Chapter 9: Morphological Image Processing

Chapter 9: Morphological Image Processing(Digital Image Processing Gonzalez/Woods)prepared by:Ahmed Daoud teaching assistant at FCI ZU developerdaoud@gmail.com

Preview Morphology a branch in biology that deals with the form and structure of animals and plants. Mathematical Morphology as a tool for extracting image components, that are useful in the representation and description of region shape What are the applications of Morphological Image Filtering?boundaries extractionskeletonsconvex hullmorphological filteringthinningPruningThe language of mathematical morphology is Set theory.Unified and powerful approach to numerous image processing problems

Structure Morphology`2

Sets in mathematical morphology represent objects in an image:binary image (0 = black, 1 = white) : the element of the set is the coordinates (x,y) of pixel belong to the object Z2gray-scaled image : the element of the set is the coordinates (x,y) of pixel belong to the object and the gray levels Z3

Image Attributes Extracted from ImageMorphological Image Processing

9.1 Basic Concepts in Set TheorySubset

Union

Intersection

disjoint / mutually exclusive Complement DifferenceReflectionTranslation

Logic Operations Involving Binary Pixels and Images The principal logic operations used in image processing are: AND, OR, NOT (COMPLEMENT).These operations are functionally complete.Logic operations are preformed on a pixel by pixel basis between corresponding pixels (bitwise).Other important logic operations : XOR (exclusive OR), NAND (NOT-AND)Logic operations are just a private case for a binary set operations, such : AND Intersection , OR Union, NOT-Complement.

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Structuring Elements (SE)Structuring Elements(SE): small sets or subimages used to probe an image under study for properties of interest.The origin of the SE is indicated by a black dot.When SE is symmetric and no dot is show the origin is at the center of symmetry.When SE is symmetric B = (reflection of B) When working with images SE must be in a Rectangular Arrays (padding with the smallest possible number of background elements).Images must be in a Rectangular Arrays (padding with the smallest possible number of background elements).For Images a background border is provided to accommodate the entire SE when its origin is on the border.

SE is look like a mask that perform filtering on an image.7

9.2.1 ErosionErosion is used for shrinking of element A by using element BErosion for Sets A and B in Z2, is defined by the following equation:

This equation indicates that the erosion of A by B is the set of all points z such that B, translated by z, is contained in A.Erosion can be used toShrinks or thins objects in binary imagesRemove image components(how?)Erosion is a morphological filtering operation in which image details smaller than the structuring elements are filtered(removed)

9.2.1 Erosion Example 1

Take B and move over all pixels in A . If B is completely contained in A the shade the new pixel, otherwise left it unshaded9

9.2.1 Erosion Example 2

Suppose that the structuring element is a 33 square Note that in subsequent diagrams, foreground pixels are represented by 1's and background pixels by 0's. The structuring element is now superimposed over each foreground pixel ( input pixel ) in the image. If all the pixels below the structuring element are foreground pixels then the input pixel retains its value. But if any of the pixels is a background pixel then the input pixel gets the background pixel value.

111111111Structuring element

Take SE and move over all pixels in A . If SE is completely contained in A then the new pixel is the foreground pixel(1), otherwise it is background pixel(0)

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9.2.2 DilationDilation is used for expanding an element A by using structuring element BDilation of A by B and is defined by the following equation:

This equation is based 0n obtaining the reflection 0f Babout its origin and shifting this reflection by z.The dilation of A by B is the set of all displacements z, such that and A overlap by at least one element. Based On this interpretation the equation of (9.2-1) can berewritten as:

Relation to Convolution mask:- Flipping- Overlapping

9.2.2 Dilation Example 1

9.2.2 Dilation Example 2

Suppose that the structuring element is a 33 square .Note that in subsequent diagrams, foreground pixels are represented by 1's and background pixels by 0's. To compute the dilation of a binary input image by this structuring element, we superimpose the structuring element on top of the input image so that the origin of the structuring element coincides with the input pixel position. If the center pixel in the structuring element coincides with a foreground pixel in the image underneath, then the input pixel is set to the foreground value.

