10 Deconvolution, Deblurring, and Restoration Image enhancement techniques are typically designed to yield \better looking" images satisfying some subjective criteria In comparison with the given image, the processed image may not be closer to the true image in any sense On the other hand, image restoration 8, 9, 11, 589, 822, 823, 824, 825, 826, 827, 828] is de ned as image quality improvement under objective evaluation criteria to nd the best possible estimate of the original unknown image from the given degraded image The commonly used criteria are LMS, MMSE, and distance measures of several types Additional constraints based upon prior and independent knowledge about the original image may also be imposed to limit the scope of the solution Image restoration may then be posed as a constrained optimization problem Image restoration requires precise information about the degrading phenomenon, and analysis of the system that produced the degraded image Typical items of information required are estimates or models of the impulse response of the degrading lter (the PSF, or equivalently, the MTF) the PSD (or ACF) of the original image and the PSD of the noise If the degrading system is shift-variant, then a model of the variation of its impulse response across the eld of imaging would be required The success of a procedure for image restoration depends upon the accuracy of the model of degradation used, and the accuracy of the functions used to represent the image degrading phenomena In this chapter, we shall explore several techniques for image restoration under varying conditions of degradation, available information, and optimization 10.1 Linear Space-invariant Restoration Filters Assuming the degrading phenomenon to be linear and shift-invariant, the simplest model of image degradation is g(x y) = h(x y) f (x y) + (x y) G(u v) = H (u v) F (u v) + (u v) © 2005 by CRC Press LLC (10.1) 857 858 Biomedical Image Analysis where f is the original image, h is the impulse response of the degrading LSI system, g is the observed (degraded) image, and is additive random noise that is statistically independent of the image-generating process The functions represented by upper-case letters represent the Fourier transforms of the image-domain functions represented by the corresponding lower-case letters A block diagram of the image degradation system as above is given in Figure 10.1 f original image linear shiftinvariant system h + g degraded image η noise FIGURE 10.1 Image degradation model involving an LSI system and additive noise The image restoration problem is de ned as follows: Given g and some knowledge of h, f , and , nd the best possible estimate of f When the degrading phenomenon can be represented by an LSI system, it is possible to design LSI lters to restore the image, within certain limits A few well-known LSI lters for image restoration are described in the following subsections 10.1.1 Inverse ltering Let us consider the degradation model expressed in matrix form (see Section 3.5) as g = hf (10.2) with no noise being present The restoration problem may be stated as follows: Given g and h, estimate f In order to develop a mathematical statement of the problem, let us consider an approximation ~f to f In the least-squares approach 9], the criterion for obtaining the optimal solution is stated as follows: Minimize the squared error between the observed response g, and the response g~ had the input been ~f The error between g and g~ is given by = g ; g~ = g ; h ~f : (10.3) The squared error is given as 2= T = (g ; h ~f )T (g ; h ~f ) (10.4) = gT g ; ~f T hT g ; gT h ~f + ~f T hT h ~f : © 2005 by CRC Press LLC Deconvolution, Deblurring, and Restoration 859 Now, we can state the image restoration problem as an optimization problem: Find ~f that minimizes Taking the derivative of the squared error in Equation 10.4 with respect to ~f , we get @ = ;2 hT g + hT h ~f : @~f Setting this expression to zero, we get ~f = (hT h);1 hT g: (10.5) (10.6) This is the least-squares or pseudo-inverse solution If h is square and nonsingular, we get ~f = h;1 g: (10.7) ; ; ; If h is circulant or block-circulant, we have h = W Dh W (see Section 3.5.5) Then, ~f = W D;h W;1 g (10.8) which leads to G (u v ) : F~ (u v) = H (10.9) (u v ) This operation represents the inverse lter, which may be expressed as LI (u v) = H (u1 v) : (10.10) It is evident that the inverse lter requires knowledge of the MTF of the degradation process see Sections 2.9, 2.12, and 10.1.6 for discussions on methods to derive this information The major drawback of the inverse lter is that it fails if H (u v) has zeros, or if h is singular Furthermore, if noise is present (as in Equation 10.1), we get F~ (u v) = F (u v) + H((uu vv)) : (10.11) Problems arise because H (u v) is usually a lowpass function, whereas (u v) is uniformly distributed over the entire spectrum then, the ampli ed noise at higher frequencies (the second component in the equation above) overshadows