Morphology

Structural processing of images

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Morphological Transformations Set theoretic methods of extracting

quantitative descriptions of image components Boundaries Skeletons Convex hull

Mainly binary, sometimes greylevel Two fundamental operations

Erode, dilate

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Thresholding Morphological operations usually

applied to bilevel images (black and white)

Generated by thresholding Greyscale Colour

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Thresholding Greyscale Images Previous lecture If g(x,y) > then g(x,y) = 1 else g(x,y) = 0

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Thresholding Colour Images Partition colour cube:

if cr(x,y) > r AND

cg(x,y) > g AND

cb(x,y) > b then c(x,y) = 1

else c(x,y) = 0

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Thresholding Colour

Images

Colour matching

222

max

,,,)),,((

0

)),,((1),(

oBoGoR ByxfGyxfRyxfCyxfd

where

otherwise

dCyxfdifyxg

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Preliminary Notation Translation of a region by x

Reflection of a region

AaxaccAx ,:

BbbxxB ,:ˆ

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Formally

Informally place the structuring element on a

pixel of the object remove that pixel if the structuring

element overlaps a non-object pixel

Binary Erode

ABxxBA :

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Binary Dilate Formally

Informally All pixels covered by structuring

element placed at all locations on region

ABxxBA ˆ:

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Binary Open and Close Erosion shrinks an object Dilation expands it Can combine operators Open = erosion then dilation Close = dilation then erosion

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Binary Open Opening smoothes regions

Removes spurs Breaks narrow lines

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Example

Binaryimage

Dilate Erode

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Binary Close Closing fills gaps

Holes in region Narrow gaps

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Processing grey scale images Same methods can be applied to

greyscale images Slight redefinition

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Greyscale Erode Set operation replaced by min

operation Output at a point is minimum of

image pixel and structuring element pixel

kjbknjmaBADG

Bkj,,,min,

,

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Greyscale Dilate Set operation replaced by max

operation Output is maximum of image and

structuring element

kjbknjmaBADGBkj

,,,max,,

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Examples

Erode Dilate

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Distance Applies to binary images For each pixel in a region

distance = minimum path to outside0 0 0 0 0

0 1 1 1 0

0 1 1 1 0

0 1 1 1 0

0 0 0 0 0

0 0 0 0 0

0 1 1 1 0

0 1 2 1 0

0 1 1 1 0

0 0 0 0 0

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Computation Use erosion

Label removed pixel with iteration number

Use relationship operator f(i,j) are neighbours of f(x,y)

jifyxfyxf

yxfyxfmm ,min,,

,,10

0

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Skeleton Reduces regions of a binary image

to lines one pixel thick Preserves

Shape Continuity

How? Uses?

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Algorithms Thinning

Repeatedly thin image Retain end points and connections

Distance Transform Skeleton lies along discontinuities Sort of local maxima or ridges

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Applications Shape representation, maintaining

topology Character recognition

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Convex Hull

Convex hull follows outline of object, except for concavities.Number and shape of regions between convex hull and object are characteristic of object shape.

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Summary Binary morphology

Erode, dilate, open, close Greyscale morphology

Erode, dilate Distance Skeleton Convex Hull

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Computers in the future may weigh no more than 1 ½ tonsPopular Mechanics, 1949