111111111Structuring element

Take SE reflect it and move over all pixels in A . If at least one pixel of the SE coincides a pixel in A then the new pixel is the foreground pixel(1), otherwise it is background pixel(0)15

9.2.2 Dilation A More interesting Example (bridging gaps)

Duality between dilation and erosionDilation and erosion are duals of each other with respect to set complementation and reflection. That,

Prove of (9.2-5)

One of the simplest uses of erosion is for eliminating irrelevant details (in terms of size) from a binary image

Erosion and Dilation summary

Erosion vs. DilationErosion:Shrinks or thins objects in binary imagesRemove image components(how?) Erode away the boundaries of regions of foreground pixels Areas of foreground pixels shrink in size, and holes within those areas become largerDilation:Grows or thickens object in a binary imageBridging gapsFill small holes of sufficiently small size

9.3 Opening And ClosingOpening smoothes contours , eliminates protrusionsClosing smoothes sections of contours, fuses narrow breaks and long thin gulfs, eliminates small holes and fills gaps in contours

These operations are dual to each other

These operations are can be applied few times, but has effect only once

9.3 Opening And ClosingOpening First erode A by B, and then dilate the result by B In other words, opening is the unification of all B objects Entirely Contained in A

9.3 Opening And ClosingClosing First dilate A by B, and then erode the result by B In other words, closing is the group of points, which the intersection of object B around them with object A is not empty

Use of opening and closing for morphological filtering

9.4 The Hit-or-Miss TransformationA basic morphological tool for shape detection.Let the origin of each shape be located at its center of gravity.If we want to find the location of a shape , say X , at (larger) image, say A :Let X be enclosed by a small window, say W.The local background of X with respect to W is defined as the set difference (W - X).Apply erosion operator of A by X, will get us the set of locations of the origin of X, such that X is completely contained in A. It may be also view geometrically as the set of all locations of the origin of X at which X found a match (hit) in A.

9.4 The Hit-or-Miss TransformationCont.Apply erosion operator on the complement of A by the local background set (W X).Notice, that the set of locations for which X exactly fits inside A is the intersection of these two last operators above. This intersection is precisely the location sought.Formally:If B denotes the set composed of X and its background B = (B1,B2) ; B1 = X , B2 = (W-X).The match (or set of matches) of B in A, denoted is:

Hit-or-Miss exp:

9.4 The Hit-or-Miss TransformationThe reason for using these kind of structuring element B = (B1,B2) is based on an assumed definition that,two or more objects are distinct only if they are disjoint (disconnected) sets.In some applications , we may interested in detecting certain patterns (combinations) of 1s and 0s. and not for detecting individual objects.In this case a background is not required. and the hit-or-miss transform reduces to simple erosion.This simplified pattern detection scheme is used in some of the algorithms for identifying characters within a text.

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9.5 Basic Morphological Algorithms(Applications) 1 Boundary Extraction2 Region Filling3 Extraction of Connected Components4 Convex Hull5 Thinning6 Thickening7 Skeletons8 Pruning

Convex Hull = frame 30

9.5.1 Boundary ExtractionFirst, erode A by B, then make set difference between A and the erosionThe thickness of the contour depends on the size of constructing object B

9.5.1 Boundary Extraction

9.5.2 Region FillingThis algorithm is based on a set of dilations, complementation and intersectionsp is the point inside the boundary(given) , with the value of 1 The process stops when The result that given by union of A and X(k), is a set contains the filled set and the boundary

9.5.2 Region Filling

9.5.3 Extraction of Connected ComponentsThis algorithm extracts a component by selecting a point on a binary object AWorks similar to region filling, but this time we use in the conjunction the object A, instead of its complement

This shows automated inspection of chicken-breast, that contains bone fragment

The original image is thresholded

We can get by using this algorithm the number of pixels in each of the connected components

Now we could know if this food contains big enough bones and prevent hazards

9.5.4 Convex HullA is said to be convex if a straight line segment joining any two points in A lies entirely within AThe convex hull H of set S is the smallest convex set containing SConvex deficiency is the set difference H-SUseful for object descriptionThis algorithm iteratively: Apply the hit-or-miss transform to A with the first B elementUnions it with ARepeat with second B element

Let The convex hull is :

