III WVC 2007 - Iris.sel.eesc.sc.usp.br - USP

WVC'2007 - III Workshop de Visão Computacional, 22 a 24 de Outubro de 2007, São José do Rio Preto, SP.48 million positions each. Since we are using complex values,each pixel demands 16 bytes to be stored in memory,so each matrix would need more than 762 MB to be stored.The method known as overlap-add was employed to tacklethis problem.4. Overlap-Add MethodThe overlap-add method [12] is used to reduce the a-mount of memory and computation required to convolvetwo functions and to perform the true linear convolution:the same requirements we have in our work. Since we areworking in 3D, we extend this method for our purposes.Suppose we want to convolve two functions, f(x, y, z)and g(x, y, z), as in Section 3. In the overlap-add method,f(x, y, z) is divided into segments f ijk (x, y, z), then thefunction f(x, y, z) can be represented by∑L 1∑L 2∑L 3f(x, y, z) =f ijk (x, y, z), (9)i=1 j=1 k=1where L 1 L 2 L 3 is the number of segments in the image.Convolving f(x, y, z) with g(x, y, z) and using the distributiveproperty of convolution, we obtainf(x, y, z) ∗ g(x, y, z) =(∑ L1∑L 2∑L 3)=f ijk (x, y, z) ∗ g(x, y, z)=i=1 j=1 k=1L 1 L 2 L 3∑ ∑ ∑(f ijk (x, y, z) ∗ g(x, y, z)). (10)i=1 j=1 k=1The segment f ijk (x, y, z) has a smaller support thanf(x, y, z), consequently the computation of the convolutionrequires less memory. The method makes the convolutionslowler, but the alternative is a series of input/outputoperations, which would be much less efficient.Once the problem of complex convolution is solved, wecan now apply the blurring model as shown in Section 2 andthen assess the restoration algorithms.5. RestorationImage restoration (or deconvolution) techniques removeor minimize some known degradations in an image [1, 10].In our case, the degradations of the VA image are caused bythe system PSF and the added Gaussian noise, both knownas we saw in the last Sections.For the purposes of this work, three well known filtersare applied: Wiener, Regularized Least Squares and GeometricMean. This choice was made because these arewidespread used filters and, moreover, there is the restrictionto work in frequency domain.The Fourier Transform of the restored image using theWiener Filter is given by[ 1 |H|ˆF 2 ]=H |H| 2 G, (11)+ Kwhere F , H and G are functions of (u, v, w), andK is aconstant. Using the Regularized Least Squares Filter[ 1 |H|ˆF 2 ]=H |H| 2 + γ|P | 2 G, (12)where γ is the regularization parameter and P (u, v, w)is the Fourier Transform of the 3D Laplacian Operator,p(x, y, z), givenby,⎡p(x, y, z) = ⎣⎡p(x, y, 2) = ⎣0 0 00 −1 00 0 0⎤⎦ ,z =1, 30 −1 0−1 6 −10 −1 0Using the Geometric Mean Filter{ [ ] H∗ α [HˆF ∗ ] } 1−α=H ∗ H H ∗ G. (13)H + γKThis last filter can be considered as an extended version ofthe Wiener Filter and, actually, its result lies between theWiener Filter and the Inverse Filter. Some initial tests withthe filters were made in Matlab and the final algorithmswere implemented in C.6. Experimental ResultsIn order to make our experimental tests, we used a digitalphantom (see Figure 3(a)), which was designed to mimicthe major features exhibited by the breast phantom shown inFigure 3(b). This phantom consists of three opaque spheresof radii 4, 3.5 and 3 mm, respectively. We are using a singleexperimental phantom instead of various phantoms or realimages, however controlled simulations and experimentsare necessary to evaluate the behavior of the method. Besides,making use of simulated images, we can control thenoise level and can evaluate the performance of the algorithmsusing images with different levels of additive andsignal independent noise.The VA image of the breast phantom is shown in Figure3(c), along with the region described by the digitalphantom.We need the object function and the PSF to form theimage, so we use a discrete version of the phantom with196×161×211 pixels. The discrete version of the phantom⎤⎦38