I noticed the S value and Adaptive Values are applied less and less agressively with each successive layer with the MultiscaleMedianTransform.
Good observation. At larger dimensional scales, the SNR is larger and hence less noise reduction is required. The successive smaller thresholds reflect the fact that the fraction of coefficients due to noise at each transformation layer is inversely proportional to the scale.
I was never sure what the adaptive parameter does. Can you expand on this?
Each layer of a MMT transform contains coefficients that represent the structures described by the layer. MMT coefficients are real numbers that can be positive, negative, or zero. This happens also in the wavelet transform.
In each layer, a number of coefficients are due to the noise. The set of noise coefficients is always characterized by relatively small absolute values. The threshold noise reduction parameter specifies a limiting value (in sigma units) to kill noise coefficients: all coefficients with absolute values less than the threshold are set to zero, so their contributions to the reconstructed image (generated by an inverse MMT transform) are eliminated. This simple procedure is known as
hard thresholding.
We don't implement hard thresholding in the ATrousWaveletTransform tool because there are other techniques, known as
soft thresholding, that work better for wavelets. However, soft thresholding doesn't work for the MMT. For the MultiscaleMedianTransform tool we implement something slightly better than standard hard thresholding using the Amount parameter: each coefficient below the threshold limit is multiplied by 1 - Amount, so when Amount=1 we have pure hard thresholding, and Amount < 1 allows for some fine control.
So in theory we could reduce or remove the noise completely at each layer by just finding the threshold value that divides the coefficients into noise and significant. However, when it comes to noise we always have uncertainty: as we get close to the threshold limit we cannot say for sure if a given coefficient is due to the noise or is supporting a significant image structure. The consequence of this uncertainty is that we always have to set the noise threshold somewhat below its optimum value, in order to be sure that nothing significant will be damaged.
So in practice some noise coefficients
always survive after thresholding. If no noise survives, then we are destroying significant data
for sure. Surviving noise can be identified as isolated bright pixels on the sky background and other dark regions of the images. These 'artificial hot pixels' are typical artifacts of the first two or three MMT layers. The Adaptive parameter controls a special adaptive noise reduction filter applied to the layer coefficients after the thresholding process. This filter looks for isolated, bright and small structures and selectively removes them. The higher the Adaptive parameter, the more aggressive adaptive filtering effect.
The noise reduction part of the MMT tool is still undergoing extensive development. Some or even all of the noise reduction parameters in MMT can change in future versions. In addition, a new algorithm will be available on this tool as an option: the wavelet-median transform, which is intended to have the best of both worlds.