the restored image An approach to address the singularity problem associated with the inverse lter is the use of the singular value decomposition (SVD) method 825] A widely used implementation of this approach is an iterative algorithm based on Bialy's theorem to solve the normal equation 829] the algorithm is also known as the Landweber iterative method 830] McGlamery 831] demonstrated the application of the inverse lter to restore images blurred by atmospheric turbulence © 2005 by CRC Press LLC 860 Biomedical Image Analysis In an interesting extension of the inverse lter to compensate for distortions or aberrations caused by abnormalities in the human eye, Alonso and Barreto 832] applied a predistortion or precompensation inverse lter to test images prior to being displayed on a computer monitor The PSF of the affected eye was estimated using the wavefront aberration function measured using a wavefront analyzer In order to overcome the limitations of the inverse lter, a weighting function similar to the parametric Wiener lter (see Section 10.1.3) was applied The subjects participating in the study indicated improved visual acuity in reading predistorted images of test-chart letters than in reading directly displayed test images Example: The original \Shapes" test image (of size 128 128 pixels) is shown in Figure 10.2 (a), along with its log-magnitude spectrum in part (b) of the gure The image was blurred via convolution with an isotropic Gaussian PSF having a radial standard deviation of two pixels The PSF and the related MTF are shown in parts (c) and (d) of the gure, respectively The blurred image is shown in part (e) of the gure Gaussian-distributed noise of variance 0:01 was added to the blurred image after normalizing the image to the range 1] the degraded, noisy image is shown in part (f) of the gure The results of application of the inverse lter to the noise-free and noisy blurred versions of the \Shapes" image are shown in Figure 10.3 in both the space and frequency domains The result of inverse ltering of the noise-free blurred image for radial frequencies up to the maximum frequency in (u v), shown in part (a) of the gure, demonstrates e ective deblurring A small amount of ringing artifact may be observed upon close inspection, due to the removal of frequency components beyond a circular region see the spectrum in part (b) of the gure] Inverse ltering of the noisy degraded image, even when limited to radial frequencies less than 0:4 times the maximum frequency in (u v), resulted in signi cant ampli cation of noise that led to the complete loss of the restored image information, as shown in part (c) of the gure Limiting the inverse lter to radial frequencies less than 0:2 times the maximum frequency in (u v) prevented noise ampli cation, but also severely curtailed the restoration process, as shown by the result in part (e) of the gure The results illustrate a severe practical limitation of the inverse lter (See also Figures 10.5 and 10.6.) 10.1.2 Power spectrum equalization Considering the degradation model in Equation 10.1, the method of power spectrum equalization (PSE) 825] takes the following approach: Find a linear transform L so as to obtain an estimate f~(x y) = L g(x y)], subject to the constraint f~(u v) = f (u v), that is, the PSD of the restored image be equal to the PSD of the original image Applying the linear transform L to © 2005 by CRC Press LLC Deconvolution, Deblurring, and Restoration 861 (a) (b) (c) (d) (e) (f) FIGURE 10.2 (a) \Shapes" test image size 128 128 pixels (b) Log-magnitude spectrum of the test image (c) PSF with Gaussian shape radial standard deviation = pixels (d) MTF related to the PSF in (c) (e) Test image blurred with the PSF in (c) (f) Blurred image in (e) after normalization to 1] and the addition of Gaussian noise with variance = 0:01 © 2005 by CRC Press LLC 862 Biomedical Image Analysis (a) (b) (c) (d) (e) (f) FIGURE 10.3 (a) Result of inverse ltering the blurred \Shapes" image in Figure 10.2 (e) Result of inverse ltering the noisy blurred \Shapes" image in Figure 10.2 (f) using the inverse lter up to the radial frequency equal to (c) 0:4 times and (e) 0:2 times the maximum frequency in (u v) The log-magnitude spectra of the images in (a), (c), and (e) are shown in (b), (d), and (f), respectively © 2005 by CRC Press LLC Deconvolution, Deblurring, and Restoration 863 Equation 10.1 as well as the constraint mentioned above to the result, we get 2 f~(u v ) = jL(u v )j jH (u v )j f (u v ) + = f (u v ) (u v ) (10.12) where L(u v) represents the MTF of the lter L Rearranging the expression above, we get LPSE (u v) = jL(u v)j = jH (u v)j2 f((uu vv)) + f 31 = jH (u v )j + (u v) f (u v) (u v ) (10.13) : (10.14) A detailed inspection of the equation above indicates the following properties of the PSE lter: The PSE lter requires knowledge of the PSDs of the original image and noise processes (or models thereof) The PSE lter tends toward the inverse lter in magnitude