MATHEMATICS OF BINARY MORPHOLOGY and APPLICATIONS IN Vision.
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Transcript of MATHEMATICS OF BINARY MORPHOLOGY and APPLICATIONS IN Vision.
MATHEMATICS OF MATHEMATICS OF BINARY BINARY
MORPHOLOGY MORPHOLOGY and and
APPLICATIONS IN APPLICATIONS IN VisionVision
In general; what is “Morphology”?
• The science of form and structure– the science of form, that of the outer form, inner structure, and
development of living organisms and their parts
– about changing/counting regions/shapes
• Among other applications it is used to pre- or post-process images– via filtering, thinning and pruning
• Smooth region edges– create line drawing of face
• Force shapes onto region edges– curve into a square • Count regions (granules)
– number of black regions
• Estimate size of regions– area calculations
What is Morphology in computer vision ?
1. Morphology generally concerned with shape and properties of objects.
2. Used for segmentation and feature extraction.
3. Segmentation = used for cleaning binary objects.
4. Two basic operations1. erosion (opening)2. dilation (closing)
Morphological operations and algebras
1. Different definitions in the textbooks
2. Different implementations in the image processing programs.
3. The original definition, based on set theory, is made by J. Serra in 1982.
4. Defined for binary images - binary operations (boolean, set-theoretical)
5. Can be used on grayscale images - multiple-valued logic operations
Morphological operations on a PC
• Various but slightly different implementations in
• Scion
• Paint Shop Pro
• Adope Photoshop
• Corel Photopaint
• mmTry them, it is a lot of fun
Binary Morphology• Morphological operators are used to prepare
binary (thresholded) images for object segmentation/recognition– Binary images often suffer from noise (specifically salt-
and-pepper noise)
– Binary regions also suffer from noise (isolated black pixels in a white region). Can also have cracks, picket fence occlusions, etc.
• Dilation and erosion are two binary morphological operations that can assist with these problems.
Goals of morphological operations:
1. Simplifies image data2. Preserves essential shape characteristics3. Eliminates noise4. Permits the underlying shape to be identified
and optimally reconstructed from their distorted, noisy forms
What is the mathematical morphology ?
1. An approach for processing digital image based on its shape
2. A mathematical tool for investigating geometric structure in image
The language of morphology is set theory.
Mathematical morphology is extension to set theory.
Importance of Shape in Processing and Analysis
Shape is a prime carrier of information in machine vision
For instance, the following directly correlate with shape:identification of objectsobject featuresassembly defects
Shape Operators Shapes are usually combined by means of :
X X X Xc2 1 1 2\
X2X1
• Set Difference based on Set intersection (occluded objects):
X X1 2X1 X2
• Set Union (overlapping objects):
Set difference
X2X1
Set intersection
Morphological Operations based on Morphological Operations based on combining base operationscombining base operations
The primary morphological operations are dilation and erosion
More complicated morphological operators can be designed by means of combining erosions and dilations
Let us illustrate them and explain how to combine
We will use combinations of union, complement, intersection, erosion, dilation, translation...
Libraries of Structuring Libraries of Structuring ElementsElements
• Application specific structuring elements created by the user
X
B
No necessarily compactnor filled
A special set :the structuring element
-2 -1 0 1 2
-2 -1 0 1 2
Origin at center in this case, but not necessarily centered nor symmetric
NotationNotation
x
y
3*3 structuring element, see next slide how it works
Dilation : x = (x1,x2) such that if we center B on them, then the so translated B intersects X.
X
B
difference
Explanation of DilationExplanation of Dilation
dilation
Notation for Dilation Notation for Dilation
Dilation : x = (x1,x2) such that if we center B on them, then the so translated B intersects X.
How to formulate this definition ?
1) Literal translation
2) Better : from Minkowski’s sum of sets
Mathematical definition of dilation
Another Mathematical definition of dilation uses the concept of Minkowski’s sum
BB is ingeneral not the same as B
Minkowski’s Sum Minkowski’s Sum
Definition of Minkowski’s sum of sets S and B :
l
l
Minkowski’s Minkowski’s Sum Sum
Comparison of Dilation and Comparison of Dilation and Minkowski sum Minkowski sum
Dilation :Dilation :
Minkowski sum Minkowski sum
Bx =
x and b are points
l
Dilation is Dilation is notnot the Minkowski’s sum the Minkowski’s sum
Minkowski’s Minkowski’s Sum Sum
l
l
bbbb l
Dilation is Dilation is notnot the Minkowski’s sum the Minkowski’s sum
DilationDilation
DilationDilation
BB is not the same as B
Dilation vs SE • Erosion shrinks• Dilation expands binary regions
• Can be used to fill in gaps or cracks in binary images
• If the point at the origin of the structuring element is set in the underlying image, then all the points that are set in the SE are also set in the image
•Basically, its like OR’ing the SE into the image
structuring Element ( SE )
Dilation fills holes
• Fills in holes.
