Optimal mean-absolute-error hit-or-miss filters: morphological representation and estimation of the binary conditional expectation

Dougherty, Edward R.; Loce, Robert P.

doi:10.1117/12.132376

Cited by 54 publications

(30 citation statements)

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“…Using that representation in the digital setting (see [16]), Dougherty and Loce [5], [6] use statistical optimization in conjunction with logic reduction to construct optimal MAE binary filters of the form…”

Section: [E F ~] = {G:e C G C Fc} a Subset Of P Is Called The Closmentioning

confidence: 99%

“…Logic reduction can be applied to reduce the number of terms in the expansion. A key part of [5] concerns thinning and thickening filters. A thinning filter is given by a set subtraction I-~, where I is the identity mapping and • is a hit-or-miss union.…”

Section: [E F ~] = {G:e C G C Fc} a Subset Of P Is Called The Closmentioning

confidence: 99%

“…Morphological thinning and thickening have been characterized in terms of it [1], and numerous other enhancement/restoration algorithms employ it. More recently, Dougherty and Loce [5], [6] have used the hit-or-miss paradigm to characterize optimal mean-absolute-error (MAE) translation-invariant, binary-image filters, in essence providing morphological estimation of the binary conditional expectation. Since binary and gray-scale morphology are closely related by means of the umbra transform, it is natural to examine filtration of gray-scale images by the hit-or-miss transform applied to image umbrae.…”

Section: Introductionmentioning

confidence: 99%

“…Given a degraded version g of an uncorrupted random signal f, we desire to find a filter g' such that g'(g) provides an optimal-MAE estimate of f. The approach employed herein is to find a binary-signal filter ~ to apply to the umbra U [g] ficulty with the approach rests on the fact that a direct construction of • based on the methods of [5] is not possible since we must require that • be umbra preserving; indeed, the main proposition of the paper concerns necessary and sufficient conditions for certain operators to be umbra-preserving.…”

Section: Introductionmentioning

confidence: 99%

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Optimal mean-absolute-error filtering of gray-scale signals by the morphological hit-or-miss transform

Dougherty

1994

J Math Imaging Vis

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Abstract. The binary hit-or-miss transform is applied to filter digital gray-scale signals. This is accomplished by applying a union of hit-or-miss transforms to an observed signal's umbra and then taking the surface of the filtered umbra as the estimate of the ideal signal. The hit-or-miss union is constructed to provide the optimal mean-absolute-error filter for both the ideal signal and its umbra. The method is developed in detail for thinning hit-or-miss filters and applies at once to the dual thickening filters. It requires the output of the umbra filter to be an umbra, which in general is not true. A key aspect of the paper is the complete characterization of umbra-preserving union-ofhit-or-miss thinning and thickening filters. Taken together, the mean-absolute-error theory and the umbra-preservation characterization provide a full characterization of binary hit-or-miss filtering as applied to digital gray-scale signals. The theory is at once applicable to hit-or-miss filtering of digital gray-scale images through the three-dimensional (3-D) binary hit-or-miss transform.

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Section: [E F ~] = {G:e C G C Fc} a Subset Of P Is Called The Closmentioning

confidence: 99%

Section: [E F ~] = {G:e C G C Fc} a Subset Of P Is Called The Closmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimal mean-absolute-error filtering of gray-scale signals by the morphological hit-or-miss transform

Dougherty

1994

J Math Imaging Vis

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“…For n observation binary random variables X 1 , X 2 ,..., X n , binary random variable Y, and Boolean function f(X 1 , X 2 ,..., X n ) to estimate Y, the mean-absolute error (MAE) of f is defined by the expected value Given that f possesses a logical sum-of-products representation, an optimal choice of f is determined by finding the representation providing minimal error. The most general approach is to assume a disjunctive-normal-form representation, find the minterms that result in minimal MAE, and then apply logic reduction to find the optimal Boolean function [5][6][7]. The image filter, ⌿, defined by the optimal Boolean function then provides the optimal image filter and we define its error by MAE〈⌿〉 = MAE〈f〉.…”

Section: Vector-valued Mappingmentioning

confidence: 99%

Logically Efficient Spatial Resolution Conversion Using Paired Increasing Operators

et al. 1997

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Foundations of Morphological Image Processing

Dougherty

2002

Encyclopedia of Imaging Science and Technology

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The basic morphological operations of erosion, dilation, opening, and closing are a part of every image processing toolbox. So too are many basic algorithms such as the watershed and top‐hat transforms. This article reviews the fundamental algebraic theory of mathematical morphology as applied to Euclidean set theory, the context in which it was originally conceived by Matheron and Serra from mathematical operations dating back to Minkowski and Hadwiger. Thorough knowledge of basic algebraic theory is required for anyone engaged in serious research in morphological image processing and is very beneficial for anyone interested in advanced applications. Since its original set‐theoretic formulation, mathematical morphology has been extended to real‐valued functions and to lattice‐valued functions. These extensions take note of the fact that the operations and propositions of the algebraic theory depend on order relations. This can be seen in the direct generalization of many binary definitions and properties to lattices based on the partial order relation among sets. In many cases, the definitions in this article are extended to lattices (including real‐valued and integral‐valued functions) by changing the subset relation ⊂ to the order relationship ≤. This article restricts attention to binary morphology.

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Optimal mean-absolute-error hit-or-miss filters: morphological representation and estimation of the binary conditional expectation

Cited by 54 publications

References 1 publication

Optimal mean-absolute-error filtering of gray-scale signals by the morphological hit-or-miss transform

Optimal mean-absolute-error filtering of gray-scale signals by the morphological hit-or-miss transform

Logically Efficient Spatial Resolution Conversion Using Paired Increasing Operators

Foundations of Morphological Image Processing

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