Harold G. Longbotham

Theory of order statistic filters and their relationship to linear FIR filters

Necessary and/or sufficient conditions on both the filter coefficients and the signal process are derived in order that nonrecursive order statistic (OS) and linear filtering are equivalent operations. The results indicate that an understanding of OS filters hinges on a better understanding of the properties of signals containing logically monotonic components. The results extend a number of previous theories characterizing the well-known median and ranked-order filters to a broader class of filters and input signals. >

The WMMR filters: a class of robust edge enhancers

The authors develop a class of filters called weighted majority of m values with minimum range (WMMR/sup m/) that have the same impulse rejection properties as the median. They demonstrate a subclass of these filters (WMMR) that may be optimized for edge enhancement in one dimension in that their output converges to the closest perfect edge. One of these filters is shown to restore a class of noisy edges to the closest perfect edge on one pass. Applications in one and two dimensions are discussed and a two-dimensional simulation is provided comparing the WMMR to other filters for smoothing and edge enhancement. >

Complete classification of roots to one-dimensional median and rank-order filters

The set of roots to the one-dimensional median filter is completely determined. Let 2N+1 be the filter window width. It has been shown that if a root contains a monotone segment of length N+1, then it must be locally monotone N+2. For roots with no monotone segment of length N+1, it is proved that the set of such roots is finite, and that each such root is periodic. The methods used are constructive, so given N, one can list all possible roots of this type. The results developed for the median filter also apply to rank-order filters. >

Nonlinear indicators of malignancy

This paper investigates the use of fractional dimension analysis and nonlinear filters for quantifying the degree of lesion diffusion in mammograms. The fractal method involves computing the fractal dimension over the entire lesion. Based on the observation that malignant lesions usually exhibit rougher intensity profiles and often have more toruous boundaries than benign lesions, the fractal dimension, which is a popular means of quantifying the degree of image/surface roughness, is proposed as a natural tool to assist in the diagnosis of malignancy. In this work, the fractal dimension of the image intensity surface is estimated using the fractional Brownian motion model. The nonlinear analysis was performed on horizontal and vertical lines (one-dimensional data) through the area of interest. These scan lines were also processed by a nonlinear (maximum) transformation as a means of reducing the dimensionality of the data, to aid in clarifying the degree of diffusion present in the data. For benign lesions little diffusion will be present, whereas malignant lesions generally display a higher degree of diffusion. Results of these techniques are applied on several malignant and benign lesions are presented, using mammogram X-rays digitized to a 512 X 512 pixel resolution and 8-bits of gray-scale resolution.

Generalized order statistic filters

The authors explore generalizations of order statistic (OS) filters, a class of finite-width discrete windowing filters defined by linearly weighting the samples in the window according to their natural ordering as real numbers. The concept can be extended by defining generalized ordering rules generating different permutations of the sample prior to weighting. Although the resulting class of generalized OS filter is very broad (including, e.g. the linear filters), local signal ordering properties relating to local signal monotonicity unify them. It is envisaged that both the framework for filter/signal description and the class of filters generated will find use in many signal shaping applications.<<ETX>>

Automated inspection of through hole solder joints utilizing X-ray imaging

Solder joint inspection has traditionally been done by manual inspection. The disadvantage of manual inspection is the large amount of time required and the decrease in efficiency as operator fatigue occurs. This has prompted the development of automated inspection systems to speed up the inspection process and increase efficiency. Automated inspection systems typically use visible light, infrared light or X-rays to illuminate the board. These systems require solder joint position information that is provided either by CAD data or by human entry of the position information. This paper describes a preliminary, automated inspection system that finds the solder joints in an X-ray image and inspects them using an artificial neural network (ANN). The identification of solder joints in the gray-scale image is done using image processing techniques; CAD data or manual registration of the solder joints is not required. The image processing techniques also yield binary maps (i.e., black and white images) showing the locations of ICs and other components, which is useful for other diagnostics.<<ETX>>

Nondestructive reverse engineering of trace maps in multilayered PCBs

The focus of this paper is on separating the layers of an unpopulated, two layered circuit card assemblies/printed circuit boards (CCA/PCB) using X-ray stereo imaging. By separating the layers, we mean that our desire is to separate the traces on each layer of any multi-layered CCA. The final end product is envisioned to be an interactive, automated trace reconstruction system which will separate individual trace layers on a multi-layered PCB. Some of the benefits of such a system are: (1) eliminates the need to have prior knowledge of the CCA via examination of its engineering data; (2) the system can be generalized to different types of CCAs which have different number of trace layers, without any modification to the software; (3) allow the examination of each trace layer of a (multi-layered) CCA separately; (4) a non-intrusive technique which would not subject the CCA to any electronic stimulation.

Statistical properties, fixed points, and decomposition with WMMR filters

WMMRmfilters weight the m ordered values in the window with minimum range. If m is not specified, it is assumed to be N + 1 for a window of length 2N + 1. Previous work has demonstrated a subclass of these filters that may be optimized for edge enhancement in that their output converges to the closest perfect edge. In this work it is shown that normalized WMMRmfilters, whose weights sum to unity, are affine equivariant. The concept of the breakpoint of a filter is discussed, and the optimality of median and WMMR filters under the breakpoint concept is demonstrated. The optimality of a WMMRmfilter and of a similar generalized-order-statistic (GOS) filter is demonstrated for various non-LPcriterion, which we call closeness measures. Fixed-point results similar to those derived by Gallagher and Wise (see N.C. Gallagher and G.L. Wise, IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-29, 1981, pp. 1136–1141) for the median filter are derived for order-statistic (OS) and WMMR filters with convex weights (weights that sum to anity and are nonnegative), i.e., we completely classify the fixed points under the assumption of a finite-length signal with constant boundaries. These fixed points are shown to be almost always the class of piecewise-constant (PICO) signals. The use of WMMR filters for signal decomposition and filtering based on the Haar basis is discussed. WMMR filters with window width 2N + 1 are shown to be linear over the PICO(N + 1) signals (minimum constant length N+1). Concepts similar to lowpass, highpass, and bandpass for filtering PICO signals are introduced. Application of the filters to 1-dimensional biological data (non-PICO) and images of printed-circuit boards is then demonstrated, as is application to images in general.

Comments on "The analog median filter" by J.P. Fitch et al

It is pointed out that in the above-mentioned paper (see ibid., vol.33, no.1, p.94-102, 1986), the necessary and sufficient conditions (if any exist) under which the properties of the discrete median filter also apply to the analog median filter remain unclear. >

Application of weighted-majority minimum-range filters in the detection and sizing of tumors in mammograms

In image processing the solution is often unique to the problem. To be more specific, the importance of the filter window and sampling pattern chosen to filter, pass, or enhance a specific shape is very specific to the problem at hand. We detect suspect tumors in mammograms using a weighted majority minimum range filter and different sampling patterns and windows as a demonstration of this fact. Several methods have been developed to automate the process of detecting tumors in mammograms. We show that traditional windowing or sampling methods may be replaced by a hexagonal method that more accurately reflects the geometry of the problem and could improve the techniques already in existence. Several theorems involving a hexagonal filter window are presented, followed by the results of our application to mammograms.