Monogenic Wavelets (Greyscale)

Previous chapters concentrate on defining local features to analyze images.
What about retrieving visual content back from features?

Wavelet transforms are known to provide reversible transformation of images, useful for analysis as well as processing. Reversibility means that the encoding of the information can be controlled by our ability to see what visual element is contained in what part of the feature set. It also means that the image can be modified accordingly to some tweaking of its features, opening the way to image enhancement for example.

A reconstruction method retrieving a whole image based on its elliptical monogenic features can be defined thanks to wavelets.

Writing an accessible and practical webpage to present the combination of the color monogenic framework with the wavelet theory is a challenge. For simplicity's sake, we choose to get back to the greyscale world - the color dimension will be added in the next chapter. This one presents how 2d amplitude and phase can be handled by the Monogenic Wavelet Transform, and how it represents visual elements through modulated wavelets. We will also give a denoising example, and an alternative filterbank based on the Radon transform.

For all the details:

Soulard & Carré, ...Monogenic maxima... IEEE TPAMI 2017

The monogenic filterbank

Among numerous variants of wavelet transforms being proposed since the 1990s, the sub-family of analytic and monogenic wavelets is of great interest here. Their particularity over classical wavelets is that they are based on a primary filterbank for frequency selectivity, and another parallel filterbank that is linked to the first one by a Hilbert or a Riesz transform.

The monogenic analysis requires isotropic filtering for the primary frequency selection, which excludes the usual "row-column" filterbanks, known for their square block artifacts and poor handling of contour orientation. A better representation is given by using 2d FFT-based IIR filters, defined by their radial Fourier response. Also, the Riesz transform of the band-pass filter can be easily processed in the FFT domain.

From the theoretical point of view, we use the undecimated version of the wavelet-frames from [Unser & Chenouard, SIAM J. Imaging Sci. 2013], which gives many degrees of freedom regarding the filter design. We have proposed the following high-pass and low-pass filters definitions:

The algorithm is described by the flowchart depicted in the figure opposite.

Different scales are obtained by iterating the filterbank, which provides as many independent monogenic analyses. In the end, for every scale i and every space location x, three wavelet coefficients are produced, to be converted into amplitude, phase and orientation features.

The advantage of using wavelets instead of any set of band-pass filters is reversibility. The monogenic reconstruction of input image from th esubband information is obtained as the average of the two reverse filterbanks. The figure opposite shows examples of synthesis monogenic wavelets. These shapes are the atoms of the monogenic wavelet representation, associated to particular amplitude, phase and orientation values. Contrary to usual wavelet transforms, our wavelet coefficients are vector-valued (amp., phase, orient), and the shape of the synthesis wavelet can be modulated by phase and orientation values. This particularity gives better sparsity to the wavelet representation.

Demo Matlab script for the greyscale Monogenic Wavelet Transform (MWT)
Demo Matlab script showing the synthesis monogenic wavelets
'face4b.png': Test image.
Dependencies.

Choosing a wavelet

The isotropic wavelet frame design allows for intuitive definition of the frequency response, and several definitions have been proposed (see references below). The Gaussian-based wavelet introduced in our article [TPAMI 2017] is well fitted for space-domain contour detection because it produces smoother amplitude. As a counterpart, we loose the reconstruction property for the 'pyramid' filterbank setting, which is not a problem since we only use the 'undecimated' setting. Amplitude smoothness will be of great importance for maxima detection.

Demo Matlab script to compare different wavelet definitions (Same dependencies as above).
'spiral_edgecol2_400.png': Test image.

References:
Held, Storath et al., "Steerable wavelet frames based on the riesz transform" IEEE TIP 2010
Unser & Chenouard, "A unifying parametric framework for 2D steerable wavelet transforms" SIAM J. Imaging Sci. 2013.
Soulard & Carré "Characterization of color images with multiscale monogenic maxima" IEEE TPAMI 2017

Monogenic denoising

Wavelet shrinkage is the most illustrative use of a wavelet transform, because it exploits its fundamental analysis property:
"high-magnitude coefficients convey the most important information"
as well as its reversibility, so that a denoised version of the input image can be reconstructed.

The figure opposite gives such an example. After applying the Monogenic Wavelet Transform, coefficients are selected by their amplitude compared to some threshold. All coefficients with low amplitude are replaced by zeros, and then the reverse monogenic filterbank is run to reconstruct the new image.

We can see that the monogenic filterbank allows for efficient wavelet-based denoising.

Demo Matlab script for greyscale monogenic image denoising (Same dependencies as above).
'sailing214x179.png': Test image.

The Radon-based filterbank

Our Radon-based filterbank is described in details in our 2015 article:

Soulard & Carré, Elliptical Monogenic Wavelets... IEEE TSP 2015

We have proposed an innovative numerical scheme for monogenic wavelet design, based on the Radon transform from Carré & Andres (see references below). In the Radon domain, isotropic filtering reduces to 1d filtering, and the monogenic analysis has a simple mathematical writing. On the other hand, the Radon transform is quite hard to define for discrete signals.

The proposed filterbank is perfectly reversible but cannot be isotropic, so it may produce directional artifacts. This research lead is still open and shows that discretization methods for the Radon transform could bring new insights on monogenic analysis.

Demo Matlab script for greyscale Radon-based monogenic filterbank.
Dependencies.

References:
Our Matlab code for the Radon transform is based on
Carré & Andres "Discrete Analytical Ridgelet Transform" Elsevier Sig. Proc. 84, 11 (2004)

Now the greyscale MWT is presented, let us introduce de color extension based on the elliptical model.