when the noise PSD tends toward zero This property may be viewed in terms of the entire noise PSD or at individual frequency samples The PSE lter performs restoration in spectral magnitude only Phase correction, if required, may be applied in a separate step In most practical cases, the degrading PSF and MTF are isotropic, and H (u v) has no phase The gain of the PSE lter is not a ected by zeros in H (u v), as long as (u v) is also not zero at the same frequencies (In most cases, the noise PSD is nonzero at all frequencies.) The gain of the PSE lter reduces to zero wherever the original image PSD is zero The noise-to-signal PSD ratio in the denominator of Equation 10.14 controls the gain of the lter in the presence of noise Models of the PSDs of the original image and noise processes may be estimated from practical measurements or experiments (see Section 10.1.6) See Section 10.2 for a discussion on extending the PSE lter to blind deblurring Examples of application of the PSE lter are provided in Sections 10.1.3 and 10.5 10.1.3 The Wiener lter Wiener lter theory provides for optimal ltering by taking into account the statistical characteristics of the image and noise processes 9, 198, 589, 833] © 2005 by CRC Press LLC 864 Biomedical Image Analysis The lter characteristics are optimized with reference to a performance criterion The output is guaranteed to be the best achievable result under the conditions imposed and the information provided The Wiener lter is a powerful conceptual tool that changed traditional approaches to signal processing The Wiener lter performs probabilistic (stochastic) restoration with the least-squares error criterion 9, 589] The basic degradation model used is g = hf + (10.15) where f and are real-valued, second-order-stationary random processes that are statistically independent, with known rst-order and second-order moments Observe that this equation is the matrix form of Equation 10.1 The approach taken to estimate the original image is to determine a linear estimate ~f = L g to f from the given image g, where L is the lter to be derived The criterion used is to minimize the MSE = E f ; ~f : (10.16) Expressing the MSE as the trace of the outer product matrix of the error vector, we have n h io = E Tr (f ; ~f ) (f ; ~f )T : (10.17) In expanding the expression above, we could make use of the following relationships: (f ; ~f ) (f ; ~f )T = f f T ; f ~f T ; ~f f T + ~f ~f T : (10.18) T T T T T T T ~f = g L = (f h + ) L : (10.19) T T T T T T ~ ff =ff h L +f L : (10.20) ~f f T = L h f f T + L f T : (10.21) ~f ~f T = L ;h f f T hT + h f T + f T hT + T LT : (10.22) Because the trace of a sum of matrices is equal to sum of their traces, the E and Tr operators may be interchanged in order We then obtain the following expressions and relationships: E f fT = f (10.23) the autocorrelation matrix of the original image h i E f ~f T = f hT LT (10.24) with the observation that E f T =0 because f and are statistically independent processes and h i E ~f f T = L h f : © 2005 by CRC Press LLC = (10.25) (10.26) Deconvolution, Deblurring, and Restoration h 865 i E ~f ~f T = L h f hT LT + L T =E LT : (10.27) (10.28) is the autocorrelation matrix of the noise process Now, the MSE may be written as: ; = Tr f ; f hT LT ; L h f + L h f hT LT + L ; = Tr f ; f hT LT + L h f hT LT + L LT : LT (10.29) (Note: Tf = f and T = because the autocorrelation matrices are symmetric, and the trace of a matrix is equal to the trace of its transpose.) At this point, the MSE is no longer a function of f, g, or , but depends only on the statistical characteristics of f and , as well as on h and L In order to derive the optimal lter L, we could set the derivative of with respect to L to zero: @ = ;2 hT + L h hT + L f f @L =0 (10.30) which leads to the optimal Wiener lter function LW = f hT ; h f hT + ;1 : (10.31) Note that this solution does not depend upon the inverses of the individual ACF matrices or h, but upon the inverse of their combination The combined matrix could be expected to be invertible even if the individual matrices are singular Considerations in implementation of the Wiener lter: Consider the matrix to be inverted in Equation 10.31: Z = h f hT + : (10.32) This matrix would be of size N N for N N images, making inversion practically impossible Inversion becomes easier if the matrix can be written as a product of diagonal and unitary matrices A condition that reduces the complexity of the problem is that h, f , and each be circulant or blockcirculant We can now make the following observations: We know, from Section 3.5, that h is block-circulant for 2D LSI operations expressed using circular convolution is a diagonal matrix if is white (uncorrelated) noise In most real images, the correlation between pixels reduces as the distance (spatial separation) increases: f is then banded and may be approximated by a block-circulant matrix © 2005 by CRC Press LLC 866 Biomedical Image Analysis Based upon the observations