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

Discrete Wavelet Transforms

The discrete wavelet transform (DWT) algorithms have a firm position in processing of signals in several areas of research and industry. As DWT provides both octave-scale frequency and spatial timing of the analyzed signal, it is constantly used to solve and treat more and more advanced problems. The present book: Discrete Wavelet Transforms: Algorithms and Applications reviews the recent progress in discrete wavelet transform algorithms and applications. The book covers a wide range of methods (e.g. lifting, shift invariance, multi-scale analysis) for constructing DWTs. The book chapters are organized into four major parts. Part I describes the progress in hardware implementations of the DWT algorithms. Applications include multitone modulation for ADSL and equalization techniques, a scalable architecture for FPGA-implementation, lifting based algorithm for VLSI implementation, comparison between DWT and FFT based OFDM and modified SPIHT codec. Part II addresses image processing algorithms such as multiresolution approach for edge detection, low bit rate image compression, low complexity implementation of CQF wavelets and compression of multi-component images. Part III focuses watermaking DWT algorithms. Finally, Part IV describes shift invariant DWTs, DC lossless property, DWT based analysis and estimation of colored noise and an application of the wavelet Galerkin method. The chapters of the present book consist of both tutorial and highly advanced material. Therefore, the book is intended to be a reference text for graduate students and researchers to obtain state-of-the-art knowledge on specific applications

Connected Attribute Filtering Based on Contour Smoothness

A new attribute measuring the contour smoothness of 2-D objects is presented in the context of morphological attribute filtering. The attribute is based on the ratio of the circularity and non-compactness, and has a maximum of 1 for a perfect circle. It decreases as the object boundary becomes irregular. Computation on hierarchical image representation structures relies on five auxiliary data members and is rapid. Contour smoothness is a suitable descriptor for detecting and discriminating man-made structures from other image features. An example is demonstrated on a very-high-resolution satellite image using connected pattern spectra and the switchboard platform

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao

Spatial and Temporal Image Prediction with Magnitude and Phase Representations

In this dissertation, I develop the theory and techniques for spatial and temporal image prediction with the magnitude and phase representation of the Complex Wavelet Transform (CWT) or the over-complete DCT to solve the problems of image inpainting and motion compensated inter-picture prediction. First, I develop the theory and algorithms of image reconstruction from the analytic magnitude or phase of the CWT. I prove the conditions under which a signal is uniquely specified by its analytic magnitude or phase, propose iterative algorithms for the reconstruction of a signal from its analytic CWT magnitude or phase, and analyze the convergence of the proposed algorithms. Image reconstruction from the magnitude and pseudo-phase of the over-complete DCT is also discussed and demonstrated. Second, I propose simple geometrical models of the CWT magnitude and phase to describe edges and structured textures and develop a spatial image prediction (inpainting) algorithm based on those models and the iterative image reconstruction mentioned above. Piecewise smooth signals, structured textures and their mixtures can be predicted successfully with the proposed algorithm. Simulation results show that the proposed algorithm achieves appealing visual quality with low computational complexity. Finally, I propose a novel temporal (inter-picture) image predictor for hybrid video coding. The proposed predictor enables successful predictive coding during fades, blended scenes, temporally decorrelated noise, and many other temporal evolutions that are beyond the capability of the traditional motion compensated prediction methods. The proposed predictor estimates the transform magnitude and phase of the desired motion compensated prediction by exploiting the temporal and spatial correlations of the transform coefficients. For the case of implementation in standard hybrid video coders, the over-complete DCT is chosen over the CWT. Better coding performance is achieved with the state-of-the-art H.264/AVC video encoder equipped with the proposed predictor. The proposed predictor is also successfully applied to image registration

Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing

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

Probabilistic methods for pose-invariant recognition in computer vision

This thesis is concerned with two central themes in computer vision, the properties of oriented quadrature filters, and methods for implementing rotation invariance in an object matching and recognition system. Objects are modeled as combinations of local features, and human faces are used as the reference object class. The topics covered include optimal design of filter banks for feature detection and object recognition, modeling of pose effects in filter responses and the construction of probability-based pose-invariant object matching and recognition systems employing oriented filters. Gabor filters have been derived as information-theoretically optimal bandpass filters, simultaneously maximizing the localization capability in space and spatial-frequency domains. Steerable oriented filters have been developed as a tool for reducing the amount of computation required in rotation invariant systems. In this work, the framework of steerable filters is applied to Gabor-type filters and novel analytical derivations for the required steering equations for them are presented. Gabor filters and some related filters are experimentally shown to be approximately steerable with low steering error, given suitable filter shape parameters. The effects of filter shape parameters in feature localization and object recognition are also studied using a complete feature matching system. A novel approach for modeling the pose variation of features due to depth rotations is introduced. Instead of manifold learning methods, the use synthetic data makes it possible to apply simpler regression modeling methods. The use of synthetic data in learning the pose models for local features is a central contribution of the work. The object matching methods considered in the work are based on probabilistic reasoning. The required object likelihood functions are constructed using feature similarity measures, and random sampling methods are applied for finding the modes of high probability in the likelihood probability distribution functions. The Population Monte Carlo algorithm is shown to solve successfully pose estimation problems in which simple Metropolis and Gibbs sampling methods give unsatisfactory performance.Tämä väitöskirja käsittelee kahta keskeistä tietokonenäön osa-aluetta, signaalin suunnalle herkkien kvadratuurisuodinten ominaisuuksia, ja näkymäsuunnasta riippumattomia menetelmiä kohteiden sovittamiseksi malliin ja tunnistamiseksi. Kohteet mallinnetaan paikallisten piirteiden yhdistelminä, ja esimerkkikohdeluokkana käytetään ihmiskasvoja. Työssä käsitellään suodinpankin optimaalista suunnittelua piirteiden havaitsemisen ja kohteen tunnistuksen kannalta, näkymäsuunnan piirteissä aiheuttamien ilmiöiden mallintamista sekä edellisen kaltaisia piirteitä käyttävän todennäköisyyspohjaisen, näkymäsuunnasta riippumattomaan havaitsemiseen kykenevän kohteidentunnistusjärjestelmän toteutusta. Gabor-suotimet ovat informaatioteoreettisista lähtökohdista johdettuja, aika- ja taajuustason paikallistamiskyvyltään optimaalisia kaistanpäästösuotimia. Nk. ohjattavat (steerable) suuntaherkät suotimet on kehitetty vähentämään laskennan määrää tasorotaatioille invarianteissa järjestelmissä. Työssä laajennetaan ohjattavien suodinten teoriaa Gabor-suotimiin ja esitetään Gabor-suodinten ohjaukseen vaadittavien approksimointiyhtälöiden johtaminen analyyttisesti. Kokeellisesti näytetään, että Gabor-suotimet ja eräät niitä muistuttavat suotimet ovat sopivilla muotoparametrien arvoilla likimäärin ohjattavia. Lisäksi tutkitaan muotoparametrien vaikutusta piirteiden havaittavuuteen sekä kohteen tunnistamiseen kokonaista kohteidentunnistusjärjestelmää käyttäen. Piirteiden näkymäsuunnasta johtuvaa vaihtelua mallinnetaan suoraviivaisesti regressiomenetelmillä. Näiden käyttäminen monisto-oppimismenetelmien (manifold learning methods) sijaan on mahdollista, koska malli muodostetaan synteettisen datan avulla. Työn keskeisiä kontribuutioita on synteettisen datan käyttäminen paikallisten piirteiden näkymämallien oppimisessa. Työssä käsiteltävät mallinsovitusmenetelmät perustuvat todennäköisyyspohjaiseen päättelyyn. Tarvittavat kohteen uskottavuusfunktiot muodostetaan piirteiden samankaltaisuusmitoista, ja uskottavuusfunktion suuren todennäköisyysmassan keskittymät löydetään satunnaisotantamenetelmillä. Population Monte Carlo -algoritmin osoitetaan ratkaisevan onnistuneesti asennonestimointiongelmia, joissa Metropolis- ja Gibbs-otantamenetelmät antavat epätyydyttäviä tuloksia.reviewe