Developed with Giuseppe Marino

Discrete wavelet transform

Wavelets can be used to analyze the frequencies of a signal (an image, in this case).

The file dwt.m implements the Haar wavelet transform using a combination of high pass and low pass filters, while idwt.m provides the inverse transform to go back to the original image. The transformed image is suitable for applying compression techniques.

The file dwt_a_trous.m implements the à trous algorithm, starting with the kernel [1 2 1] which is then upsampled, filled with zeros. This is in contrast to the downsampling of the signal performed in dwt.m. The output is redundant, and thus more suitable for analysis and manipulation. The implementation doesn't use the standard convolution algorithm, due to the sparsity of the kernels.

Usage

The functions work on grayscale images only.

t = dwt(img);
figure; imshow(t);

it = idwt(t);
figure; imshow(uint8(it));

at = dwt_a_trous(img, 8);
figure; imshow(at(:, :, 1), []);
figure; imshow(at(:, :, 5), []);
figure; imshow(at(:, :, 9), []);

iat = sum(at, 3);
figure; imshow(uint8(iat));

The à trous algorithm also requires a second parameter k which represents the number of iterations you want to perform. The array returned is of size [n n k+1], as the last plane is the residual.

DWT	À trous, first wavelet plane
À trous, fifth wavelet plane	À trous, residual plane