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Computing the Fourier transform of the Laplacian result in spatial domain is equivalent to multiplying the F(u, v), Fourier transformed function of f(x, y) an input image of size M*N, and H(u, v), the filter used for implementing Laplacian in frequency domain. This dual relationship is expressed as Fourier transform pair notation given by_____________(a) ∇^2 f(x,y)↔[(u –M/2)^2+ (v –N/2)^2]F(u,v)(b) ∇^2 f(x,y)↔-[(u+M/2)^2– (v+N/2)^2]F(u,v)(c) ∇^2 f(x,y)↔-[(u –M/2)^2+ (v –N/2)^2]F(u,v)(d) ∇^2 f(x,y)↔[(u+M/2)^2– (v+N/2)^2]F(u,v)This question was posed to me by my school principal while I was bunking the class.The origin of the question is Laplacian in Frequency Domain topic in section Image Enhancement of Digital Image Processing

Answer»

The CORRECT choice is (c) ∇^2 f(x,y)↔-[(U –M/2)^2+ (V –N/2)^2]F(u,v)

Easiest explanation: The Fourier TRANSFORM of the Laplacian result in spatial domain is equivalent to multiplying the F(u, v) and H(u, v). This dual RELATIONSHIP is expressed as Fourier transform pair notation given by:∇^2 f(x,y)↔-[(u – M/2)^2+ (v – N/2)^2]F(u,v), for an image of size M*N.



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