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.functions f e (x, y, z) and g e (x, y, z) is defined byFigure 2. Theoretical PSF magnitude for asector array transducerof interest) and ĥ (VA PSF) in the second equation, respectively.The final model used in this paper can be written asfollows, using notation commonly in restoration:i(x, y, z) =o(x, y, z) ∗ h(x, y, z)+n(x, y, z). (6)With the information above, we have the necessary apparatusto obtain a degraded image by blurring and noise, inorder to apply restoration techniques as presented in Section5. Since we are not aware of works in the literature applyingrestoration algorithms to VA images, our initial aimis to analyze the behavior of these images to a restorationproblem. Thus, we use as input the VA image with noiseand try to restore it to its original state using three filters(Wiener, Regularized Least Squares and Geometric Mean).We obtain acceptable results after restoration using these filters.The complex nature of the VA PSF and, therefore, of theconvolution is noteworthy, since the problem involves anamplitude and a phase in each position. Then, we have thefirst problem of applying restoration filters to VA images:the use of complex convolution. This fact considerably increasesthe computational costs of both implementing andusing restoration algorithms. Next Section presents the definitionof the 3D complex convolution.3. 3D Complex ConvolutionLet us define the 3D discrete real convolution of twofunctions f(x, y, z) and g(x, y, z). In that case, f and gare defined as matrices of dimensions A × B × C andD × E × F (3D array of numbers), respectively. The functionsare extended with zeros to dimensions M = A+D−1,N = B + E − 1 and O = C + F − 1, so that they are periodicwith period M, N and O in the x, y and z directions[3, 8]. The resulting circular convolution is linear withthese extensions [2]. The 3D convolution of the extendedf e (x, y, z) ∗ g e (x, y, z) = (7)1 ∑f e (m, n, o)g e (x − m, y − n, z − o),MNOm,n,o=0for x =0, 1, 2,...,M − 1, y =0, 1, 2,...,N − 1 and z =0, 1, 2,...,O− 1, where∗ denotes the spatial convolution.However, the convolution used in this paper has complexnature, then let us consider f e and g e as complex functions.Let R[f e ] and I[f e ] be the real and imaginary parts ofthe function f e and the same for the function g e (R[g e ] andI[g e ]). Then,f e (x, y, z) ∗ g e (x, y, z) =1 ∑=(R[f e ]+iI[f e ])(R[g e ]+iI[g e ])MNOm,n,o=01 ∑={R[fe ]R[g e ]+i(R[f e ]I[g e ]MNO+ I[f e ]R[g e ]) −I[f e ]I[g e ]}1{∑ [∑=R[fe ]R[g e ]+i R[fe ]I[g e ]MNO+ ∑ ]I[f e ]R[g e ] − ∑ }I[f e ]I[g e ]= R[f e ] ∗R[g e ]+i(R[f e ] ∗I[g e ]+ I[f e ] ∗R[g e ]) −I[f e ] ∗I[g e ] (8)where i = √ −1.One can see in equation (8) that the complex convolutioncan be written in terms of the convolutions of the real andimaginary parts of the complex functions, which are realfunctions. At first, for implementation purposes, this equationseems very interesting and useful but, because of thecharacteristics of the problem we are leading with, usingthis equation in order to form the VA image becomes impracticable.The size of the PSF we used in the simulationof the VA system is 80 × 80 × 512. To compute the convolutionand then the blurring model in space domain usingsuch function is unfeasible in standard PCs.However we know, by the convolution theorem, that theconvolution of two functions in the space domain is theproduct of their Fourier Transforms in the frequency domain.Moreover, using the Fast Fourier Transform (FFT)the problem is much more tractable. The complex convolutionin the frequency domain (product of Fourier Transforms)is the same of the real convolution, since the FourierTransform of any function, complex or not, is always complex.For those reasons, all the implementations of this workwere made in frequency domain.Even working in the frequency domain, our problemstill requires huge computer memory resources (more than4 GB). The PSF and the phantom have 80 × 80 × 512 and196×161×211 coordinates, respectively, so they both needto be extended to objects with 275 × 240 × 722 positions;we therefore have to work with two matrices with almost37