• Smoothes object boundaries.
• Adds an extra outer ring of pixels onto object boundary, ie, object becomes slightly larger.
Possible problems with Morphological Operators
• Erosion and dilation clean image but leave objects either smaller or larger than their original size.
• Opening and closing perform same functions as erosion and dilation but object size remains the same.
Dilation explained pixed by pixel
•
••
•
•
••
•••
••
••
••B
A BA
Denotes origin of B i.e. its (0,0)
Denotes origin of A i.e. its (0,0)
Dilation explained by shape of A
•
••
•
•
••
•••
••
••
••B
A BA
Shape of A repeated without shift
Shape of A repeated with shift
Properties of Dilation
• 1. fills in valleys between spiky regions
• 2. increases geometrical area of object
• 3. sets background pixels adjacent to object's contour to object's value
• 4. smoothes small negative grey level regions
Dilation does the following:
objects are light (white in binary)
Image Structuring Element Result
Structuring Element for DilationStructuring Element for Dilation
Length 5Length 6
Image Structuring Element Result
Structuring Element for DilationStructuring Element for Dilation
Single point in Image replaced with this in the Result
Definition of Dilation: Mathematically
Dilation is the operation that combines two sets using vector addition of set elements.
Let A and B are subsets in 2-D space. A: image undergoing analysis, B: Structuring element, denotes dilation
},{ 2 BbAasomeforbacZcBA
Dilation versus translation
Let A be a Subset of and . The translation of A by x is defined as:
The dilation of A by B can be computed as the union of translation of A by the elements of B
2Z2Zx
},{)( 2 AasomeforxacZcA x
Aa
aBb
b BABA
)()(
x is a vector
Dilation versus translation, illustrated
•
••
•
•
•
••
•
•
••
•••
••
••
)0,0(A )1,0(A
BA
•• B
Shift vector (0,0)
Shift vector (0,1)
Element (0,0)
Dilation using Union Formula
xB)(
BA
A
Aa
aBb
b BABA
)()(
Center of the circle
This circle will create one point
This circle will create no point
Example of Dilation with various sizes of structuring
elements
Pablo Picasso, Pass with the Cape, 1960
StructuringElement
Mathematical Properties of Dilation
Commutative
Associative
Extensivity
Dilation is increasing
BAABif ,0
DBDAimpliesBA
ABBA
CBACBA )()( Illustrated in next slide
Illustration of Extensitivity of Dilation
•
•
•
•
••
••
••
••
B
ABA
••
BAABif ,0
Here 0 does not belong to B and A is not included in A B
Replaced with
More Properties of Dilation
Translation Invariance
Linearity
Containment
Decomposition of structuring element
xx BABA )()(
)()()( CBCACBA
)()()( CBCACBA
)()()( CABACBA
Dilation
1. The dilation operator takes two pieces of data as input
1. A binary image, which is to be dilated
2. A structuring element (or kernel), which determines the behavior of the morphological operation
2. Suppose that X is the set of Euclidean coordinates of the input image, and K is the set of coordinates of the structuring element
3. Let Kx denote the translation of K so that its origin is at x.
4. The DILATION of X by K is simply the set of all points x such that the intersection of Kx with X is non-empty
DilationExample: Suppose that the structuring element is a 3x3 square with the origin at its center
{ (-1,-1), (0,-1), (1,-1), (-1,0), (0,0), (1,0), ( 1,1), (0,1), (1,1) }
111
111
111
X =
K =
Dilation
Example: Suppose that the structuring element is a 3x3 square with the origin at its center
Note: Foreground pixels are represented by a color and background pixels are empty
Erosion : x = (x1,x2) such that if we center B on them, then the so translated B is contained in X.
Example of ErosionExample of Erosion
difference
erosion
Notation for ErosionNotation for Erosion
2) Better : from Minkowski’s substraction of sets
Erosion : x = (x1,x2) such that if we center B on them, then the so translated B is contained in X.