listed above, we can write (see Section 3.5) h = W Dh W;1 (10.33) ; (10.34) f = W Df W and = W D W ;1 (10.35) where the matrices D are diagonal matrices resulting from the application of the DFT to the corresponding block-circulant matrices, with the subscripts indicating the related entity (f : original image, h : degrading system, or : noise) Then, we have Z = W Dh Df Dh W;1 + W D W;1 (10.36) = W (Dh Df Dh + D ) W;1 : The Wiener lter is then given by LW = W Df Dh (Dh Df Dh + D );1 W;1 : (10.37) The optimal MMSE estimate is given by ~f = LW g = W Df Dh (Dh Df Dh + D );1 W;1 g: (10.38) Interpretation of the Wiener lter: With reference to Equation 10.38, we can make the following observations that help in interpreting the nature of the Wiener lter W;1 g is related to G(u v) the Fourier transform of the given degraded image g(x y) Df is related to the PSD f (u v) of the original image f (x y) D is related to the PSD (u v) of the noise process Dh is related to the transfer function H (u v) of the degrading system Then, the output of the Wiener lter before the nal inverse Fourier transform is given by F~ (u v) = H (u v)f (u(uv) vH) H(u(uv) vG)(+u v) (u v) f H ( u v ) =4 (u v) G(u v ) jH (u v )j2 + f (u v) =4 jH (u j H (u © 2005 by CRC Press LLC v)j2 v)j2 + (u v) f (u v) G(u v) H (u v) : (10.39) 940 Biomedical Image Analysis Unfortunately, myocardial SPECT images possess poor statistics, because only a small fraction of the injected activity will accumulate in the myocardium Furthermore, as the peak photon energy of 201 Tl is about 80 keV , scattering has a serious e ect on image quality Boulfelfel et al 86, 751] applied prereconstruction restoration and post-reconstruction restoration techniques using the Wiener, PSE, and Metz lters to myocardial SPECT images examples from their works are presented and discussed in the following paragraphs In the procedure to acquire myocardial SPECT images of human patients, 74 MBq (2 mCi) of 201 Tl was injected into the body After accumulation of the tracer in the myocardium, 44 planar projections, each of size 64 64 pixels, spanning the full range of 0o 180o ], were acquired The time for the acquisition of each projection was 30 s Each projection image had a total count in the range 10 000 to 20 000 The projections were acquired in an elliptic trajectory with the average distance from the heart being about 20 cm Two energy peaks were used in the acquisition of the projections in order to perform scatter correction using the dual-energy-window subtraction technique No attenuation correction was performed as the organ is small Given the imaging protocol as above, and the fact that the organ being imaged is small, it is valid to assume that the blurring function is nearly shift-invariant hence, it becomes possible to apply shift-invariant lters, such as the Wiener, PSE, and Metz lters, for the restoration of myocardial planar and SPECT images Figures 10.31 and 10.32 show one projection image each of two patients Transverse SPECT images were reconstructed, and oblique slices perpendicular to the long axis of the heart were then computed from the 3D data available Figures 10.31 and 10.32 show one representative oblique section image in each case, along with several restored versions of the images The parts of the myocardium with reduced activity (cold spots) are seen more clearly in the restored images than in the original images The results of prereconstruction restoration applied to the planar images are better than those of post-reconstruction restoration ltering in terms of noise content as well as improvement in sharpness and clarity SPECT images of the brain: Radionuclide brain scanning has been used extensively in the study of neurological and psychiatric diseases The main area of application is the detection of pathology in the cerebral hemispheres and the cerebellum A number of radiopharmaceuticals are used for brain scanning however, 99m Tc-based materials are most widely used An advantage in brain imaging is that the patient's head can be positioned at the center of rotation of the camera, which allows imaging over 360o , rather than over only 180o as in the case of myocardial imaging Although the brain may be considered to be a large organ, the homogeneity of the (scattering) medium and the ability to image it from a short distance using a circular orbit allow the use of geometric averaging of the planar images as a preprocessing © 2005 by CRC Press LLC Deconvolution, Deblurring, and Restoration FIGURE 10.31 941 Top: A sample planar projection image of a patient (a) Short-axis SPECT image showing the myocardium of the left ventricle in cross-section Results of post-reconstruction restoration applied to the SPECT image using (b) the Wiener, (c) the PSE, and (d) Metz lters Results of prereconstruction restoration applied to the planar images using (e) the Wiener, (f) the PSE, and (g) Metz lters Images courtesy of D Boulfelfel, L.J Hahn, and R Kloiber, Foothills