The authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters, allowing one to adaptively steer a filter to any orientation, and to determine analytically the filter output as a function of orientation. Steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation. The authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase, angularly adaptive filtering, edge detection, and shape from shading. One can also build a self-similar steerable pyramid representation. The same concepts can be generalized to the design of 3-D steerable filters. >

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The design and use of steerable filters

In computer vision and image processing, edge detection concerns the localization of significant variations of the grey level image and the identification of the physical phenomena that originated them. This information is very useful for applications in 3D reconstruction, motion, recognition, image enhancement and restoration, image registration, image compression, and so on. Usually, edge detection requires smoothing and differentiation of the image. Differentiation is an ill-conditioned problem and smoothing results in a loss of information. It is difficult to design a general edge detection algorithm which performs well in many contexts and captures the requirements of subsequent processing stages. Consequently, over the history of digital image processing a variety of edge detectors have been devised which differ in their mathematical and algorithmic properties. This paper is an account of the current state of our understanding of edge detection. We propose an overview of research in edge detection: edge definition, properties of detectors, the methodology of edge detection, the mutual influence between edges and detectors, and existing edge detectors and their implementation.

Edge Detection Techniques-An Overview

Why and how one should study a scale-space is prescribed by the universal physical law of scale invariance, expressed by the so-called Pi-theorem. The fact that any image is a physical observable with an inner and outer scale bound, necessarily gives rise to a ‘scale-space representation’, in which a given image is represented by a one-dimensional family of images representing that image on various levels of inner spatial scale. An early vision system is completely ignorant of the geometry of its input. Its primary task is to establish this geometry at any available scale. The absence of geometrical knowledge poses additional constraints on the construction of a scale-space, notably linearity, spatial shift invariance and isotropy, thereby defining a complete hierarchical family of scaled partial differential operators: the Gaussian kernel (the lowest order, resettling operator) and its linear partial derivatives. They enable local image analysis through the detection of local differential structure in a robust way, while at the same time capturing global features through the extra scale degree of freedom. In this paper we show why the operations of scaling and differentiation cannot be separated. This framework permits us to construct in a systematic way multiscale, cartesian differential invariants, i.e. true image descriptors that exhibit manifest invariance with respect to a change of cartesian coordinates. The scale-space operators closely resemble the receptive field profiles found in mammalian frontend visual systems.

Scale and the differential structure of images

dix A. Even if you don’t choose to memorize them this system aids in reference and retreival of important formulas. The book was compiled from notes developed during eight years of teaching a graduate course on the subject and was used as a text. Thus it has been student tested. Appendix F contains a number of problems, grouped to be used on a chapter by chapter basis The problems are designed to illustrate practical applications of the text material. The parts of this book of most interest and value to the EMC engineer will be the chapters on Thermal Noise, Antennas, Propagation and Transmission Lines, and  Reflection and Refraction. This is not  to downpade the chapters on Statistics and Its Applications, Signal Processing and Detection, and Some System Characteristics which also contain much potentially useful materials. Additional plus values for the book include a list of 40 references, a table of symbols used throughout the book, and a subject index. Some readers may find the condensed type and close line spacing hard to read. It was apparently set up by typewriter using an elite type face with single line spacing. When reduced down to a 6 by 9 5 inch size page it is too crowded for easy reading. In spite of this shortcoming your reviewer recommends this book as a worthwhile reference in this field of interest.

http://ieeexplore.ieee.org/iel5/5/31269/01455368.pdf

Signal analysis

We present a new method for planning footstep placements for a robot walking on uneven terrain with obstacles, using a mixed-integer quadratically-constrained quadratic program (MIQCQP). Our approach is unique in that it handles obstacle avoidance, kinematic reachability, and rotation of footstep placements, which typically have required non-convex constraints, in a single mixed-integer optimization that can be efficiently solved to its global optimum. Reachability is enforced through a convex inner approximation of the reachable space for the robot's feet. Rotation of the footsteps is handled by a piecewise linear approximation of sine and cosine, designed to ensure that the approximation never overestimates the robot's reachability. Obstacle avoidance is ensured by decomposing the environment into convex regions of obstacle-free configuration space and assigning each footstep to one such safe region. We demonstrate this technique in simple 2D and 3D environments and with real environments sensed by a humanoid robot. We also discuss computational performance of the algorithm, which is currently capable of planning short sequences of a few steps in under one second or longer sequences of 10–30 footsteps in tens of seconds to minutes on common laptop computer hardware. Our implementation is available within the Drake MATLAB toolbox [1].

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Footstep planning on uneven terrain with mixed-integer convex optimization

Generalized and Separable Sobel Operators

This paper addresses the problem on how to evaluate various operators used for estimation of derivatives in images. Such operators are extremely commonly used, for instance to detect edges. For bandlimited correctly sampled signals ideal derivative operators are easy to define. For 2D signals the first derivative operators take the form of a rotation invariant pair. Rotation invariance is also a natural requirement for the non-ideal practically implementable operators of which the Sobel operator is one example. Second degree derivators do not form a set of rotation invariant operators by itself. Instead, certain linear combinations of these derivators display this property. The Fourier domain is used extensively in the analysis, partly because the angular variation in the form of circular harmonics is preserved over the Fourier transform. Circular harmonics expansion is used for evaluation of practically implementable operator kernels. A quantity called total harmonic distortion (THD) is defined to capture the overall deviation from rotation invariance. In a comparison with two other simple kernels it seems that the Sobel operator does fairly well. Errors in magnitude and orientation estimation follow the THD-values quite closely. For all pairs of operators (convolution kernels), which are to be employed for orientation estimation, the paper presents a method to evaluate their quality. Besides, the paper brings about new insights in the interconnected concepts of edge detectors, line detectors, derivative operators, rotation invariance, and rotational symmetry.

Rotation invariance in gradient and higher order derivative detectors

The MATLAB/C program take version 3.1 is a program for simulation of X-ray projections from 3D volume data. It is based on an older C version by Muller-Merbach as well as an extended C version by T ...

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The MATLAB/C program take - a program for simulation of X-ray projections from 3D volume data. Demonstration of beam-hardening artefacts in subsequent CT reconstruction.

Modeling of the point-spread function of laser scanning microscopes using canonical transforms

This paper gives an overview of operator-based models that seem to be especially interesting for laser scanning microscopy. These methods were developed by Nazarathy et. al. in a series of papers about 10 years ago. In these papers they demonstrated that both the operators and Gaussian beams can be represented by matrices. We implemented the operator algebra in Mathematic and we also show by some simple examples how to analyze paraxial systems. A number of empirical experiments have also been performed to verify the validity of the model.

Olle Seger

Papers

Generalized and Separable Sobel Operators

Rotation invariance in gradient and higher order derivative detectors

The MATLAB/C program take - a program for simulation of X-ray projections from 3D volume data. Demonstration of beam-hardening artefacts in subsequent CT reconstruction.

Modeling of the point-spread function of laser scanning microscopes using canonical transforms

Experiments with an operator model of a confocal microscope