how to use the filter in the time domain to filtering the image?

Question

I get some wavelets in the frequency domain in Matlab. Then I transfer it to wavelet in the time domain by Fourier transformsifft2. The size of the wavelet in the time domain and in the frequency domain is the same as the image. And both of them are matrix.

The question is:

how to use the filter in the time domain to filtering the image，or transfer the filter in the time domain to the something that could multiply the image？ Normally，the filter in the time domain filters the image by convolution. But the size of the filters I get by transfer from wavelet in the frequency domain is the same as the image. It seems that it can not convolution.

X = imread('barbara.jpg');
shearletSystem = SLgetShearletSystem2D(0,size(X,1),size(X,2),scales);  
%the size of {shearlets} and {shearlets_timedomain} are [size(X,1),size(X,2),nshear],nshear is the number of the shearlets
%shearlets is the filter in the frequency domain,shearlets_timedomain is the filter in the time domain
shearlets_timedomain=fftshift(ifft2(ifftshift(shearletSystem.shearlets(:,:,:))));

Answer 1

the FWT can completely be done in the frequency domain, see an explanation how in this paper. If I understand correctly your question is not why a convolved signal has the same size as the original (see @ChrisLuengo's reply), but rather how to do the downsampling (selecting either the even or odd positions) in the frequency domain.

As @Adriaan says, convolution in the time domain is pointwise multiplication in the frequency domain and vice versa. For subsampling a signal x of length n [x(1) ... x(n) in MatLab], and its Fourier transform X, taking x(1:2:n) in the time domain corresponds to taking X(1:(n/2)) in the frequency domain -- just for completeness, taking x(2:2:n) in the time domain corresponds to taking X((n/2+1):n) in the frequency domain.