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

Symmetric measures via moments

Algebraic tools in statistics have recently been receiving special attention and a number of interactions between algebraic geometry and computational statistics have been rapidly developing. This paper presents another such connection, namely, one between probabilistic models invariant under a finite group of (non-singular) linear transformations and polynomials invariant under the same group. Two specific aspects of the connection are discussed: generalization of the (uniqueness part of the multivariate) problem of moments and log-linear, or toric, modeling by expansion of invariant terms. A distribution of minuscule subimages extracted from a large database of natural images is analyzed to illustrate the above concepts.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6144 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Altered Perceptual Sensitivity to Kinematic Invariants in Parkinson's Disease

Ample evidence exists for coupling between action and perception in neurologically healthy individuals, yet the precise nature of the internal representations shared between these domains remains unclear. One experimentally derived view is that the invariant properties and constraints characterizing movement generation are also manifested during motion perception. One prominent motor invariant is the “two-third power law,” describing the strong relation between the kinematics of motion and the geometrical features of the path followed by the hand during planar drawing movements. The two-thirds power law not only characterizes various movement generation tasks but also seems to constrain visual perception of motion. The present study aimed to assess whether motor invariants, such as the two thirds power law also constrain motion perception in patients with Parkinson's disease (PD). Patients with PD and age-matched controls were asked to observe the movement of a light spot rotating on an elliptical path and to modify its velocity until it appeared to move most uniformly. As in previous reports controls tended to choose those movements close to obeying the two-thirds power law as most uniform. Patients with PD displayed a more variable behavior, choosing on average, movements closer but not equal to a constant velocity. Our results thus demonstrate impairments in how the two-thirds power law constrains motion perception in patients with PD, where this relationship between velocity and curvature appears to be preserved but scaled down. Recent hypotheses on the role of the basal ganglia in motor timing may explain these irregularities. Alternatively, these impairments in perception of movement may reflect similar deficits in motor production

New human action recognition scheme with geometrical feature representation and invariant discretization for video surveillance

Human action recognition is an active research area in computer vision because of its immense application in the field of video surveillance, video retrieval, security systems, video indexing and human computer interaction. Action recognition is classified as the time varying feature data generated by human under different viewpoint that aims to build mapping between dynamic image information and semantic understanding. Although a great deal of progress has been made in recognition of human actions during last two decades, few proposed approaches in literature are reported. This leads to a need for much research works to be conducted in addressing on going challenges leading to developing more efficient approaches to solve human action recognition. Feature extraction is the main tasks in action recognition that represents the core of any action recognition procedure. The process of feature extraction involves transforming the input data that describe the shape of a segmented silhouette of a moving person into the set of represented features of action poses. In video surveillance, global moment invariant based on Geometrical Moment Invariant (GMI) is widely used in human action recognition. However, there are many drawbacks of GMI such that it lack of granular interpretation of the invariants relative to the shape. Consequently, the representation of features has not been standardized. Hence, this study proposes a new scheme of human action recognition (HAR) with geometrical moment invariants for feature extraction and supervised invariant discretization in identifying actions uniqueness in video sequencing. The proposed scheme is tested using IXMAS dataset in video sequence that has non rigid nature of human poses that resulting from drastic illumination changes, changing in pose and erratic motion patterns. The invarianceness of the proposed scheme is validated based on the intra-class and inter-class analysis. The result of the proposed scheme yields better performance in action recognition compared to the conventional scheme with an average of more than 99% accuracy while preserving the shape of the human actions in video images

Study of object recognition and identification based on shape and texture analysis

The objective of object recognition is to enable computers to recognize image patterns without human intervention. According to its applications, it is mainly divided into two parts: recognition of object categories and detection/identification of objects. My thesis studied the techniques of object feature analysis and identification strategies, which solve the object recognition problem by employing effective and perceptually important object features. The shape information is of particular interest and a review of the shape representation and description is presented, as well as the latest research work on object recognition. In the second chapter of the thesis, a novel content-based approach is proposed for efficient shape classification and retrieval of 2D objects. Two object detection approaches, which are designed according to the characteristics of the shape context and SIFT descriptors, respectively, are analyzed and compared. It is found that the identification strategy constructed on a single type of object feature is only able to recognize the target object under specific conditions which the identifier is adapted to. These identifiers are usually designed to detect the target objects which are rich in the feature type captured by the identifier. In addition, this type of feature often distinguishes the target object from the complex scene. To overcome this constraint, a novel prototyped-based object identification method is presented to detect the target object in the complex scene by employing different types of descriptors to capture the heterogeneous features. All types of descriptors are modified to meet the requirement of the detection strategy’s framework. Thus this new method is able to describe and identify various kinds of objects whose dominant features are quite different. The identification system employs the cosine similarity to evaluate the resemblance between the prototype image and image windows on the complex scene. Then a ‘resemblance map’ is established with values on each patch representing the likelihood of the target object’s presence. The simulation approved that this novel object detection strategy is efficient, robust and of scale and rotation invariance

The computational magic of the ventral stream: sketch of a theory (and why some deep architectures work).

This paper explores the theoretical consequences of a simple assumption: the computational goal of the feedforward path in the ventral stream -- from V1, V2, V4 and to IT -- is to discount image transformations, after learning them during development

Space-time Zernike Moments and Pyramid Kernel Descriptors for Action Classification

Action recognition in videos is a relevant and challenging task of automatic semantic video analysis. Most successful approaches exploit local space-time descriptors. These descriptors are usually carefully engineered in order to obtain feature invariance to photometric and geometric variations. The main drawback of space-time descriptors is high dimensionality and efficiency. In this paper we propose a novel descriptor based on 3D Zernike moments computed for space-time patches. Moments are by construction not redundant and therefore optimal for compactness. Given the hierarchical structure of our descriptor we propose a novel similarity procedure that exploits this structure comparing features as pyramids. The approach is tested on a public dataset and compared with state-of-the art descriptors