Hospital, Calgary 86] © 2005 by CRC Press LLC 942 FIGURE 10.32 Biomedical Image Analysis Top: A sample planar projection image of a patient (a) Short-axis SPECT image showing the myocardium of the left ventricle in cross-section Results of post-reconstruction restoration applied to the SPECT image using (b) the Wiener, (c) the PSE, and (d) Metz lters Results of prereconstruction restoration applied to the planar images using (e) the Wiener, (f) the PSE, and (g) Metz lters Images courtesy of D Boulfelfel, L.J Hahn, and R Kloiber, Foothills Hospital, Calgary 86] © 2005 by CRC Press LLC Deconvolution, Deblurring, and Restoration 943 step to reduce both attenuation and shift-variance of the blur before restoration Brain images are also not as low in statistics as myocardial images In the procedure for nuclear medicine imaging of the brain, after the 99m Tcchloride administered to the patient had accumulated in the brain, 44 planar projections, each of size 64 64 pixels, were acquired The time for the acquisition of each projection was 30 s The projections were acquired over the full range of 360o in a circular trajectory with the radius of rotation of 20 cm Two energy peaks were used in the acquisition of the projections to perform scatter correction using the dual-energy-window subtraction technique Figures 10.33 and 10.34 show a set of opposing projection images as well as their geometric mean for two patients Transverse SPECT images were reconstructed after performing geometric averaging of conjugate projections and (prereconstruction) restoration using the Wiener, PSE, and Metz lters 86] Figures 10.33 and 10.34 show one representative SPECT image in each case, along with several restored versions The results show that averaging of conjugate projections improves the quality of the restored images, which are sharper than the images restored without averaging Images of a resolution phantom: Boulfelfel et al 86, 87, 749, 750, 935] conducted several restoration experiments with SPECT images of a \resolution" phantom The phantom contains nine pairs of hot spots of diameters 39 22 17 14 12 and mm in the \hot lesion" insert (Nuclear Associates), with a total diameter of 200 mm see Figure 3.68 for related illustrations The phantom was lled with mCi of 201 Tl-chloride, centered at the axis of rotation of the gamma camera at a distance of 217 mm, and 120 projections, each of size 128 128 pixels, were acquired over 360o SPECT images of di erent transaxial slices were reconstructed using the Siemens Micro-Delta software Given the large size of the phantom, it would be inappropriate to assume that the degradation phenomena are shift-invariant Boulfelfel et al 750, 948] applied the Kalman lter for restoration of SPECT images of the resolution phantom Figure 10.35 shows a representative SPECT image of the phantom, along with (post-reconstruction) restoration of the image using the Kalman lter the results of application of the shift-invariant Wiener and PSE lters are also shown for comparison It is evident that the shift-variant Kalman lter has provided better results than the other lters: the Kalman-restored image clearly shows seven of the nine pairs of hot spots, whereas the results of the Wiener and PSE lters show only four or ve pairs For the sake of comparison, the results of prereconstruction restoration of the resolution phantom image obtained by applying the shift-invariant Wiener, PSE, and Metz lters after geometric averaging of conjugate projections are shown in Figure 10.36 Observe that the orientation of these results is di erent from that of the images in Figure 10.35 due to the alignment procedure required for averaging Although the results show some of the hot spots with more clarity than the original image in Figure 10.35 (a), they are of lower quality than the result of Kalman ltering, shown in Figure 10.35 (d) © 2005 by CRC Press LLC 944 FIGURE 10.33 Biomedical Image Analysis Top row: A sample pair of conjugate projections of a patient, along with their geometric mean (a) SPECT image showing the brain in cross-section Results of prereconstruction restoration applied to the planar images using (b) the Wiener, (c) the PSE, and (d) Metz lters Results of geometric averaging and prereconstruction restoration applied to the planar images using (e) the Wiener, (f) the PSE, and (g) Metz lters The orientation of the images in (e) { (g) is di erent from that of the images in (a) { (d) due to the alignment of conjugate projection images for geometric averaging Images courtesy of D Boulfelfel, L.J Hahn, and R Kloiber, Foothills Hospital, Calgary 86] © 2005 by CRC Press LLC Deconvolution, Deblurring, and Restoration FIGURE 10.34 945 Top row: A sample pair of conjugate projections of a patient, along with their geometric mean (a) SPECT image showing the brain in cross-section Results of prereconstruction restoration applied to the planar images using (b) the Wiener, (c) the PSE, and (d) Metz lters Results of geometric