Signal Modeling for Two-Dimensional Image Structures and Scale-Space Based Image Analysis

Model based image representation plays an important role in many computer vision tasks. Consequently, it is of high significance to model image structures with more powerful representation capabilities. In the literature, there exist bulk of researches for intensity based modeling. However, most of them suffer from the illumination variation. On the other hand, phase information, which carries most essential structural information of the original signal, has the advantage of being invariant to the brightness change. Therefore, phase based image analysis is advantageous when compared to purely intensity based approaches. This thesis aims to propose novel image representations for 2D image structures, from which useful local features can be extracted, which are useful for phase based image analysis. The first approach presents a 2D rotationally invariant quadrature filter. This model is able to handle superimposed intrinsically two-dimensional (i2D) patterns with flexible angles of intersection. Hence, it can be regarded as an extension of the structure multivector. The second approach is the monogenic curvature tensor. Coupling methods of differential geometry, tensor algebra, monogenic signal and quadrature filter, we can design a general model for 2D structures as the monogenic extension of a curvature tensor. Based on it, local representations for the intrinsically one-dimensional (i1D) and i2D structures are derived as the monogenic signal and the generalized monogenic curvature signal, respectively. From them, independent features of local amplitude, phase and orientation are simultaneously extracted. Besides, a generalized monogenic curvature scale-space can be built by applying a Poisson kernel to the monogenic curvature tensor. Compared with other related work, the remarkable advantage of our approach lies in the rotationally invariant phase evaluation of 2D structures in a multi-scale framework, which delivers access to phase-based processing in many computer vision tasks. To demonstrate the efficiency and power of the theoretic framework, some computer vision applications are presented, which include the phase based image reconstruction, detecting i2D image structures using local phase and monogenic curvature tensor for optical flow estimation

A Novel Representation for Two-dimensional Image Structures

This paper presents a novel approach towards two-dimensional (2D) image structures modeling. To obtain more degrees of freedom, a 2D image signal is embedded into a certain geometric algebra. Coupling methods of differential geometry, tensor algebra, monogenic signal and quadrature filter, we can design a general model for 2D structures as the monogenic extension of a curvature tensor. Based on it, a local representation for the intrinsically two-dimensional (i2D) structure is derived as the monogenic curvature signal. From it, independent features of local amplitude, phase and orientation are simultaneously extracted. Besides, a monogenic curvature scale-space can be built by applying a Poisson kernel to the monogenic curvature signal. Compared with the other related work, the remarkable advantage of our approach lies in the rotationally invariant phase evaluation of 2D structures in a multi-scale framework, which delivers access to phase-based processing in many computer vision tasks

Algebraic Representation and Geometric Interpretation of Hilbert Transformed Signals