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9.5.5 Thinning The thinning of a set A by a structuring element B, can be defined by terms of the hit-and-miss transform: 9.5-6A more useful expression for thinning A symmetrically is based on a sequence of structuring elements:{B}={B1, B2, B3, , Bn}Where Bi is a rotated version of Bi-1. Using this concept we define thinning by a sequence of structuring elements:

9.5.5 Thinning contThe process is to thin by one pass with B1 , then thin the result with one pass with B2, and so on until A is thinned with one pass with Bn. The entire process is repeated until no further changes occur.Each pass is preformed using the equation:

9.5.5 Thinning example

9.5.6 ThickeningThickening is a morphological dual of thinning.Definition of thickening .As in thinning, thickening can be defined as a sequential operation:

the structuring elements used for thickening have the same form as in thinning, but with all 1s and 0s interchanged.

9.5.6 Thickening - contA separate algorithm for thickening is often used in practice, Instead the usual procedure is to thin the background of the set in question and then complement the result.

In other words, to thicken a set A, we form C=Ac , thin C and than form Cc.

depending on the nature of A, this procedure may result in some disconnected points. Therefore thickening by this procedure usually require a simple post-processing step to remove disconnected points.

9.5.6 Thickening example previewWe will notice in the next example 9.22(c) that the thinned background forms a boundary for the thickening process, this feature does not occur in the direct implementation of thickening

This is one of the reasons for using background thinning to accomplish thickening.

9.5.6 Thickening example

9.5.7 SkeletonThe notion of a skeleton of a set A :S(A) is intuitively defined, we deduce from this figure that:

If z is a point of S(A) and (D)z is the largest disk centered at z and contained in A (one cannot find a larger disk that fulfils this terms) this disk is called maximum disk.

The disk (D)z touches the boundary of A at two or more different places.

9.5.7 SkeletonThe skeleton of A is defined by terms of erosions and openings:

withWhere B is the structuring element and indicates k successive erosions of A:

k times, and K is the last iterative step before A erodes to an empty set, in other words:in conclusion S(A) can be obtained as the union of skeleton subsets Sk(A).

9.5.7 Skeleton Example

9.5.7 SkeletonA can be also reconstructed from subsets Sk(A) by using the equation:

Where denotes k successive dilations of Sk(A) that is:

9.5.8 PruningRemoving parasitic component Complement to thinning and skeletonizing by successive elimination its end point.

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Morphological ReconstructionInvolve two images and SE Marker(F): starting point for transformation.Mask(G): constrains the transformation.SE: define connectivity.F and G are both binary images and

Geodesic Dilation & Erosion

Morphological reconstruction by dilation & erosionMorphological reconstruction by dilation of a mask image G from a marker image F

Morphological reconstruction by erosion of a mask image G from a marker image F

Example:

Sample Applications(1) Opening by reconstruction:Restore exactly the shapes of the objects that remain after erosion.The opening by reconstruction of size n of an image F is the reconstruction by dilation of F from the erosion of size n of F:

(2) Filling holesIf I(x,y) is a binary image A marker image F is defined:

is a binary image equal to I with all holes filled.

Sample Apps. Cont.(3) Border clearing:Removing objects that touch (are connected to)the borderSubtract from original image

Summary

9.6 Gray-Scale Imagesf(x,y): the input imageb(x,y): a structuring element (a subimage function)(x,y) are integers.f(x,y) and b(x,y) are functions that assign a gray-level value (real number or real integer) to each distinct pair of coordinate (x,y)For example the domain of gray values can be 0-255, whereas 0 is black, 255- is white.

All what we have seen are applied to binary images. Now lets start dealing with gray-scaled images63

9.6 Gray-Scale Images

9.6.1 Erosion Gray-ScaleThe erosion of f by a flat structuring element b at any location (x,y) is defined as the minimum value of the image in the region coincides with b when the origin of b is at (x,y).

The erosion of f by a non flat structuring element :

9.6.1 Erosion Gray-Scale example 1

9.6.1 Erosion Gray-Scale (cont)General effect of performing an erosion in grayscale images:If all elements of the structuring element are positive, the output image tends to be darker than the input image.

The effect of bright details in the input image that are smaller in area than the structuring element is reduced, with the degree of reduction being determined by the grayscale values surrounding by the bright detail and by shape and amplitude values of the structuring element itself.