How to formulate this definition ?
1) Literal translation
ErosionErosion
Minkowski’s substraction
BINARY MORPHOLOGY Notation for ErosionNotation for Erosion
ErosionErosion
Minkowski’s substraction of sets
Did not belong to X
Erosion with other structuring elements
When the new SE is included in old SE then a larger area is created
Common structuring elements Common structuring elements shapesshapes
= origin
x
y
Note that here :
circledisk
segments 1 pixel wide
points
Problem in BINARY MORPHOLOGY Problem in BINARY MORPHOLOGY using Minkowski Sum using Minkowski Sum
First we calculate the operation in parentheses to obtain a diamond
Implementation of dilation: Implementation of dilation: very low computational costvery low computational cost
0
1 (or >0)
Logical or
Erosion1. (Minkowski subtraction)
2. The contraction of a binary region (aka, region shrinking)
3. Use a structuring element on binary image data to produce a new binary image
4. Structuring elements (SE) are simply patterns that are matched in the image
5. It is useful to explain operation of erosion and dilation in different ways.
Typical Uses of Erosion
1. Removes isolated noisy pixels.
2. Smoothes object boundary.
3. Removes the outer layer of object pixels, ie, object becomes slightly smaller.
Properties of Erosion:
• Erosion removes spiky edges– objects are light (white in binary)
– decreases geometrical area of object
– sets contour pixels of object to background value
– smoothes small positive grey level regions
How It Works?
• During erosion, a pixel is turned on at the image pixel under the structuring element origin only when the pixels of the structuring element match the pixels in the image
• Both ON and OFF pixels should match.
• This example erodes regions horizontally from the right.
Image Structuring Element Result
Structuring Element in Erosion Structuring Element in Erosion ExampleExample
Image Structuring Element Result
Structuring Element in Erosion Structuring Element in Erosion ExampleExample
Image Structuring Element Result
Structuring Element in Erosion Structuring Element in Erosion ExampleExample
Image Structuring Element Result
Structuring Element in Erosion Structuring Element in Erosion ExampleExample
Image Structuring Element Result
Structuring Element in Erosion Structuring Element in Erosion ExampleExample
Image Structuring Element Result
Structuring Element in Erosion Structuring Element in Erosion ExampleExample
Mathematical Definition of Erosion
1. Erosion is the morphological dual to dilation.
2. It combines two sets using the vector subtraction of set elements.
3. Let denotes the erosion of A by BBA
){
}..,{2
2
BbeveryforAbxZx
baxtsAaanexistBbeveryforZxBA
Erosion in terms of other operations:
Erosion can also be defined in terms of translation
In terms of intersection
))({ 2 ABZxBA x
Bb
bABA
)(
Observe that vector here is negative
Erosion illustrated in terms of intersection and negative translation
•
•
•
•
••••
••
BA
• •••
•
•
•
•
••••)1,0(1A )0,0(A
Observe negative translation
Because of negative shift the origin is here
Erosion formula and intuitive example
xB)(
A
BA
))({ 2 ABZxBA x
Center of B is here and adds a point
Center here will not add a point to the Result
Pablo Picasso, Pass with the Cape, 1960
Structuring
Element
Example of Erosions Example of Erosions with various sizes of with various sizes of structuring elementsstructuring elements
Properties of Erosion
Erosion is not commutative!
Extensivity
Erosion is dereasing
Chain rule
ABBA
ABABif ,0
)...)(...()...( 11 kk BBABBA
CABAimpliesCBBCBAimpliesCA ,
The chain rule is as sharp operator in Cube Calculus used in logic synthesis. There are more similarities of these algebras
Properties of Erosion Translation Invariance
Linearity
Containment
Decomposition of structuring element
)()()( CBCACBA
)()()( CBCACBA
)()()( CABACBA
xxxx BABABABA )(,)(
Duality Relationship between erosion and dilationDuality Relationship between erosion and dilation
Dilation and Erosion transformation bear a marked similarity, in that what one does to image foreground and the other does for the image background.
, the reflection of B, is defined as
Erosion and Dilation Duality Theorem
2ZBB
},{ bxBbsomeforxB
BABA cc )(
Observe negative value which is mirror image reflection of B
Similar but not identical to De Morgan rule in Boolean Algebra
Erosion as Dual of Dilation
• Erosion is the dual of dilation
– i.e. eroding foreground pixels is equivalent to dilating the background pixels.