This thesis covers a fundamental problem of local phase based signal processing: the isotropic generalization of the classical one dimensional analytic signal (D. Gabor) to higher dimensional signal domains. The classical analytic signal extends a real valued one dimensional signal to a complex valued signal by means of the classical 1D Hilbert transform. This signal extension enables the complete analysis of local phase and local amplitude information for each frequency component in the sense of Fourier analysis. In case of two dimensional signal domains, e.g. for images, additional geometric information is required to characterize higher dimensional signals locally. The local geometric information is called orientation, which consists of the main orientation and apex angle for two superimposed one dimensional signals. The problem of two dimensional signal analysis is the fact that in general those signals could consist of an unlimited number of superimposed one dimensional signals with individual orientations. Local phase, amplitude and additional orientation information can be extracted by the monogenic signal (M. Felsberg and G. Sommer) which is always restricted to the subclass of intrinsically one dimensional signals, i.e. the class of signals which only make use of one degree of freedom within the embedding signal domain. In case of 2D images the monogenic signal enables the rotationally invariant analysis of lines and edges. In contrast to the 1D analytic signal the monogenic signal extends all real valued signals of dimension n to a (n+1) - dimensional vector valued monogenic signal by means of the generalized first order Hilbert transform, which is also known as the Riesz transform. The analytic signal and the monogenic signal show that a direct relation between analytical signals and their algebraic representation exists. This fact has motivated the work and the results of this thesis, namely the extension of the 2D monogenic signal to more general 2D analytic signals, their algebraic representation, and their most geometric embedding. In case of more general 2D signals the geometric algebra will be shown to be a natural representation, and the conformal space as the geometric embedding for the signal interpretation. In this thesis we present 2D analytic signals as generalizations of the 2D monogenic signal which now extend the original 2D signal to a multi-vector valued signal in homogeneous conformal space by means of higher order Hilbert transforms, and by means of a so called hybrid matrix geometric algebra representation. The 2D analytic signal and the more general multi-vector signal will be interpreted in conformal space which delivers a descriptive geometric interpretation and algebraic embedding of signals. In case of 2D image signals the 2D analytic signal and the multi-vector signal enable the rotationally invariant analysis of lines, edges, corners and junctions in one unified framework. Furthermore, additional local curvature can be determined by first order generalized Hilbert transforms without the need of derivatives. This so called conformal monogenic signal can be defined for any signal domain

On the Analysis and Decomposition of Intrinsically One-Dimensional Signals and their Superpositions

Computer and machine vision tasks can roughly be divided into a hierarchy of processing steps applied to input signals captured by a measuring device. In the case of image signals, the first stage in this hierarchy is also referred to as low-level vision or low-level image processing. The field of low-level image processing includes the mathematical description of signals in terms of certain local signal models. The choice of the signal model is often task dependent. A common task is the extraction of features from the signal. Since signals are subject to transformations, for example camera movements in the case of image signals, the features are supposed to fulfill the properties of invariance or equivariance with respect to these transformations. The chosen signal model should reflect these properties in terms of its parameters. This thesis contributes to the field of low-level vision. Local signal structures are represented by (sinusoidal) intrinsically one-dimensional signals and their superpositions. Each intrinsically one-dimensional signal consists of certain parameters such as orientation, amplitude, frequency and phase. If the affine group acts on these signals, the transformations induce a corresponding action in the parameter space of the signal model. Hence, it is reasonable, to estimate the model parameters in order to describe the invariant and equivariant features. The first and main contribution studies superpositions of intrinsically one-dimensional signals in the plane. The parameters of the signal are supposed to be extracted from the responses of linear shift invariant operators: the generalized Hilbert transform (Riesz transform) and its higher-order versions and the partial derivative operators. While well known signal representations, such as the monogenic signal, allow to obtain the local features amplitude, phase and orientation for a single intrinsically one-dimensional signal, there exists no general method to decompose superpositions of such signals into their corresponding features. A novel method for the decomposition of an arbitrary number of sinusoidal intrinsically one-dimensional signals in the plane is proposed. The responses of the higher-order generalized Hilbert transforms in the plane are interpreted as symmetric tensors, which allow to restate the decomposition problem as a symmetric tensor decomposition. Algorithms, examples and applications for the novel decomposition are provided. The second contribution studies curved intrinsically one-dimensional signals in the plane. This signal model introduces a new parameter, the curvature, and allows the representation of curved signal structures. Using the inverse stereographic projection to the sphere, these curved signals are locally identified with intrinsically one-dimensional signals in the three-dimensional Euclidean space and analyzed in terms of the generalized Hilbert transform and partial derivatives therein. The third contribution studies the generalized Hilbert transform in a non-Euclidean space, the two-sphere. The mathematical framework of Clifford analysis proposes a further generalization of the generalized Hilbert transform to the two-sphere in terms of the corresponding Cauchy kernel. Nonetheless, this transform lacks an intuitive interpretation in the frequency domain. A decomposition of the Cauchy kernel in terms of its spherical harmonics is provided. Its coefficients not only provide insights to the generalized Hilbert transform on the sphere, but also allow for fast implementations in terms of analogues of the convolution theorem on the sphere

