The development of the quaternion wavelet transform

The purpose of this article is to review what has been written on what other authors have called quaternion wavelet transforms (QWTs): there is no consensus about what these should look like and what their properties should be. We briefly explain what real continuous and discrete wavelet transforms and multiresolution analysis are and why complex wavelet transforms were introduced; we then go on to detail published approaches to QWTs and to analyse them. We conclude with our own analysis of what it is that should define a QWT as being truly quaternionic and why all but a few of the “QWTs” we have described do not fit our definition

Vector extension of monogenic wavelets for geometric representation of color images

14 pagesInternational audienceMonogenic wavelets offer a geometric representation of grayscale images through an AM/FM model allowing invariance of coefficients to translations and rotations. The underlying concept of local phase includes a fine contour analysis into a coherent unified framework. Starting from a link with structure tensors, we propose a non-trivial extension of the monogenic framework to vector-valued signals to carry out a non marginal color monogenic wavelet transform. We also give a practical study of this new wavelet transform in the contexts of sparse representations and invariant analysis, which helps to understand the physical interpretation of coefficients and validates the interest of our theoretical construction

Modulation Domain Image Processing

The classical Fourier transform is the cornerstone of traditional linearsignal and image processing. The discrete Fourier transform (DFT) and thefast Fourier transform (FFT) in particular led toprofound changes during the later decades of the last century in howwe analyze and process 1D and multi-dimensional signals.The Fourier transform represents a signal as an infinite superpositionof stationary sinusoids each of which has constant amplitude and constantfrequency. However, many important practical signals such as radar returnsand seismic waves are inherently nonstationary. Hence, more complextechniques such as the windowed Fourier transform and the wavelet transformwere invented to better capture nonstationary properties of these signals.In this dissertation, I studied an alternative nonstationary representationfor images, the 2D AM-FM model. In contrast to thestationary nature of the classical Fourier representation, the AM-FM modelrepresents an image as a finite sum of smoothly varying amplitudesand smoothly varying frequencies. The model has been applied successfullyin image processing applications such as image segmentation, texture analysis,and target tracking. However, these applications are limitedto \emph{analysis}, meaning that the computed AM and FM functionsare used as features for signal processing tasks such as classificationand recognition. For synthesis applications, few attempts have been madeto synthesize the original image from the AM and FM components. Nevertheless,these attempts were unstable and the synthesized results contained artifacts.The main reason is that the perfect reconstruction AM-FM image model waseither unavailable or unstable. Here, I constructed the first functionalperfect reconstruction AM-FM image transform that paves the way for AM-FMimage synthesis applications. The transform enables intuitive nonlinearimage filter designs in the modulation domain. I showed that these filtersprovide important advantages relative to traditional linear translation invariant filters.This dissertation addresses image processing operations in the nonlinearnonstationary modulation domain. In the modulation domain, an image is modeledas a sum of nonstationary amplitude modulation (AM) functions andnonstationary frequency modulation (FM) functions. I developeda theoretical framework for high fidelity signal and image modeling in themodulation domain, constructed an invertible multi-dimensional AM-FMtransform (xAMFM), and investigated practical signal processing applicationsof the transform. After developing the xAMFM, I investigated new imageprocessing operations that apply directly to the transformed AM and FMfunctions in the modulation domain. In addition, I introduced twoclasses of modulation domain image filters. These filters produceperceptually motivated signal processing results that are difficult orimpossible to obtain with traditional linear processing or spatial domainnonlinear approaches. Finally, I proposed three extensions of the AM-FMtransform and applied them in image analysis applications.The main original contributions of this dissertation include the following.- I proposed a perfect reconstruction FM algorithm. I used aleast-squares approach to recover the phase signal from itsgradient. In order to allow perfect reconstruction of the phase function, Ienforced an initial condition on the reconstructed phase. The perfectreconstruction FM algorithm plays a critical role in theoverall AM-FM transform.- I constructed a perfect reconstruction multi-dimensional filterbankby modifying the classical steerable pyramid. This modified filterbankensures a true multi-scale multi-orientation signal decomposition. Such adecomposition is required for a perceptually meaningful AM-FM imagerepresentation.- I rotated the partial Hilbert transform to alleviate ripplingartifacts in the computed AM and FM functions. This adjustment results inartifact free filtering results in the modulation domain.- I proposed the modulation domain image filtering framework. Iconstructed two classes of modulation domain filters. I showed that themodulation domain filters outperform traditional linear shiftinvariant (LSI) filters qualitatively and quantitatively in applicationssuch as selective orientation filtering, selective frequency filtering,and fundamental geometric image transformations.- I provided extensions of the AM-FM transform for image decompositionproblems. I illustrated that the AM-FM approach can successfullydecompose an image into coherent components such as textureand structural components.- I investigated the relationship between the two prominentAM-FM computational models, namely the partial Hilbert transformapproach (pHT) and the monogenic signal. The established relationshiphelps unify these two AM-FM algorithms.This dissertation lays a theoretical foundation for future nonlinearmodulation domain image processing applications. For the first time, onecan apply modulation domain filters to images to obtain predictableresults. The design of modulation domain filters is intuitive and simple,yet these filters produce superior results compared to those of pixeldomain LSI filters. Moreover, this dissertation opens up other research problems.For instance, classical image applications such as image segmentation andedge detection can be re-formulated in the modulation domain setting.Modulation domain based perceptual image and video quality assessment andimage compression are important future application areas for the fundamentalrepresentation results developed in this dissertation

Overcomplete Image Representations for Texture Analysis

Advisor/s: Dr. Boris Escalante-Ramírez and Dr. Gabriel Cristóbal. Date and location of PhD thesis defense: 23th October 2013, Universidad Nacional Autónoma de México.In recent years, computer vision has played an important role in many scientific and technological areas mainlybecause modern society highlights vision over other senses. At the same time, application requirements and complexity have also increased so that in many cases the optimal solution depends on the intrinsic charac-teristics of the problem; therefore, it is difficult to propose a universal image model. In parallel, advances in understanding the human visual system have allowed to propose sophisticated models that incorporate simple phenomena which occur in early stages of the visual system. This dissertation aims to investigate characteristicsof vision such as over-representation and orientation of receptive fields in order to propose bio-inspired image models for texture analysis