V-Cube 7: Difference between revisions

Revision as of 17:09, 22 January 2014

In mathematical morphology, hit-or-miss transform is an operation that detects a given configuration (or pattern) in a binary image, using the morphological erosion operator and a pair of disjoint structuring elements. The result of the hit-or-miss transform is the set of positions, where the first structuring element fits in the foreground of the input image, and the second structuring element misses it completely.

Mathematical definition

In binary morphology, an image is viewed as a subset of an Euclidean space $ℝ^{d}$ or the integer grid $ℤ^{d}$ , for some dimension d. Let us denote this space or grid by E.

A structuring element is a simple, pre-defined shape, represented as a binary image, used to probe another binary image, in morphological operations such as erosion, dilation, opening, and closing.

Let $C$ and $D$ be two structuring elements satisfying $C \cap D = \emptyset$ . The pair (C,D) is sometimes called composite structuring element. The hit-or-miss transform of a given image A by B=(C,D) is given by:

A ⊙ B = (A ⊖ C) \cap (A^{c} ⊖ D)

,

where $A^{c}$ is the set complement of A.

That is, a point x in E belongs to the hit-or-miss transform output if C translated to x fits in A, and D translated to x misses A (fits the background of A).

Some applications

Thinning

Let $E = Z^{2}$ , and consider the eight composite structuring elements, composed by:

C_{1} = {(0, 0), (- 1, - 1), (0, - 1), (1, - 1)}

and

D_{1} = {(- 1, 1), (0, 1), (1, 1)}

,

C_{2} = {(- 1, 0), (0, 0), (- 1, - 1), (0, - 1)}

and

D_{2} = {(0, 1), (1, 1), (1, 0)}

and the three rotations of each by $9 0^{o}$ , $18 0^{o}$ , and $27 0^{o}$ . The corresponding composite structuring elements are denoted $B_{1}, \dots, B_{8}$ .

For any i between 1 and 8, and any binary image X, define

X \otimes B_{i} = X ∖ (X ⊙ B_{i})

,

where $∖$ denotes the set-theoretical difference.

The thinning of an image A is obtained by cyclically iterating until convergence:

A \otimes B_{1} \otimes B_{2} \otimes \dots \otimes B_{8} \otimes B_{1} \otimes B_{2} \otimes \dots

.

Other applications

Pattern detection. By definition, the hit-or-miss transform indicates the positions where a certain pattern (characterized by the composite structuring element B) occurs in the input image.

Pruning. The hit-or-miss transform can be used to identify the end-points of a line to allow this line to be shrunk from each end to remove unwanted branches.

Computing the Euler number.

Bibliography

An Introduction to Morphological Image Processing by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)

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+<!-- [[File:Giunzioni e endpoint.png|thumb|Binarization hit-or-miss transform and branch points and end points detection]] -->
+In [[mathematical morphology]], '''hit-or-miss transform''' is an operation that detects a given configuration (or pattern) in a [[binary image]], using the morphological [[erosion (morphology)|erosion]] operator and a pair of [[Disjoint sets|disjoint]] [[structuring element]]s. The result of the hit-or-miss transform is the set of positions, where the first [[structuring element]] fits in the foreground of the input image, and the second structuring element misses it completely.
+== Mathematical definition ==
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+In binary morphology, an image is viewed as a [[subset]] of an [[Euclidean space]] <math>\mathbb{R}^d</math> or the integer grid <math>\mathbb{Z}^d</math>, for some dimension ''d''. Let us denote this space or grid by ''E''.
+A structuring element is a simple, pre-defined shape, represented as a binary image, used to probe another binary image, in morphological operations such as [[erosion (morphology)|erosion]], [[dilation (morphology)|dilation]], [[opening (morphology)|opening]], and [[closing (morphology)|closing]].
+Let <math>C</math> and <math>D</math> be two structuring elements satisfying <math>C\cap D=\emptyset</math>. The pair (''C'',''D'') is sometimes called ''composite structuring element''. The hit-or-miss transform of a given image ''A'' by ''B''=(''C'',''D'') is given by:
+::<math>A\odot B=(A\ominus C)\cap(A^c\ominus D)</math>,
+where <math>A^c</math> is the [[set complement]] of ''A''.
+That is, a point ''x'' in ''E'' belongs to the hit-or-miss transform output if ''C'' translated to ''x'' fits in ''A'', and ''D'' translated to ''x'' misses ''A'' (fits the background of ''A'').
+== Some applications ==
+===Thinning===
+Let <math>E=Z^2</math>, and consider the eight composite structuring elements, composed by:
+:<math>C_1=\{(0,0),(-1,-1),(0,-1),(1,-1)\}</math> and <math>D_1=\{(-1,1),(0,1),(1,1)\}</math>,
+:<math>C_2=\{(-1,0),(0,0),(-1,-1),(0,-1)\}</math> and <math>D_2=\{(0,1),(1,1),(1,0)\}</math>
+and the three rotations of each by <math>90^o</math>, <math>180^o</math>, and <math>270^o</math>. The corresponding composite structuring elements are denoted <math>B_1,\ldots,B_8</math>.
+For any ''i'' between 1 and 8, and any binary image ''X'', define
+::<math>X\otimes B_i=X\setminus (X\odot B_i)</math>,
+where <math>\setminus</math> denotes the [[set minus|set-theoretical difference]].
+The thinning of an image ''A'' is obtained by cyclically iterating until convergence:
+:<math>A\otimes B_1\otimes B_2\otimes\ldots\otimes B_8\otimes B_1\otimes B_2\otimes\ldots</math>.
+===Other applications===
+* '''[[Pattern detection]]'''. By definition, the hit-or-miss transform indicates the positions where a certain pattern (characterized by the composite structuring element ''B'') occurs in the input image.
+* '''[[Pruning (morphology)|Pruning]]'''. The hit-or-miss transform can be used to identify the end-points of a line to allow this line to be shrunk from each end to remove unwanted branches.
+* Computing the '''[[Euler number (topology)|Euler number]]'''.
+==Bibliography==
+* ''An Introduction to Morphological Image Processing'' by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)
+[[Category:Mathematical morphology]]
+[[Category:Digital geometry]]

V-Cube 7: Difference between revisions

Revision as of 17:09, 22 January 2014

Contents

Mathematical definition

Some applications

Thinning

Other applications

Bibliography

Navigation menu

V-Cube 7: Difference between revisions

Revision as of 17:09, 22 January 2014

Mathematical definition

Some applications

Thinning

Other applications

Bibliography

Navigation menu

Search