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319 12 POINT AND SPATIAL IMAGE RESTORATION TECHNIQUES A common defect in imaging systems is unwanted nonlinearities in the sensor and display systems. Post processing correction of sensor signals and pre-processing correction of display signals can reduce such degradations substantially (1). Such point restoration processing is usually relatively simple to implement. One of the most common image restoration tasks is that of spatial image restoration to compen- sate for image blur and to diminish noise effects. References 2 to 6 contain surveys of spatial image restoration methods. 12.1. SENSOR AND DISPLAY POINT NONLINEARITY CORRECTION This section considers methods for compensation of point nonlinearities of sensors and displays. 12.1.1. Sensor Point Nonlinearity Correction In imaging systems in which the source degradation can be separated into cascaded spatial and point effects, it is often possible directly to compensate for the point deg- radation (7). Consider a physical imaging system that produces an observed image field according to the separable model (12.1-1) F O xy,() F O xy,()O Q O D Cxyλ,,(){}{}= Digital Image Processing: PIKS Inside, Third Edition. William K. Pratt Copyright © 2001 John Wiley & Sons, Inc. ISBNs: 0-471-37407-5 (Hardback); 0-471-22132-5 (Electronic) 320 POINT AND SPATIAL IMAGE RESTORATION TECHNIQUES where is the spectral energy distribution of the input light field, represents the point amplitude response of the sensor and denotes the spatial and wavelength responses. Sensor luminance correction can then be accomplished by passing the observed image through a correction system with a point restoration operator ideally chosen such that (12.1-2) For continuous images in optical form, it may be difficult to implement a desired point restoration operator if the operator is nonlinear. Compensation for images in analog electrical form can be accomplished with a nonlinear amplifier, while digital image compensation can be performed by arithmetic operators or by a table look-up procedure. Figure 12.1-1 is a block diagram that illustrates the point luminance correction methodology. The sensor input is a point light distribution function C that is con- verted to a binary number B for eventual entry into a computer or digital processor. In some imaging applications, processing will be performed directly on the binary representation, while in other applications, it will be preferable to convert to a real fixed-point computer number linearly proportional to the sensor input luminance. In the former case, the binary correction unit will produce a binary number that is designed to be linearly proportional to C, and in the latter case, the fixed-point cor- rection unit will produce a fixed-point number that is designed to be equal to C. A typical measured response B versus sensor input luminance level C is shown in Figure 12.1-2a, while Figure 12.1-2b shows the corresponding compensated response that is desired. The measured response can be obtained by scanning a gray scale test chart of known luminance values and observing the digitized binary value B at each step. Repeated measurements should be made to reduce the effects of noise and measurement errors. For calibration purposes, it is convenient to regard the binary-coded luminance as a fixed-point binary number. As an example, if the luminance range is sliced to 4096 levels and coded with 12 bits, the binary represen- tation would be B = b 8 b 7 b 6 b 5 b 4 b 3 b 2 b 1 . b –1 b –2 b –3 b –4 (12.1-3) FIGURE 12.1-1. Point luminance correction for an image sensor. Cxyλ,,() O Q · {} O D · {} O R · {} O R O Q · {}{}1= B ˜ C ˜ SENSOR AND DISPLAY POINT NONLINEARITY CORRECTION 321 The whole-number part in this example ranges from 0 to 255, and the fractional part divides each integer step into 16 subdivisions. In this format, the scanner can pro- duce output levels over the range (12.1-4) After the measured gray scale data points of Figure 12.1-2a have been obtained, a smooth analytic curve (12.1-5) is fitted to the data. The desired luminance response in real number and binary num- ber forms is FIGURE 12.1-2. Measured and compensated sensor luminance response. 255.9375 B 0.0≤≤ CgB{}= 322 POINT AND SPATIAL IMAGE RESTORATION TECHNIQUES (12.1-6a) (12.1-6b) Hence, the required compensation relationships are (12.1-7a) (12.1-7b) The limits of the luminance function are commonly normalized to the range 0.0 to 1.0. To improve the accuracy of the calibration procedure, it is first wise to perform a rough calibration and then repeat the procedure as often as required to refine the cor- rection curve. It should be observed that because B is a binary number, the corrected luminance value will be a quantized real number. Furthermore, the corrected binary coded luminance will be subject to binary roundoff of the right-hand side of Eq. 12.1-7b. As a consequence of the nonlinearity of the fitted curve and the amplitude quantization inherent to the digitizer, it is possible that some of the corrected