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

STRUCTURE DETECTION WITH SECOND ORDER RIESZ TRANSFORMS

A frequently applied indicator of tubular structures is based on the eigenvalues of the Hessian matrix of the original image convolved with a Gaussian, whose standard derivation depends on the size of the tubes. Hence the tube size must either be known in advance or a whole scale of standard deviations has to be tested resulting in higher computational costs – a serious obstacle for data with varying tube thickness. In this paper, we propose to modify the structure indicator by replacing the derivatives of the Gaussian smoothed function by the Riesz transform. We show by various numerical examples that the resulting structure indicator is scale independent. Smoothing with a Gaussian is just necessary to cope with the noise in the image, but is not related to the size of the tubular structures. We apply the novel structure indicator for the fiber orientation analysis of fibrous materials and for the segmentation of leather. The latter one was a special challenging application since all scales are present in the microstructure of leather

Linearized Riesz Transform and Quasi-Monogenic Shearlets

The only quadrature operator of order two on $L_2 (\mathbb{R}^2)$ which covaries with orthogonal transforms, in particular rotations is (up to the sign) the Riesz transform. This property was used for the construction of monogenic wavelets and curvelets. Recently, shearlets were applied for various signal processing tasks. Unfortunately, the Riesz transform does not correspond with the shear operation. In this paper we propose a novel quadrature operator called linearized Riesz transform which is related to the shear operator. We prove properties of this transform and analyze its performance versus the usual Riesz transform numerically. Furthermore, we demonstrate the relation between the corresponding optical filters. Based on the linearized Riesz transform we introduce finite discrete quasi-monogenic shearlets and prove that they form a tight frame. Numerical experiments show the good fit of the directional information given by the shearlets and the orientation ob- tained from the quasi-monogenic shearlet coefficients. Finally we provide experiments on the directional analysis of textures using our quasi-monogenic shearlets

Linearized Riesz Transform and Quasi-Monogenic Shearlets

The only quadrature operator of order two on $L_2 (\mathbb{R}^2)$ which covaries with orthogonal transforms, in particular rotations is (up to the sign) the Riesz transform. This property was used for the construction of monogenic wavelets and curvelets. Recently, shearlets were applied for various signal processing tasks. Unfortunately, the Riesz transform does not correspond with the shear operation. In this paper we propose a novel quadrature operator called linearized Riesz transform which is related to the shear operator. We prove properties of this transform and analyze its performance versus the usual Riesz transform numerically. Furthermore, we demonstrate the relation between the corresponding optical filters. Based on the linearized Riesz transform we introduce finite discrete quasi-monogenic shearlets and prove that they form a tight frame. Numerical experiments show the good fit of the directional information given by the shearlets and the orientation ob- tained from the quasi-monogenic shearlet coefficients. Finally we provide experiments on the directional analysis of textures using our quasi-monogenic shearlets