• Easily visualized on binary image• Template created with known origin
• Template stepped over entire image– similar to correlation
• Dilation– if origin == 1 -> template unioned
– resultant image is large than original
• Erosion– only if whole template matches image
– origin = 1, result is smaller than original
1 *1 1
Duality Relationship between erosion and dilation
Another look at duality
One more view at Erosion with One more view at Erosion with examplesexamples
1. To compute the erosion of a binary input image by the structuring element
2. For each foreground pixel superimpose the structuring element
3. If for every pixel in the structuring element, the corresponding pixel in the image underneath is a foreground pixel, then the input pixel is left as it is
4. Otherwise, if any of the corresponding pixels in the image are background, however, the input pixel is set to background value
Erode and Dilate in terms of more general neighborhoods
Yet another loook at Duality Relationship Yet another loook at Duality Relationship between erosion and dilationbetween erosion and dilation
Erode and Binary Contour in Matlab
Erosion can be used to find contour
Dilation can be also used for it - think how?
Edge detectionDilate - original This subtraction is set
theoretical
Now you need to invert the image
There are more methods for edge detection
Opening & Closing
1. Opening and Closing are two important operators from mathematical morphology
2. They are both derived from the fundamental operations of erosion and dilation
3. They are normally applied to binary images
Open and CloseClose = Dilate next ErodeOpen = Erode next Dilate
OpenClose
Original image dilated
eroded
dilated
eroded
Opening :
OPENING OPENING
Supresses :• small islands• ithsmus (narrow unions)• narrow caps
difference
also
OPENING OPENING
Open• An erosion followed by a dilation
• It serves to eliminate noise
• Does not significantly change an object’s size
Comparison of Opening and Erosion
1. Opening is defined as an erosion followed by a dilation using the same structuring element
2. The basic effect of an opening is similar to erosion
3. Tends to remove some of the foreground pixels from the edges of regions of foreground pixels
4. Less destructive than erosion
5. The exact operation is determined by a structuring element.
Opening Example
Original• What combination of
erosion and dilation gives:– cleaned binary image
– object is the same size as in original
Opening Example Cont
• Erode original image.
• Dilate eroded image.
• Smooths object boundaries, eliminates noise (isolated pixels) and maintains object size.
DilateOriginal Erode
One more example of Opening1. Erosion can be used to eliminate small clumps of
undesirable foreground pixels, e.g. “salt noise”
2. However, it affects all regions of foreground pixels indiscriminately
3. Opening gets around this by performing both an erosion and a dilation on the image
Closing :
EXAMPLE OF CLOSING EXAMPLE OF CLOSING
Supresses :• small lakes (holes)• channels (narrow separations)• narrow bays
also
BINARY MORPHOLOGY Closing previous image with other Closing previous image with other structuring elementsstructuring elements
With bigger rectangle like this
With smaller cross like this
Application : shape smoothing and noise filteringApplication : shape smoothing and noise filtering
Papilary lines recognition
Application : Application : segmentation of microstructures (Matlab Help)segmentation of microstructures (Matlab Help)
original
negated
threshold
disk radius 6
closing
opening
and withthreshold
PROPERTIES IN BINARY MORPHOLOGY PROPERTIES IN BINARY MORPHOLOGY
Properties
• all of them are increasing :
• opening and closing are idempotent :
• dilation and closing are extensive operations• erosion and opening are anti-extensive operations:
EXTENSIVE VERSUS ANTI-EXTENSIVE EXTENSIVE VERSUS ANTI-EXTENSIVE OPERATIONS OPERATIONS
DUALITIES OF MORPHOLOGICAL OPERATORS DUALITIES OF MORPHOLOGICAL OPERATORS
• duality of erosion-dilation, opening-closing,...
Decomposition of structuring elements Decomposition of structuring elements
operations with big structuring elements can be doneby a succession of operations with small s.e’s
Hit-or-miss :
HIT-OR-MISSHIT-OR-MISS
“Hit” part(white)
“Miss” part(black)
Bi-phase structuring element
Looks for pixel configurations :
HIT or MISS FOR ISOLATED POINTSHIT or MISS FOR ISOLATED POINTS
doesn’t matter
background
foreground
Close • Dilation followed by erosion
• Serves to close up cracks in objects and holes due to pepper noise
• Does not significantly change object size
More examples of Closing
Original• What combination of
erosion and dilation gives:– cleaned binary image
– object is the same size as in original
More examples of Closing cont
• Dilate original image.
