the fi rst Bril- louin zone [96].

On a rectangular W -dimensional grid, this results in the simple condi- tion that the maximum wave number at which the image spectrum is not equal to zero must be restricted to less than half of the grid constants of the reciprocal grid:

 

Theorem 10 (Sampling theorem) If the spectrum gˆ ( k ) of a continuous function g( x ) is band-limited, i. e.,

gˆ ( k ) = 0 ∀ |k_w|≥ □ k_w/2,                                           (9.6)

then it can be reconstructed exactly from samples with a distance

∆ x_w = 1/□ k_w.                                                   (9.7)

In other words, we will obtain a periodic structure correctly only if we take at least two samples per wavelength. The maximum wave number that can be sampled without errors is called the Nyquist or limiting wave number. In the following, we will often use dimensionless wave numbers which are scaled to the limiting wave number. We denote this scaling with a tilde:

w =          = w      w

k˜       k w          2k  ∆ x  .                                    (9.8)

□ k_w/2

In this scaling all the components of the wave number k˜ w fall into the

−

] 1, 1[ interval.

Now we can explain the Moiré and aliasing eff ects. We start with a periodic structure that does not meet the sampling condition. The

238                                                        9 Digitization, Sampling, Quantization

 

original spectrum contains a single peak, which is marked with the long vector k in Fig. 9.3.

Because of the periodic replication of the sampled spectrum, there is exactly one peak, at k ^', which lies in the central cell. Figure 9.3 shows that this peak has not only another wavelength but in general another direction, as observed in Fig. 9.1.

The observed wave number k ^' diff ers from the true wave number k by a grid translation vector r ˆ _{p, q} on the reciprocal grid.  The indices p and q must be chosen to meet the condition

|k₁ + p □ k₁|  <          □ k₁/2

|k₂ + q □ k₂|   <  □ k₂/2.

According to this condition, we obtain an aliased wave number

 

(9.9)

k'1  = k₁ − □ k₁ = 9/10 □ k₁ − □ k₁ = − 1/10 □ k₁                           (9.10) for the one-dimensional example in Fig. 9.2, as we just observed.

The sampling theorem, as formulated above, is actually too strict a requirement. A suffi cient and necessary condition is that the periodic replications of the image spectra must not overlap.

 

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