# Filters (Wavelet): De-noising BLS GSM

## Background

The wavelet transform or wavelet analysis is a solution to overcome the shortcomings of the Fourier transform. In wavelet analysis, the modulated window is shifted along the signal, and for every position, the spectrum is calculated. Then, this process is repeated many times with a slightly shorter (or longer) window for every new cycle. In the end, the result appears as a collection of time-frequency representations of the signal, all with different resolutions. Since the modulated window is fully scalable, this solves the signal-cutting problem which arises in the Fourier transform.

Because the result appears as a collection of time-frequency representations of the signal, we can speak of a multi-resolution analysis. However, in the case of wavelets, we normally do not speak about time-frequency representations, but about time-scale representations, scale being in a way the opposite of frequency, because the term frequency is reserved for the Fourier transform.

## Image types

This algorithm only applies to 2D black and white images.

## Applying the Wavelet BLS GSM algorithm

• Open an image of interest.
• Navigate to Algorithms > Filters (wavelet) > Denoising BLS GSM.
• The Denoising Bayesian Least Squares Gray Scale Mixture dialog box appears.
• Complete the dialog box. For more information about the dialog box options, refer to Denoising Bayesian Least Squares Gray Scale Mixture dialog box.
• Specify where you wish the de-noised image to appear - in a new image frame or in the same frame replacing the existing image.
• Press OK.
• The algorithm begins to run and the progress window appears with the status. When the algorithm finishes running, the de-noised image appears in the designated image frame.

## Wavelet Thresholding dialog box

Denoising Bayesian Least Squares Gray Scale Mixture

Orthogonal wavelet - this is very fast technique, but of relatively poor denoising performance, because it is not translation invariant. For the Orthogonal wavelet technique the following filter options available:

• Daubechies 2,3,4
• Haar
• Quadrative Mirror Filter 5, 8, 9, 12, 13, 16

Numbers after the filter name refer to a highest number A of vanishing moments for given support width N=2A used in the filter.