DIGITAL IMAGE PROCESSING 4th phần 5

Ngoài ra, hình thức ma trận đầu ra cho khu vực chồng chất hữu hạn có liên quan đến mở rộng ma trận hình ảnh KE Số lượng các hoạt động tính toán cần thiết để có được kế bằng cách xử lý chuyển đổi tên miền được đưa ra bởi các phân tích trước đó cho M = N = J. trực tiếp chuyển đổi chuyển đổi nhanh: | TRANSFORM DOMAIN SUPERPOSITION 223 Also the matrix form of the output for finite-area superposition is related to the extended image matrix Ke by Q S1J K S1 M T For sampled image superposition G S2JM KE S2M The number of computational operations required to obtain kE by transform domain processing is given by the previous analysis for M N J. Direct transformation 3 J4 Fast transformation J2 4 J2 log2 J If C is sparse many of the J2 filter multiplication operations can be avoided. From the discussion above it can be seen that the secret to computationally efficient superposition is to select a transformation that possesses a fast computational algorithm that results in a relatively sparse transform domain superposition filter matrix. As an example consider finite-area convolution performed by Fourier domain processing 2 3 . Referring to Figure let AK2 AK AK where AK _-W x-1 y-1 JK with W exp -Ị I K for x y 1 2 . K. Also let h denote the K2 X 1 vector representation of the extended spatially invariant impulse response array of Eq. for J K. The Fourier transform of h is denoted as A A h These transform components are then inserted as the diagonal elements of a K2 X K2 matrix Hm diag h 1 . h K K2 224 LINEAR PROCESSING TECHNIQUES Then it can be shown after considerable manipulation that the Fourier transform domain superposition matrices for finite area and sampled image convolution can be written as 4 D HM HỤ PD for N M - L 1 and Pb Pb H N 1-W -1 L-1 1- WM-1 - WNv-1 1-W-1 L-1 1 - W M - 1 - W N v - 1 B where N M L 1 and PD u v j JM PB u v Ị JN Thus the transform domain convolution operators each consist of a scalar weighting matrix H ỊC and an interpolation matrix p p that performs the dimensionality con version between the N2 - element input vector and the M2 - element output vector. Generally the interpolation matrix is relatively sparse and therefore transform domain superposition is .