M\"obius Invariants of Shapes and Images

Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformations of the object and to find invariants to the action of that group, meaning that the object has the same values of the invariants despite the action of the group. In this paper we study the invariants of planar shapes and images under the M\"obius group $\mathrm{PSL}(2,\mathbb{C})$, which arises in the conformal camera model of vision and may also correspond to neurological aspects of vision, such as grouping of lines and circles. We survey properties of invariants that are important in applications, and the known M\"obius invariants, and then develop an algorithm by which shapes can be recognised that is M\"obius- and reparametrization-invariant, numerically stable, and robust to noise. We demonstrate the efficacy of this new invariant approach on sets of curves, and then develop a M\"obius-invariant signature of grey-scale images

Model-based recognition of curves and surfaces using tactile data

Model-based object recognition has mostly been studied over inputs including images and range data. Though such data are global, cameras and range sensors are subject to occlusions and clutters, which often make recognition difficult and computationally expensive. In contrast, touch by a robot hand is free of occlusion and clutter issues, and recognition over tactile data can be more efficient.;In this thesis, we investigate model-based recognition of two and three dimensional curved objects from tactile data. The recognition of 2D objects is an invariant-based approach. We have derived differential and semi-differential invariants for quadratic curves and special cubic curves that are found in applications. These invariants, independent of translation and rotation, can be computed from local geometry of a curve. Invariants for quadratic curves are the functions in terms of the curvature and its derivative with respect to arc length. For cubic curves, the derived invariants also involve a slope in their expressions. Recognition of a curve reduces to invariant verification with its canonical parametric form determined along the way. In addition, the contact locations with the robot hand are found on the curve, thereby localizing it relative to the touch sensor. We have verified the correctness of all invariants by simulations. We have also shown that the shape parameters of the recognized curve can be recovered with small errors. The byproduct is a procedure that reliably estimates curvature and its derivative from real tactile data. The presented work distinguishes itself from traditional model-based recognition in its ability to simultaneously recognize and localize a shape from one of several classes, each consisting of a continuum of shapes, by the use of local data.;The recognition of 3D objects is based on registration and consists of two steps. First, a robotic hand with touch sensors samples data points on the object\u27s surface along three concurrent curves. The two principal curvatures at the curve intersection point are estimated and then used in a table lookup to find surface points that have similar local geometries. Next, starting at each such point, a local search is conducted to superpose the tactile data onto the surface model. Recognition of the model is based on the quality of this registration. The presented method can recognize algebraic as well as free-form surfaces, as demonstrated via simulations and robot experiments. One difference in the recognition of these two sets of shapes lies in the principal curvature estimation, which are calculated from the close forms and estimated through fitting, respectively. The other difference lies in data registration, which is carried out by nonlinear optimization and a greedy algorithm, respectively

Shape description and matching using integral invariants on eccentricity transformed images

Matching occluded and noisy shapes is a problem frequently encountered in medical image analysis and more generally in computer vision. To keep track of changes inside the breast, for example, it is important for a computer aided detection system to establish correspondences between regions of interest. Shape transformations, computed both with integral invariants (II) and with geodesic distance, yield signatures that are invariant to isometric deformations, such as bending and articulations. Integral invariants describe the boundaries of planar shapes. However, they provide no information about where a particular feature lies on the boundary with regard to the overall shape structure. Conversely, eccentricity transforms (Ecc) can match shapes by signatures of geodesic distance histograms based on information from inside the shape; but they ignore the boundary information. We describe a method that combines the boundary signature of a shape obtained from II and structural information from the Ecc to yield results that improve on them separately

Shape localization, quantification and correspondence using Region Matching Algorithm

We propose a method for local, region-based matching of planar shapes, especially as those shapes that change over time. This is a problem fundamental to medical imaging, specifically the comparison over time of mammograms. The method is based on the non-emergence and non-enhancement of maxima, as well as the causality principle of integral invariant scale space. The core idea of our Region Matching Algorithm (RMA) is to divide a shape into a number of “salient” regions and then to compare all such regions for local similarity in order to quantitatively identify new growths or partial/complete occlusions. The algorithm has several advantages over commonly used methods for shape comparison of segmented regions. First, it provides improved key-point alignment for optimal shape correspondence. Second, it identifies localized changes such as new growths as well as complete/partial occlusion in corresponding regions by dividing the segmented region into sub-regions based upon the extrema that persist over a sufficient range of scales. Third, the algorithm does not depend upon the spatial locations of mammographic features and eliminates the need for registration to identify salient changes over time. Finally, the algorithm is fast to compute and requires no human intervention. We apply the method to temporal pairs of mammograms in order to detect potentially important differences between them