binary-coded luminance values may be unoccupied. In other words, the image histogram of may possess gaps. To minimize this effect, the number of output levels can be limited to less than the number of input levels. For example, B may be coded to 12 bits and coded to only 8 bits. Another alternative is to add pseudorandom noise to to smooth out the occupancy levels. Many image scanning devices exhibit a variable spatial nonlinear point lumi- nance response. Conceptually, the point correction techniques described previously could be performed at each pixel value using the measured calibrated curve at that point. Such a process, however, would be mechanically prohibitive. An alternative approach, called gain correction, that is often successful is to model the variable spatial response by some smooth normalized two-dimensional curve G(j, k) over the sensor surface. Then, the corrected spatial response can be obtained by the operation (12.1-8) where and represent the raw and corrected sensor responses, respec- tively. Figure 12.1-3 provides an example of adaptive gain correction of a charge cou- pled device (CCD) camera. Figure 12.1-3a is an image of a spatially flat light box surface obtained with the CCD camera. A line profile plot of a diagonal line through the original image is presented in Figure 12.1-3b. Figure 12.3-3c is the gain-cor- rected original, in which is obtained by Fourier domain low-pass filtering of C ˜ C= B ˜ B max CC min – C max C min – = C ˜ gB{}= B ˜ B max gB{} C min – C max C min – = C ˜ B ˜ CgB{}= B ˜ B ˜ B ˜ F ˜ jk,() Fjk,() Gjk,() = Fjk,() F ˜ jk,() Gjk,() SENSOR AND DISPLAY POINT NONLINEARITY CORRECTION 323 the original image. The line profile plot of Figure 12.1-3d shows the “flattened” result. 12.1.2. Display Point Nonlinearity Correction Correction of an image display for point luminance nonlinearities is identical in principle to the correction of point luminance nonlinearities of an image sensor. The procedure illustrated in Figure 12.1-4 involves distortion of the binary coded image luminance variable B to form a corrected binary coded luminance function so that the displayed luminance will be linearly proportional to B. In this formulation, the display may include a photographic record of a displayed light field. The desired overall response is (12.1-9) Normally, the maximum and minimum limits of the displayed luminance function are not absolute quantities, but rather are transmissivities or reflectivities FIGURE 12.1-3. Gain correction of a CCD camera image. ( a ) Original ( c ) Gain corrected ( d ) Line profile of gain corrected ( b ) Line profile of original B ˜ C ˜ C ˜ B C ˜ max C ˜ min – B max - C ˜ min += C ˜ 324 POINT AND SPATIAL IMAGE RESTORATION TECHNIQUES normalized over a unit range. The measured response of the display and image reconstruction system is modeled by the nonlinear function (12.1-10) Therefore, the desired linear response can be obtained by setting (12.1-11) where is the inverse function of . The experimental procedure for determining the correction function will be described for the common example of producing a photographic print from an image display. The first step involves the generation of a digital gray scale step chart over the full range of the binary number B. Usually, about 16 equally spaced levels of B are sufficient. Next, the reflective luminance must be measured over each step of the developed print to produce a plot such as in Figure 12.1-5. The data points are then fitted by the smooth analytic curve , which forms the desired trans- formation of Eq. 12.1-10. It is important that enough bits be allocated to B so that the discrete mapping can be approximated to sufficient accuracy. Also, the number of bits allocated to must be sufficient to prevent gray scale contouring as the result of the nonlinear spacing of display levels. A 10-bit representation of B and an 8-bit representation of should be adequate in most applications. Image display devices such as cathode ray tube displays often exhibit spatial luminance variation. Typically, a displayed image is brighter at the center of the dis- play screen than at its periphery. Correction techniques, as described by Eq. 12.1-8, can be utilized for compensation of spatial luminance variations. FIGURE 12.1-4. Point luminance correction of an image display. CfB{}= B ˜ gB C ˜ max C ˜ min – B max C ˜ min +    = g · {} f · {} g · {} BgC{}= g · {} B ˜ B ˜ CONTINUOUS IMAGE SPATIAL FILTERING RESTORATION 325 12.2. CONTINUOUS IMAGE SPATIAL FILTERING RESTORATION For the class of imaging systems in which the spatial degradation can be modeled by a linear-shift-invariant impulse response and the noise is additive, restoration of continuous images can be performed by linear filtering techniques. Figure 12.2-1 contains a block diagram for the analysis of such techniques. An ideal image passes through a linear spatial degradation system with an impulse response and is combined with additive noise . The noise is assumed to be uncorrelated with the ideal image. The image field observed can be represented by the