A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

Supervised Content based Image Retrieval using Fuzzy Texton and Shearlet Transform

In this paper we proposed, a novel framework to assist and automate the diagnosis of diseases from computer-based image analysis method using Content-based image retrieval (CBIR). CBIR is the process of retrieving related images from large database collections by using low level image features such as color, texture and shape etc. we have used fuzzy texton and discrete shearlet transform to extract texture and shape features. The aim is to support decision making by retrieving and displaying relevant past cases visually similar to the one under examination with relevance feedback using Support Vector Machines

3D steerable wavelets and monogenic analysis for bioimaging

ABSTRACT In this paper we introduce a 3D wavelet frame that has the key property of steerability. The proposed wavelet frame relies on the combination of a 3D isotropic wavelet transform with the 3D Riesz operator which brings steerability to the pyramid. The novel transform enjoys self reversibility and exact steering of the basis functions in any 3D direction by linear combination of the primary coefficients. We exploit the link between the Riesz transform and the directional Hilbert transform to define a multiresolution monogenic signal analysis in 3D which achieves multiscale AM/FM signal decomposition. We give an example of application of the 3D monogenic wavelet frame in biological imaging with the enhancement of anisotropic structures in 3D fluorescence microscopy

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

Local Geometric Transformations in Image Analysis

The characterization of images by geometric features facilitates the precise analysis of the structures found in biological micrographs such as cells, proteins, or tissues. In this thesis, we study image representations that are adapted to local geometric transformations such as rotation, translation, and scaling, with a special emphasis on wavelet representations. In the first part of the thesis, our main interest is in the analysis of directional patterns and the estimation of their location and orientation. We explore steerable representations that correspond to the notion of rotation. Contrarily to classical pattern matching techniques, they have no need for an a priori discretization of the angle and for matching the filter to the image at each discretized direction. Instead, it is sufficient to apply the filtering only once. Then, the rotated filter for any arbitrary angle can be determined by a systematic and linear transformation of the initial filter. We derive the Cramér-Rao bounds for steerable filters. They allow us to select the best harmonics for the design of steerable detectors and to identify their optimal radial profile. We propose several ways to construct optimal representations and to build powerful and effective detector schemes; in particular, junctions of coinciding branches with local orientations. The basic idea of local transformability and the general principles that we utilize to design steerable wavelets can be applied to other geometric transformations. Accordingly, in the second part, we extend our framework to other transformation groups, with a particular interest in scaling. To construct representations in tune with a notion of local scale, we identify the possible solutions for scalable functions and give specific criteria for their applicability to wavelet schemes. Finally, we propose discrete wavelet frames that approximate a continuous wavelet transform. Based on these results, we present a novel wavelet-based image-analysis software that provides a fast and automatic detection of circular patterns, combined with a precise estimation of their size