Proof of proposition 1
Substitute the atomic signal:
wi(x,y)=Ae2iπ(Xix+Yiy) (0≤x<M,0≤y<N) into the discrete Fourier transform:
F(u,v)=M×N1x=0∑M−1y=0∑N−1wi(x,y)e2iπ(Mxu+Nyv) We have
F(u,v)=M×NAx=0∑M−1y=0∑N−1e2iπ(Xix+Yiy)e2iπ(Mxu+Nyv) Moreover
==x=0∑M−1y=0∑N−1e2iπ(Xix+Yiy)e2iπ(Mxu+Nyv)x=0∑M−1y=0∑N−1e2iπx(Xi1+Mu)e2iπy(Yi1+Nv)(x=0∑M−1e2iπx(Xi1+Mu))(x=0∑N−1e2iπx(Yi1+Nv)) Using geometric sum formulae:
===1−e2iπ(Xi1+Mu)1−e2iπM(Xi1+Mu)1−e2iπ(Yi1+Nv)1−e2iπN(Yi1+Nv)1−e2iπ(Xi1+Mu)1−e2iπXiM+2iπu1−e2iπ(Yi1+Nv)1−e2iπYiN+2iπv1−e2iπ(Xi1+Mu)1−e2iπXiM1−e2iπ(Yi1+Nv)1−e2iπYiN Hence:
F(u,v)=M×NA1−e2iπ(Xi1+Mu)1−e2iπXiM1−e2iπ(Yi1+Nv)1−e2iπYiN