convolution operation as (12.2-1a) or (12.2-1b) The restoration system consists of a linear-shift-invariant filter defined by the impulse response . After restoration with this filter, the reconstructed image becomes (12.2-2a) or (12.2-2b) FIGURE 12.1-5. Measured image display response. F I xy,() H D xy,() Nxy,() F O xy,() F I αβ,()H D x α– y β–,()αd βd ∞ – ∞ ∫ ∞ – ∞ ∫ Nxy,()+= F O xy,()F I xy,() ᭺ ء H D xy,()Nxy,()+= H R xy,() F ˆ I xy,() F O αβ,()H R x α– y β–,()αd βd ∞ – ∞ ∫ ∞ – ∞ ∫ = F ˆ I xy,()F O xy,() ᭺ ء H R xy,()= 326 POINT AND SPATIAL IMAGE RESTORATION TECHNIQUES Substitution of Eq. 12.2-lb into Eq. 12.2-2b yields (12.2-3) It is analytically convenient to consider the reconstructed image in the Fourier trans- form domain. By the Fourier transform convolution theorem, (12.2-4) where , , , , are the two-dimen- sional Fourier transforms of , , , , , respec- tively. The following sections describe various types of continuous image restoration filters. 12.2.1. Inverse Filter The earliest attempts at image restoration were based on the concept of inverse fil- tering, in which the transfer function of the degrading system is inverted to yield a restored image (8–12). If the restoration inverse filter transfer function is chosen so that (12.2-5) then the spectrum of the reconstructed image becomes (12.2-6) FIGURE 12.2-1. Continuous image restoration model. F ˆ I xy,() F I xy,() ᭺ ء H D xy,()Nxy,()+[] ᭺ ء H R xy,()= F ˆ I ω x ω y ,()F I ω x ω y ,()H D ω x ω y ,()N ω x ω y ,()+[]H R ω x ω y ,()= F I ω x ω y ,()F ˆ I ω x ω y ,()N ω x ω y ,()H D ω x ω y ,()H R ω x ω y ,() F I xy,()F ˆ I xy,( ) Nxy,()H D xy,()H R xy,() H R ω x ω y ,() 1 H D ω x ω y ,() = F ˆ I ω x ω y ,()F I ω x ω y ,() N ω x ω y ,() H D ω x ω y ,() += CONTINUOUS IMAGE SPATIAL FILTERING RESTORATION 327 Upon inverse Fourier transformation, the restored image field (12.2-7) is obtained. In the absence of source noise, a perfect reconstruction results, but if source noise is present, there will be an additive reconstruction error whose value can become quite large at spatial frequencies for which is small. Typically, and are small at high spatial frequencies, hence image quality becomes severely impaired in high-detail regions of the recon- structed image. Figure 12.2-2 shows typical frequency spectra involved in inverse filtering. The presence of noise may severely affect the uniqueness of a restoration esti- mate. That is, small changes in may radically change the value of the esti- mate . For example, consider the dither function added to an ideal image to produce a perturbed image (12.2-8) There may be many dither functions for which FIGURE 12.2-2. Typical spectra of an inverse filtering image restoration system. F ˆ I xy,()F I xy,() 1 4π 2 N ω x ω y ,() H D ω x ω y ,() i ω x x ω y y+(){}exp ω x d ω y d ∞ – ∞ ∫ ∞ – ∞ ∫ += H D ω x ω y ,() H D ω x ω y ,()F I ω x ω y ,() Nxy,() F ˆ I xy,() Zxy,() F Z xy,()F I xy,()Zxy,()+= 328 POINT AND SPATIAL IMAGE RESTORATION TECHNIQUES (12.2-9) For such functions, the perturbed image field may satisfy the convolution integral of Eq. 12.2-1 to within the accuracy of the observed image field. Specifi- cally, it can be shown that if the dither function is a high-frequency sinusoid of arbitrary amplitude, then in the limit (12.2-10) For image restoration, this fact is particularly disturbing, for two reasons. High-fre- quency signal components may be present in an ideal image, yet their presence may be masked by observation noise. Conversely, a small amount of observation noise may lead to a reconstruction of that contains very large amplitude high-fre- quency components. If relatively small perturbations in the observation result in large dither functions for a particular degradation impulse response, the convolution integral of Eq. 12.2-1 is said to be unstable or ill conditioned. This potential instability is dependent on the structure of the degradation impulse response function. There have been several ad hoc proposals to alleviate noise problems inherent to inverse filtering. One approach (10) is to choose a restoration filter with a transfer function (12.2-11) where has a value of unity at spatial frequencies for which the expected magnitude of the ideal image spectrum is greater than the expected magnitude of the noise spectrum, and zero elsewhere. The reconstructed image spectrum is then (12.2-12) The result is a compromise between noise suppression and loss of high-frequency image detail. Another fundamental difficulty with inverse filtering is that the transfer function of the degradation may have zeros in its passband. At such points in the frequency spectrum, the inverse filter is not physically realizable, and therefore the filter must be approximated by a large value response at such points. Z αβ,()H D x α– y β–,()αd βd ∞ – ∞ ∫ ∞ – ∞ ∫ Nxy,()< F Z xy,() n αβ+(){}sin H D x α– y β–,()αd βd ∞ – ∞ ∫ ∞ – ∞ ∫    n ∞→ lim 0= F I xy,() Nxy,() H R ω x ω y ,() H K ω x ω y ,() H D ω x ω y ,() = H K ω x ω y ,() F ˆ I ω x ω y ,()F I ω x ω y ,()H K ω x ω y ,() N ω x ω y ,()H K ω x ω y ,() H D ω x ω y ,() +=