averaging and prereconstruction restoration applied to the planar images using (e) the Wiener, (f) the PSE, and (g) Metz lters The orientation of the images in (e) { (g) is di erent from that of the images in (a) { (d) due to the alignment of conjugate projection images for geometric averaging Images courtesy of D Boulfelfel, L.J Hahn, and R Kloiber, Foothills Hospital, Calgary 86] © 2005 by CRC Press LLC 946 Biomedical Image Analysis (a) (b) (c) (d) FIGURE 10.35 (a) Acquired SPECT image (128 128 pixels) of the resolution phantom Post-reconstruction restored versions using (b) the Wiener lter (c) the PSE lter and (d) the Kalman lter The images (a) { (c) were enhanced by gamma correction with = 0:8 the image (d) was enhanced with = 0:3 (see Section 4.4.3) See also Figure 3.68 Reproduced with permission from D Boulfelfel, R.M Rangayyan, L.J Hahn, R Kloiber, and G.R Kuduvalli, \Restoration of single photon emission computed tomography images by the Kalman lter", IEEE Transactions on Medical Imaging, 13(1): 102 { 109, 1994 c IEEE © 2005 by CRC Press LLC Deconvolution, Deblurring, and Restoration 947 FIGURE 10.36 Prereconstruction restoration of the SPECT image of the resolution phantom shown in Figure 10.35 (a) after geometric averaging of conjugate projection images, using (a) the Wiener, (b) the PSE, and (c) the Metz lters The orientation of the images in this gure is di erent from that of the images in Figure 10.35 due to the alignment of conjugate projection images for geometric averaging Images courtesy of D Boulfelfel, L.J Hahn, and R Kloiber, Foothills Hospital, Calgary 86] SPECT images of the liver and spleen: Liver and spleen images are di cult to restore because of their large size and irregular shape The liver and spleen are imaged together when radiopharmaceuticals that are trapped by the reticulo-endothelial cell system are used The most commonly used radiopharmaceutical for this purpose is a 99m Tc-based label In the procedure for imaging the liver and spleen, mCi of a 99m Tc-based radiopharmaceutical was given to the patient After the isotope accumulated in the liver and spleen, 44 projections, each of size 64 64 pixels, were acquired The time for the acquisition of each projection was 40 s The projections were acquired over the full range of 360o in a circular trajectory, with the average radius of rotation of 25 cm Two energy peaks were used in the acquisition of the projections in order to perform scatter correction using the dual-energywindow subtraction technique Transverse SPECT images were reconstructed after averaging and correcting for attenuation using the Siemens Micro-Delta processor The Chang algorithm was used for attenuation correction Figures 10.37 and 10.38 show a sample SPECT slice of the liver and spleen of two patients, along with its restored version using the Kalman lter The restored images demonstrate the full outlines of the liver and spleen with improved clarity, and show a few cold spots within the organs with increased contrast as compared to the original images The clinical validity of this observation was not rmed 3D restoration of SPECT images: Boulfelfel et al 86, 750, 948, 935] applied 3D lters for the restoration of SPECT images, including 3D extensions of the Wiener, PSE, and Metz lters, as well as a combination of a 2D Kalman lter in the SPECT plane and a 1D Metz lter in the inter-slice direction Figures 10.39 and 10.40 show a sample planar image of the liver and © 2005 by CRC Press LLC 948 Biomedical Image Analysis (a) (b) FIGURE 10.37 (a) Acquired SPECT image of the liver and spleen of a patient (b) Restored image obtained by the application of the Kalman lter Images courtesy of D Boulfelfel, L.J Hahn, and R Kloiber, Foothills Hospital, Calgary 86] (a) FIGURE 10.38 (b) (a) Acquired SPECT image of the liver and spleen of a patient (b) Restored image obtained by the application of the Kalman lter Images courtesy of D Boulfelfel, L.J Hahn, and R Kloiber, Foothills Hospital, Calgary 86] © 2005 by CRC Press LLC Deconvolution, Deblurring, and Restoration 949 spleen each of two patients, a sample SPECT image in each case, and restored images of the SPECT slices after 3D restoration of the entire SPECT volumes using the Wiener, PSE, and Metz lters Figures 10.41 and 10.42 show a sample SPECT image and the corresponding restored image after 3D restoration of the entire SPECT volume using the 2D Kalman lter in the SPECT plane and a 1D Metz lter in the inter-slice direction The restored images show more cold spots within the liver, with increased contrast however, the clinical validity of this observation was not rmed A sample SPECT image and the corresponding restored version after 3D restoration of the entire SPECT volume using the Kalman-Metz lter combination as above are shown in Figure 10.43 Compared to the result of 2D ltering shown in Figure 10.35, the 3D ltering procedure appears to have yielded a better image 10.6 Remarks The widespread occurrence of image degradation