Utilisation de signaux hypercomplexes en estimation du mouvement et recalage multimodal

L'imagerie médicale est d'une nécessité certaine pour aider les médecins à comprendre et interpréter les comportements mécaniques et fonctionnels du corps humain. Les différentes modalités existantes fournissent des informations complémentaires qui peuvent améliorer cette compréhension. En particulier, la déformation d'organes ou de tissus peut fournir une indication sur la présence ou non d'une pathologie. Cette appréciation qualitative est facile à effectuer à l'œil nu, mais une estimation automatisée et précise de cette déformation peut être nécessaire. Le choix le plus naturel pour traiter les images est de se baser sur l'intensité des pixels. Cependant, certaines approches d'estimation du mouvement décomposent d'abord l'image en différents descripteurs, tels que la phase spatiale, qui porte l'information structurelle de l'image. L'objectif de cette thèse est d'évaluer l'apport de ce type de descripteurs dans le cadre de séquences ultrasonores (US) et de recalage multimodal entre images par résonance magnétique (IRM) et US. Pour cela, nous avons d'abord montré que pour des images US, une approche basée sur la phase issue du signal monogène constituait un bon compromis vis-à-vis de techniques de mise en correspondance de blocs ou de flux optique basé sur la phase extraite du signal analytique complexe 2D. Nous avons ensuite poursuivi cette étude en considérant les différentes informations issues du signal monogène, avec son extension au cas 3D. Cela nous a permis de proposer un estimateur de translations basé sur un autre descripteur : l'orientation principale locale. Nous avons ensuite évalué l'apport de la phase dans le cadre du recalage IRM-US basé sur l'information mutuelle. Nous avons remarqué que dans ce cas, la phase donnait de meilleurs résultats que l'intensité dans la direction latérale mais pas axiale. Finalement, nous présentons les enjeux cliniques du prolapsus génito-urinaire chez la femme. Nous avons ainsi introduit un estimateur de mise en correspondance de blocs déformables basé sur la phase, que nous avons appliqué à des séquences échographiques in vivo. Bien que cet estimateur ait tendance à minimiser le stade du prolapsus, il permet un meilleur suivi des tissus au fil de la séquence que l'estimateur de blocs déformables initial basé sur l'intensitéNowadays, medical imaging is necessary to help doctors to understand and interpret the mechanical and functional behavior of the human body. The different existing modalities provide complementary information, which can improve this comprehension. In particular, the tissue deformation provide an indication on the presence of a pathology. This qualitative appreciation is easy to perform for the human eye, but it would be useful to get an automatic and accurate estimation of this deformation. The most natural choice to process images is to use the intensities of the pixels. However, some approaches estimate the motion decomposing the image in several descriptors, such as spatial phase, which is a strucural information of the image. The aim of this thesis is to evaluate the contribution of this kind of descriptors, when they are used for motion estimation on ultrasound (US) sequences and multimodal registration, between a magnetic resonance images (MRI) and US images. For this, we first showed that for ultrasound images, an approach based on the monogenic spatial phase was a good compromise, facing block matching technics or optical flow estimation based on 2D analytic complex signal. Then, we continued this study, considering all the features extracted from the 3D monogenic signal. It allowed us to propose a translation estimator based on another descriptor : the main local orientation. Afterward, we evaluated the contribution of the phase for MR-US registration based on the mutual information. We noted that, in this case, the spatial phase gave more accurate results than the intensity-based approach in the lateral direction, but not in the axial direction. Finally, we present the clinical issues of the pelvic organ prolaps. Thus, we introduced a phase-based block deformable block matching estimator. We applied this estimator on in vivo US sequences. Although this estimator tends to minimize the degree of the pelvic floor disorders, it allows a better tissues monitoring than the intensity-based block deformable estimator all along the sequenc