• Erode dilated image.
• Smooths object boundaries, eliminates noise (holes) and maintains object size.
ErodeDilateOriginal
Closing as dual to Opening
1. Closing, like its dual operator opening, is derived from the fundamental operations of erosion and dilation.
2. Normally applied to binary images
3. Tends to enlarge the boundaries of foreground regions
4. Less destructive of the original boundary shape
5. The exact operation is determined by a structuring element.
Closing is opening in revers
• Closing is opening performed in reverse.
• It is defined simply as a dilation followed by an erosion using the same
Mathematical Definitions of Opening and Closing
Opening and closing are iteratively applied dilation and erosion
Opening
Closing
BBABA )(
BBABA )(
Opening and Closing are Opening and Closing are idempotentidempotent
Their reapplication has not further effects to the previously transformed result
BBABA )(
BBABA )(
Properties of Opening and Closing Translation invariance
Antiextensivity of opening
Extensivity of closing
Duality
BABA x )( BABA x )(
ABA
BAA
BABA cc )(
Pablo Picasso, Pass with the Cape, 1960
StructuringElement
Example of Openings Example of Openings with various sizes of with various sizes of structuring elementsstructuring elements
Example of Closing
StructuringElement
Example of Closings Example of Closings with various sizes of with various sizes of structuring elementsstructuring elements
Thinning :
Thinning and ThickeningThinning and Thickening
Thickenning :
Depending on the structuring elements (actually, seriesof them), very different results can be achieved :
• Prunning• Skeletons• Zone of influence• Convex hull• ...
Prunning at 4 connectivity : remove end points by a sequence of thinnings
Prunning at 4 connectivityPrunning at 4 connectivity
1 iteration =
This point is removed with dark green neighbors
IDEMPOTENCE shown as a result of thinning IDEMPOTENCE shown as a result of thinning
1st iteration
2nd iteration 3rd iteration: idempotence
Contours of binary regions :
USING EROSION TO FIND CONTOURSUSING EROSION TO FIND CONTOURS
difference erosion
Contour found with larger mask
CONTOURS with different connectivity patternsCONTOURS with different connectivity patterns
4-connectivity
8-connectivitycontour
8-connectivity
4-connectivitycontour
Important for perimeter computation.
ii. Convex hull : union of thickenings, each up to idempotence
Use of thickening: Use of thickening: Convex hullConvex hull
Original shaper
Thickening with first mask
Union of four thickenings
iii. Skeleton :iii. Skeleton :
Maximal disk : disk centered at x, Dx, such that Dx X and no other Dy contains it .
Skeleton : union of centers of maximal disks.
PROBLEMS with skeletonsPROBLEMS with skeletons
Problems :
• Instability : infinitessimal variations in the border of X cause large deviations of the skeleton
• not necessarily connex even though X connex
• good approximations provided by thinning with special series of structuring elements
Thinning Thinning with thickeningwith thickening 20 iterations
of thinning color white
40 iterationsThickening color whiteSome sort of
region clustering
BINARY MORPHOLOGY
Application : skeletonization for OCR by graph matching
Skeletonization for OCR Skeletonization for OCR
skeletonization vectorization
Application : skeletonization for OCR by graph matching
skeletonizationskeletonization
and 3 rotations
Hit-or-Miss
Calculation of Geodesic zones of Calculation of Geodesic zones of influence (GZI)influence (GZI)
1. X set of n connex components {Xi}, i=1..n .2. The zone of influence of Xi , Z(Xi) , is the set of points closer
to some point of Xi than to a point of any other component. 3. Also, Voronoi partition.4. Dual to skeleton.
Calculating and using Geodesic Zones of InfluenceCalculating and using Geodesic Zones of Influence
thr erosion7x7
GZI opening5x5
and
thr erosion7x7
dist
Calculating and using Geodesic Zones of Influence (cont)Calculating and using Geodesic Zones of Influence (cont)
Example – Morphological Processing of Example – Morphological Processing of Handwritten DigitsHandwritten Digits
thresholding
opening
thinning
smoothing
Morphological Filtering
Main idea of Morphological Filtering:Examine the geometrical structure of an image by matching it
with small patterns called structuring elements at various locations
By varying the size and shape of the matching patterns, we can extract useful information about the shape of the different parts of the image and their interrelations.