9.6.1 Dilation Gray-ScaleThe Dilation of f by a flat structuring element b at any location (x,y) is defined as the maximum value of the image in the region outlined by when the origin of is at (x,y).

The Dilation of f by a non flat structuring element :

9.6.1 Dilation Gray-Scale (cont)The general effects of performing dilation on a gray scale image is twofold:

If all the values of the structuring elements are positive than the output image tends to be brighter than the input.

Dark details either are reduced or eliminated, depending on how their values and shape relate to the structuring element used for dilation

9.6.1 Dilation Gray-Scale example

9.6.1 Dilation & Erosion Gray-ScaleSimilar to binary image grayscale erosion and dilation are duals with respect to function complementation and reflection.

9.6.2 Opening And ClosingSimilar to the binary algorithmsOpening Closing In the opening of a gray-scale image, we remove small light details, while relatively undisturbed overall gray levels and larger bright featuresIn the closing of a gray-scale image, we remove small dark details, while relatively undisturbed overall gray levels and larger dark features

9.6.2 Opening And ClosingOpening a G-S picture is describable as pushing object B under the scan-line graph, while traversing the graph according the curvature of B

9.6.2 Opening And ClosingClosing a G-S picture is describable as pushing object B on top of the scan-line graph, while traversing the graph according the curvature of BThe peaks are usuallyremains in their original form

9.6.2 Opening And Closing

Opening: Decreased sizeof small bright details. Nochanges to dark regionClosing: Decreased sizeof small dark details. Nochanges to bright region

9.6.3 Some Applications of Gray-Scale MorphologyMorphological smoothingMorphological gradientTop-hat & bottom-hat transformationTextural segmentationGranulometry

9.6.3 Some Applications of Gray-Scale MorphologyMorphological smoothingPerform opening followed by a closingThe net result of these two operations is to remove or attenuate both bright and dark artifacts or noise.

Morphological gradientDilation and erosion are use to compute the morphological gradient of an image, denoted g:

It uses to highlights sharp gray-level transitions in the input image.Obtained using symmetrical structuring elements tend to depend less on edge directionality.

9.6.3 Some Applications of Gray-Scale MorphologyMorphological smoothing

Morphological gradient

9.6.3 Some Applications of Gray-Scale MorphologyTop-hat transformationCalled (white top-hat) and Is defined as:

Detect structures of a certain size.Light objects on a dark background.Bottom-hat transformationCalled (black top-hat )and Is defined as:

Detect structures of a certain size.Dark objects on a bright background

9.6.3 Some Applications of Gray-Scale MorphologyTop-hat transformation

Bottom-hat transformation

9.6.3 Some Applications of Gray-Scale MorphologyTextural segmentationThe objective is to find the boundary between different image regions based on their textural content.Performing closing with successive larger structuring element till removing the small blobsPerforming Simple openingPerforming Simple thresholding

9.6.3 Some Applications of Gray-Scale MorphologyGranulometryGranulometry is a field that deals principally with determining the size distribution of particles in an image.Because the particles are lighter than the background, we can use a morphological approach to determine size distribution. To construct at the end a histogram of it.Based on the idea that opening operations of particular size have the most effect on regions of the input image that contain particles of similar size.This type of processing is useful for describing regions with a predominant particle-like character.

9.6.3 Some Applications of Gray-Scale MorphologyGranulometry

9.6.4 Gray-Scale Morphological ReconstructionLet f and g denote the marker and mask images respectively and The geodesic erosion of size 1 of f with respect to g:

where V denotes the point-wise maximum operatorThe geodesic erosion of size n of f with respect to g:

WithReconstruction by erosion of g by f:

9.6.4 Gray-Scale Morphological ReconstructionLet f and g denote the marker and mask images respectively and The geodesic dilation of size 1 of f with respect to g:

where ^ denotes the point-wise minimum operatorThe geodesic dilation of size n of f with respect to g:

With Reconstruction by dilation of g by f:

9.6.4 Gray-Scale Morphological ReconstructionOpening by reconstruction of size n of an image f:

Closing by reconstruction of size n of an image f:

Top-hat by reconstruction:Subtracting from an image opening by reconstruction

The End!