# filterbank

Shearlet system filters

## Syntax

psi = filterbank(sls)
[psi,scale] = filterbank(sls)
[psi,scale,shear] = filterbank(sls)
[psi,scale,shear,cone] = filterbank(sls)

## Description

example

psi = filterbank(sls) returns the Fourier transforms of the shearlet filters defined by the shearlet system sls as a 3-D real-valued array. The array size is M-by-N-by-K, where M and N are the values of the two-element row vector ImageSize of sls. K is the number of shearlets including the lowpass filter, K = numshears(sls) + 1.
[psi,scale] = filterbank(sls) returns the scale parameters of the shearlet filters as a K-by-1 integer vector. K is the number of shearlets including the lowpass filter, K = numshears(sls) + 1.
[psi,scale,shear] = filterbank(sls) returns the shearing parameters of the shearlet filters as a K-by-1 integer vector. K is the number of shearlets including the lowpass filter, K = numshears(sls) + 1.

example

[psi,scale,shear,cone] = filterbank(sls) returns the frequency cones of the shearlet filters as a K-by-1 cell array of characters. K is the number of shearlets including the lowpass filter, K = numshears(sls) + 1.

## Examples

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Load an image. Create a shearlet system that can be used with the image.

load clown sls = shearletSystem('ImageSize',size(X))
sls = shearletSystem with properties: ImageSize: [200 320] NumScales: 4 PreserveEnergy: 0 TransformType: 'real' FilterBoundary: 'periodic' Precision: 'double' 

Obtain the shearlet filters defined by the shearlet system.

psi = filterbank(sls);

Obtain the shearlet transform of the image.

cfs = sheart2(sls,X);

Confirm that the size of the third dimension of psi is equal to the size of the third dimension of cfs.

[size(psi,3) size(cfs,3)]
ans = 1×2 41 41 

Create a complex-valued shearlet system for 256-by-256 images with truncated boundaries and four scales.

sls = shearletSystem('TransformType','complex',... 'ImageSize',[256 256],... 'FilterBoundary','truncated',... 'NumScales',4);

Obtain the shearlet filters and their geometric interpretations.

[psi,scale,shear,cone] = filterbank(sls);

Different scales can have a different number of shears. The scales in the shearlet system range from 0 to 3 inclusive. Find the range of shears per scale.

for k=0:3 ind = find(scale==k); fprintf('scale: %d shear range: [%d:%d]\n',... k,min(shear(ind)),max(shear(ind))) end
scale: 0 shear range: [-1:1] scale: 1 shear range: [-2:2] scale: 2 shear range: [-2:2] scale: 3 shear range: [-3:3] 

Display the unique frequency cone labels.

unique(cone)
ans = 5x1 cell {'B'} {'L'} {'R'} {'T'} {'X'} 

Find the shearlet filter at scale 2 with shear parameter equal to 1 and whose support is in the 'R' frequency cone.

vScale = 2; vShear = 1; vCone = 'R'; ind = (scale'==vScale)&(shear'==vShear)&([cone{:}]==vCone); shFilter = psi(:,:,ind);

Plot the shearlet filter.

omegax = -1/2:1/256:1/2-1/256; omegay = omegax; surf(omegax,flip(omegay),shFilter,'EdgeColor','none') view(0,90) xlabel('\omega_x') ylabel('\omega_y') str = sprintf('Shearlet Filter: Scale %d / Shear %d / Cone %c',... vScale,vShear,vCone); title(str) axis equal axis tight

Plot the supports of all scale 2 shearlet filters with shear parameters equal to $±$2. Areas where filter supports overlap have maximum brightness.

vScale = 2; vShear = 2; ind = find((scale==vScale).*(abs(shear)==vShear)); tmp = zeros(size(psi,1),size(psi,2)); for k=1:length(ind) tmp = tmp+(psi(:,:,ind(k))~=0); end surf(omegax,flip(omegay),double(tmp),'EdgeColor','none') view(0,90) xlabel('\omega_x') ylabel('\omega_y') str = sprintf('Filter Supports: Scale %d / Abs(Shear) %d',... vScale,vShear); title(str) axis equal axis tight colormap gray colorbar

## Input Arguments

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Shearlet system, specified as a shearletSystem object.

## Output Arguments

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Fourier transforms of the shearlet filters defined by the shearlet system, returned as a 3-D real-valued array. The array size is M-by-N-by-K, where M and N are the first and second elements, respectively, of the ImageSize value of sls. K is the number of shearlets including the lowpass filter, K = numshears(sls) + 1. The first element in the third dimension of psi corresponds to the lowpass filter. The subsequent elements correspond to the shearlets.

The data type of psi matches the Precision value of the shearlet system.

Data Types: single | double

Scale parameters of the shearlet filters defined by the shearlet system, returned as a K-by-1 integer vector. K is the number of shearlets including the lowpass filter, K = numshears(sls) + 1. The first element of scale, −1, corresponds to the lowpass filter. The remaining elements of scale correspond to the shearlet filters.

The data type of scale matches the Precision value of the shearlet system.

Data Types: single | double

Shearing parameters of the shearlet filters defined by the shearlet system, returned as a K-by-1 integer vector. K is the number of shearlets including the lowpass filter, K = numshears(sls) + 1. The shear values at scale S range from −ceil(2^(S/2)) to ceil(2^(S/2)) inclusive. The first element of shear, 0, corresponds to the lowpass filter. The remaining elements of shear denote the shears for the corresponding shearlet filters.

The data type of shear matches the Precision value of the shearlet system.

Data Types: single | double

Frequency cones of the shearlet filters defined by the shearlet system, returned as a K-by-1 cell array of characters. K is the number of shearlets including the lowpass filter, K = numshears(sls) + 1. The first element of cone, 'X', corresponds to the lowpass filter. The remaining elements of cone are the frequency cones in which the corresponding shearlet filters have their frequency support.

For complex-valued shearlets, the frequency plane is divided into four cones: 'R' (right), 'T' (top), 'L' (left), and 'B' (bottom). For real-valued shearlets, the frequency cones are 'H' (horizontal) and 'V' (vertical).