Combine set-theoretical and morphological operations:
Example 1: Morphological filtering
Noisy image will break down OCR systems
Clean original image Noisy image
Summary on Morphological Approaches
Mathematical morphology is an approach for processing digital image based on its shape
The language of morphology is set theory The basic morphological operations are erosion and
dilation Morphological filtering can be developed to extract useful
shape information Methods can be extended to more values and more
dimensions Nice mathematics can be formulated - non-linear
• Segmentation separates an image into regions.
• Use of histograms for brightness based segmentation.– Peak corresponds to object.
– Height of peak corresponds to size of object.
• If global image histogram is multimodal, local image region histogram may be bimodal.
• Local thresholds can give better segmentation.
ConclusionConclusion
• Postprocessing uses morphological operators.
• Same as convolution only use Boolean operators instead of multiply and add.– Erosion clears noise, makes smaller.
– Dilation fills in holes, makes larger.
• Postprocessing– Opening and closing to clean binary images.
– Repeated erosion with special rule produces skeleton.
ConclusionConclusion
Problems 1 - 6• 1. Write LISP or C++ program for dilation of binary
images
• 2. Modify it to do erosions (few types)
• 3. Modify it to perform shift and exor operation and shift and min operation
• 4. Generalize to multi-valued algebra
• 5. Create a comprehensive theory of multi-valued morphological algebra and its algorithms (publishable).
• 6. Write a program for inspection of Printed Circuit Boards using morphological algebra.
Problem 7.• Electric Outlet Extraction has been done Electric Outlet Extraction has been done
using a combination of Canny Edge using a combination of Canny Edge Detection and Hough TransformsDetection and Hough Transforms
• Write a LISP program that will use only basic morphological methods for this application.
Image Processing for electric outlet, how?• Currently there are many, many ways to
approach this problem– Segmentation
– Edge Detection
– DPC compression
– FFT
– IFFT
– DFT
– Thinning
– Growing
– Haar Transform
– Hex Rotate
Alpha filtering
DPC compression
Perimeter
Fractal
Gaussian Filter
Band Pass Filter
Homomorphic Filtering
Contrast
Sharper
Least Square Restoration
Warping
Dilation
Image Processing, how?
• Create morphological equivalents of other image processing methods.
• New, publishable, use outlet problem as example to illustrate
Problem 8. Openings and Closings as examples.
• The solution here is to follow up one operation with the other.
• An opening is defined as an erosion operation followed by dilation using the same structuring element.
• Similarly, a closing is dilation followed by erosion.• Define and implement other combined operations.
Problems 9 - 12.
• 9. Generalize binary morphological algebra from 2 dimensional to 3 dimensional images. What are the applications.
• 10. Write software for 9.
• 11. Generalize your generalized multi-valued morphological algebra to 3 or more dimensions, theoretically, find properties and theorems like those from this lecture.
• 12. Write software for 11.
Problem 13
• Mathematical morphology uses the concept of structuring elements to analyze image features.
• A structuring element is a set of pixels in some arrangement that can extract shape information from an image.
• Typical structuring elements include rectangles, lines, and circles.
• Think about other structuring elements and their applications.
Morphological Operations: Matlab
BWMORPH Perform morphological operations on binary image.
BW2 = BWMORPH(BW1,OPERATION) applies a specific morphological operation to the binary image BW1.
BW2 = BWMORPH(BW1,OPERATION,N) applies the operation N times. N can be Inf, in which case the operation is repeated until the image no longer changes.
OPERATION is a string that can have one of these values:
'close' Perform binary closure (dilation followed by erosion)
'dilate' Perform dilation using the structuring elementones(3)
'erode' Perform erosion using the structuring elementones(3)
'fill' Fill isolated interior pixels (0's surrounded by1's)
'open' Perform binary opening (erosion followed bydilation)
'skel' With N = Inf, remove pixels on the boundariesof objects without allowing objects to break apart
demos/demo9morph/
SourcesD.A. Forsyth, University of New Mexico,Qigong Zheng, Language and Media Processing LabCenter for Automation ResearchUniversity of Maryland College ParkOctober 31, 2000John MillerMatt RoachJ. W. V. Miller and K. D. WhiteheadThe University of Michigan-DearbornSpencer LustorLight Works Inc. C. Rössl, L. Kobbelt, H.-P. Seidel, Max-Planck Institute for, Computer Science, Saarbrücken, Germany LBA-PC4
Howard Schultz
Shreekanth MandayamECE DepartmentRowan University
D.A. Forsyth, University of New Mexico