in even the most sophisticated and expensive imaging systems has continually frustrated and challenged researchers in imaging and image processing The eld of image restoration has attracted a high level of activity from researchers with several di erent perspectives 8, 11, 822, 823, 824, 825, 952, 953, 954, 955] In this chapter, we have studied a small selection of techniques that are among the popular approaches to this intriguing problem Most of the restoration techniques require detailed and speci c information about the original undegraded image and the degradation phenomena Several additional constraints may also be applied, based upon a priori and independent knowledge about the desired image However, it is often di cult to obtain accurate information as above The quality of the result obtained is a ected by the accuracy of the information provided and the appropriateness of the constraints applied The nature of the problem is characterized very well by the title of a special meeting held on this subject: \Signal recovery and synthesis with incomplete information and partial constraints" 954, 955] Regardless of the di culties and challenges involved, researchers in the eld of image restoration have demonstrated that a good understanding of the problem can often lead to usable solutions © 2005 by CRC Press LLC 950 FIGURE 10.39 Biomedical Image Analysis Top: A sample planar projection image of a patient (a) SPECT image showing the liver and spleen Results of post-reconstruction 3D restoration applied to the entire SPECT volume using (b) the Wiener, (c) the PSE, and (d) Metz lters Images courtesy of D Boulfelfel, L.J Hahn, and R Kloiber, Foothills Hospital, Calgary 86] © 2005 by CRC Press LLC Deconvolution, Deblurring, and Restoration FIGURE 10.40 951 Top: A sample planar projection image of a patient (a) SPECT image showing the liver and spleen Results of post-reconstruction 3D restoration applied to the entire SPECT volume using (b) the Wiener, (c) the PSE, and (d) Metz lters Images courtesy of D Boulfelfel, L.J Hahn, and R Kloiber, Foothills Hospital, Calgary 86] © 2005 by CRC Press LLC 952 Biomedical Image Analysis (a) FIGURE 10.41 (b) (a) Acquired SPECT image of the liver and spleen of a patient (b) Restored image obtained by the application of the 3D Kalman-Metz combined lter Images courtesy of D Boulfelfel, L.J Hahn, and R Kloiber, Foothills Hospital, Calgary 86] (a) FIGURE 10.42 (b) (a) Acquired SPECT image of the liver and spleen of a patient (b) Restored image obtained by the application of the 3D Kalman-Metz combined lter Images courtesy of D Boulfelfel, L.J Hahn, and R Kloiber, Foothills Hospital, Calgary 86] © 2005 by CRC Press LLC Deconvolution, Deblurring, and Restoration (a) FIGURE 10.43 953 (b) (a) Acquired SPECT image of the resolution phantom (b) Restored image obtained by the application of the 3D Kalman-Metz combined lter Images courtesy of D Boulfelfel, L.J Hahn, and R Kloiber, Foothills Hospital, Calgary 86] 10.7 Study Questions and Problems Selected data les related to some of the problems and exercises are available at the site www.enel.ucalgary.ca/People/Ranga/enel697 Using mathematical expressions and operations as required, explain how a degraded image of an edge may be used to derive the MTF ( ) of an imaging system State clearly any assumptions made, and explain their relevance or signi cance Given g = hf + and ~f = Lg, where g is a degraded image, f is the original image, is the noise process, h is the PSF of the blurring sys~ tem, image, and L is the restoration lter, expand = n fh is the restored io T Tr (f ; ~f ) (f ; ~f ) and simplify the result to contain only h, L, and the ACF matrices of f and Give the reasons for each step, and explain the signi cance and implications of the assumptions made in deriving the Wiener lter With reference to the Wiener lter for image restoration (deblurring), explain the role of the signal-to-noise spectral ratio How does this ratio control the performance of the lter? Prove that the PSE lter is the geometric mean of the inverse and Wiener lters List the various items of information required in order to implement the Wiener lter for deblurring a noisy image Explain how you would derive each item in practice H u v E © 2005 by CRC Press LLC 954 Biomedical Image Analysis 10.8 Laboratory Exercises and Projects Create or acquire a test image including components with sharp edges Blur the image by convolution with a Gaussian PSF Add Gaussian-distributed random noise to the blurred image Derive the MTF of the blurring function and the PSD of the noise Pay attention to the scale factors involved in the Fourier transform Restore the degraded image using the inverse, Wiener, and PSE lters You may have to restrict the inverse lter to a certain frequency limit in order to prevent the ampli cation of noise How would you derive or model the ideal object PSD required in the design of the Wiener and PSE lters? Using a camera that is not in focus, capture