In order to explore the use of signal processing package in image processing, @Samir Khan and I created this application that performs discrete Haar wavelet transform on images to achieve both lossy (irreversible) and lossless (reversible) compression.

Haar wavelet compression modifies the image matrix to increase the number of zero entries in the matrix, which results in a sparse matrix that can be stored more efficiently, thus reducing the file size. A Haar wavelet transform groups adjacent items in the matrix, stores the average and difference of the two numbers. Notice that this computation is reversible since knowing the values of a, b and given that x1-x2 = a; (x1+x2)/2 = b, we can solve for x1 and x2. Haar wavelet compression is taking advantage of the property that neighboring pixels in an image usually share very similar value; hence recursively applying Haar wavelet transform to the rows and columns of an image matrix significantly increases the number of zero entries. In the application we achieved a compression ratio of 1.296 (number of non-zero entries in original: number of non-zero entries in processed matrix).

The fact that Haar wavelet transform is reversible means that we can use it to perform lossless image compression: the decompressed image will be exactly the same as the image before compression. Transmission and temporary storage of the data would be made more efficient without any loss of details in the image.

But what if the file size is still too big or we simply don’t need that many details in the image? That is when lossy compression comes into use. By omitting some details/fidelity, lossy compression is able to achieve notably smaller file size. In this application, we apply a filter to the transformed image matrix, setting entries that are close to zeros to actual zeros (i.e. pick a positive number ϵ such that all x < ϵ are changed to 0 in the matrix). The value of ϵ directly impacts the quality of the decompressed image so should be chosen carefully in practice. In this application, we chose ϵ = 0.01, which results in a compression ratio of 19.38, but instead produces a very blurry image after reversing the compression.

(left: Original image, right: lossy compression with ϵ = 0.01)

The application can be accessed here for more details.