This example shows how to take the shearlet transform of an image and reconstruct the image using only coefficients corresponding to zero shearing.

Load and display an image of a circle.

load circleGS
imagesc(circleGS)
colormap gray
axis equal
axis tight

Create a shearlet system that can be used with the image. Obtain the shearlet filters defined by the system, as well as their geometric interpretations.

[numRows,numCols] = size(circleGS);
sls = shearletSystem('ImageSize',[numRows numCols],'FilterBoundary','truncated');
[psi,scale,shear,cone] = filterbank(sls);

Obtain the shearlet transform of the image.

cfs = sheart2(sls,circleGS);

Find the indices of the shearlet filters that correspond to zero shearing. Keep in mind that the lowpass filter also corresponds to zero shearing.

ind = find((shear==0).*(scale~=-1))'

ind = 1×10

     3     6    10    15    20    25    31    38    46    55

Plot one of the shearlets in the frequency plane. Because the shearlet corresponds to zero shearing, confirm the frequency response is concentrated along either the horizontal or vertical axis.

sh = 31;
omegax = -1/2:1/numCols:1/2-1/numCols;
omegay = omegax;
figure
surf(omegax,flip(omegay),psi(:,:,sh),'EdgeColor','none')
view(0,90)
xlabel('\omega_x')
ylabel('\omega_y')
axis equal
axis tight
title(['Zero Shear Shearlet: Scale = ',num2str(scale(sh)),', Cone - ',cone{sh}])

Create an array that only contains the shearlet coefficients that correspond to the zero shearing filters.

cfsx = zeros(size(cfs));
for k=1:length(ind)
    cfsx(:,:,ind(k)) = cfs(:,:,ind(k));
end

Reconstruct the image using the new coefficients array. Because the only nonzero shearlet coefficients are those that correspond to zero shearing, the horizontal and vertical portions of the circle are emphasized in the reconstruction.

rec = isheart2(sls,cfsx);
imagesc(rec)
axis equal
axis tight
colormap gray
title('Reconstruction')