a blurred image of a test image containing a sharp line Derive the PSF and the MTF of the imaging system Using a camera that is not in focus, capture a blurred image of a scene, such as your laboratory, including a person and some equipment Ensure that the scene includes an object with a sharp edge (for example, the edge of a door frame or a blackboard), as well as a uniform area (for example, a part of a clean wall or board with no texture) Derive the PSF and MTF of the imaging system by manual segmentation of the edge spread function and further analysis as required Estimate the noise PSD by using segments of areas expected to be uniform Design the Wiener and PSE lters and restore the image How would you derive or model the ideal PSD of the original scene? Restore the image in the preceding exercise by designing the blind deblurring version of the PSE lter © 2005 by CRC Press LLC [...]... of the imaging (blurring) system, we may analyze the Fourier transform of g(x y) , as follows G(u v) = = 1 Z Z 1 ;1 ;1 Z 1 Z 1 g(x y) exp ;j 2 (ux + vy)] dx dy (Z ;1 ;1 ) T f x ; (t) y ; (t)] dt exp ;j 2 (ux + vy)] 0 dx dy: (10.54) Reversing the order of integration with respect to t and (x y) , we get G(u v) = T Z 0 dt: Z 1 Z 1 ;1 ;1 f x ; (t) y ; (t)] exp ;j 2 (ux + vy)] dx dy (10.55) The expression... Reproduced with permission from A.C.G Martins and R.M Rangayyan, \Complex cepstral ltering of images and echo removal in the Radon domain", Pattern Recognition, 30(11):1931{1938, 1997 c Pattern Recognition Society Published by Elsevier Science Ltd Let fe (x y) be an image given by fe (x y) = f (x y) d(x y) (10.79) where d(x y) = (x y) + a (x ; x0 y ; y0 ) (10.80) with a being a scalar weighting factor In... less commonly used.) In practice, the Fourier transform is used in place of the z -transform Given g(x y) = h(x y) f (x y) , it follows that G^ (u v) = H^ (u v) + F^ (u v) (10.77) © 2005 by CRC Press LLC 886 Biomedical Image Analysis and furthermore, that the complex cepstra of the signals are related simply as g^(x y) = h^ (x y) + f^(x y) : (10.78) Here, the ^ symbol over a function of frequency indicates... y sin ; t) dx dy d d +a +1 Z +1 Z ;1 ;1 f( ) Z +1 Z +1 ;1 ;1 (x ; x0 ) ; (10.81) y; ) (y ; y0 ) ; ] (x cos + y sin ; t) dx dy d d : (10.82) Here, t is the displacement between the projection samples (rays) in the Radon domain Using the properties of the function (see Section 2.9), we get p (t) = +a Z +1 Z +1 Z ;1 ;1 +1 Z +1 ;1 ;1 f( © 2005 by CRC Press LLC f( ) ( cos + sin ; t) d ) (x0 + ) cos + (y0 ... could label f (x y) as the basic element image, d(x y) as a eld of impulses at the positions of the echoes (including the original element), and fe (x y) as the composite image Applying the Radon transform (see Section 9.1), we have the projection p (t) of the composite image fe (x y) at angle given by p (t) = Z which leads to +1 ;1 Z +1 Z ;1 +1 Z fe (x y) (x cos + y sin +1 Z ; t) dx dy +1 Z +1 p (t)... homomorphic lter for convolved signals The symbol at the input or output of each block indicates the operation that combines the signal components at the corresponding step Reproduced with permission from Reproduced with permission from R.M Rangayyan, Biomedical Signal Analysis: A Case-Study Approach, IEEE Press and Wiley, New York, NY, 2002 c IEEE 887 © 2005 by CRC Press LLC 888 Phase Exponential window... (intensity) than the original object, they have been maintained at the same intensity as the original in this image The test image is of size 101 101 pixels the radius of each circle is 10 pixels, and the intensity of each circle is 100 on an inverted gray scale The Radon-domain homomorphic lter was applied to the test image The image was multiplied by a weighting function given by y3 , where y is the... by blurring due to motion during the period of exposure (imaging) In some cases, it may be appropriate to assume that the motion is restricted to within the plane of the image Furthermore, it may also be assumed that the velocity is constant over the (usually short) period of exposure Under such conditions, it becomes © 2005 by CRC Press LLC 876 Biomedical Image Analysis possible to derive an analytical... Paranjape, and R.M Rangayyan, \Iterative method for blind deconvolution", Journal of Electronic Imaging, 3(3):245{250, 1994 c SPIE © 2005 by CRC Press LLC Deconvolution, Deblurring, and Restoration 885 10.3 Homomorphic Deconvolution Consider the case where we have an image that is given by the convolution of two component images, as expressed by the relation g(x y) = h(x y) f (x y) : (10.74) Similar to... thresholding Reproduced with permission from A.C.G Martins and R.M Rangayyan, \Complex cepstral ltering of images and echo removal in the Radon domain", Pattern Recognition, 30(11):1931{1938, 1997 c Pattern Recognition Society Published by Elsevier Science Ltd images are not stationary, and, at best, may be described as locally stationary furthermore, in practice, the PSD of the uncorrupted original image