Chapter 5 Mathematical Morphology

5.1 Introduction

mathematical morphology works on shape

shape: prime carrier of information in machine vision

morphological operations: simplify image data,

preserve essential shape characteristics, eliminate irrelevancies

shape: correlates directly with decomposition of objects, object features,

object surface defects, assembly defects

5.2 Binary Morphology

set theory: language of binary mathematical morphology

sets in mathematical morphology: represent shapes

Euclidean -space:

discrete Euclidean -space:

: hexagonal grid, square grid

dilation, erosion: primary morphological operations

opening, closing: composed from dilation, erosion

opening, closing: related to shape representation, decomposition, primitive

extraction

5.2.1 Binary Dilation

dilation: combines two sets by vector addition of set elements

dilation of by :

addition commutative dilation commutative:

binary dilation: Minkowski addition

=====Fig. 5.1=====

: referred as set, image

: structuring element: kernel

dilation by disk: isotropic swelling or expansion

=====Fig. 5.2=====

dilation by kernel without origin: might not have common pixels
with

translation of dilation: always can contain

=====lena.bin.128=====

=====lena.bin.dil=====

for noise removal

: set of four 4-neighbors of (0,0) but not (0,0)

4-isolated pixels removed

only points in with at least one of its 4-neighbors remain

: translation of by the point

dilation: union of translates of kernel

addition associative dilation associative

associativity of dilation: chain rule: iterative rule

dilation of translated kernel: translation of dilation

dilation distributes over union

dilating by union of two sets: the union of the dilation

dilating by kernel with origin guaranteed to contain

extensive: operators whose output contains input

dilation extensive when kernel contains origin

dilation preserves order

increasing: preserves order

=====

5.2.2 Binary Erosion

erosion: morphological dual of dilation

erosion of by : set of all s.t. for every

erosion: shrink: reduce

=====Fig. 5.3=====

=====lena.bin.ero=====

erosion of by : set of all for which translated to contained in

if translated to contained in , then in

erosion: difference of elements and

dilation: union of translates

erosion: intersection of negative translates

=====Fig. 5.4=====

Minkowski subtraction: close relative to erosion

Minkowski subtraction:

erosion: shrinking of the original image

antiextensive: operated set contained in the original set

erosion antiextensive: if origin contained in kernel

if then
because:

if
then for every , since
thus

eroding by kernel without origin can have nothing in common with

=====Fig. 5.5=====

possible:
and

e.g.
, then
,
yet

dilating translated set results in a translated dilation

eroding by translated kernel results in negatively translated erosion

dilation, erosion: increasing

eroding by larger kernel produces smaller result

dilation, erosion similar that one does to foreground, the other to background

similarity: duality

dual: negation of one equals to the other on negated variables

DeMorgan's law: duality between set union and intersection

negation of a set: complement

negation of a set in two possible ways in morphology

- logical sense: set complement
- geometric sense: reflection: reversing of set orientation

: reflection about the origin of

: symmetrical set of with respect to origin [Matheron 1975]

: transpose [Serra 1982]

complement of erosion: dilation of the complement by reflection

Theorem 5.1: Erosion Dilation Duality

=====Fig. 5.6=====

Corollary 5.1:

erosion of intersection of two sets: intersection of erosions

=====Fig. 5.7=====

erosion of a kernel of union of two sets: intersection of erosions

erosion of kernel of intersection of two sets: contains union of erosions

no stronger

=====Fig. 5.8=====

chain rule for erosion holds when kernel decomposable through dilation

duality does not imply cancellation on morphological equalities

containment relationship holds

genus : number of connected components minus number of holes of

4-connected for object, 8-connected for background

8-connected for object, 4-connected for background

=====Fig. 5.9=====

5.2.3 Hit-and-Miss Transform

hit-and-miss: selects corner points, isolated points, border points

hit-and-miss: performs template matching, thinning, thickening, centering

hit-and-miss: intersection of erosions

kernels satisfy

hit-and-miss of set by

hit-and-miss: to find upper right-hand corner

=====Fig. 5.11=====

locates all pixels with south, west neighbors part of

locates all pixels of with south, west neighbors in

and displaced from one another

hit-and-miss: locate particular spatial patterns

then
:
set of all 4-isolated pixels

hit-and-miss: to compute genus of a binary image

=====Fig. 5.10=====

hit-and-miss: thickening and thinning

hit-and-miss: counting

hit-and-miss: template matching

5.2.4 Dilation and Erosion Summary

=====dilation and erosion summary, p. 173=====

=====Garfield 17:5=====

5.2.5 Opening and Closing

dilation and erosions: usually employed in pairs

: opening of image by kernel

: closing of image by kernel

open under : open w.r.t. :

closed under : closed w.r.t. :

morphological opening, closing: no relation to topologically open, closed sets

opening characterization theorem

: selects points covered by some translation of , entirely contained in

=====lena.bin.open=====

opening with disk kernel: smoothes contours, breaks narrow isthmuses

opening with disk kernel: eliminates small islands, sharp peaks, capes

=====lena.bin.close=====

closing by disk kernel: smoothes contours, fuses narrow breaks, long, thin gulfs

closing with disk kernel: eliminates small holes, fill gaps on the contours

unlike erosion and dilation: opening invariant to kernel translation

opening antiextensive

like erosion and dilation: opening increasing

: those pixels covered by sweeping kernel all over inside of

: shape with body and handle

: small disk structuring element with radius just larger than handle width

extraction of the body and handle by opening and taking the residue

=====Fig. 5.16=====

extraction of trunk and arms with vertical and horizontal kernels

=====Fig. 5.17=====

extraction of base, trunk, horizontal and vertical areas

=====Fig. 5.18=====

noisy background line segment removal

=====Fig. 5.19=====

decomposition into parts

=====Fig. 5.20==========Fig. 5.21=====

closing: dual of opening

like opening: closing invariant to kernel translation

closing extensive

like dilation, erosion, opening: closing increasing

opening idempotent

closing idempotent

if not necessarily follows that

=====Fig. 5.22=====

closing may be used to detect spatial clusters of points

=====Fig. 5.23=====

5.2.6 Morphological Shape Feature Extraction

morphological pattern spectrum: shape-size histogram [Maragos 1987]

5.2.7 Fast Dilations and Erosions

decompose kernels to make dilations and erosions fast

5.3 Connectivity

morphology and connectivity: close relation

5.3.1 Separation Relation

separation if and only if symmetric, exclusive, hereditary, extensive

5.3.2 Morphological Noise Cleaning and Connectivity

images perturbed by noise can be morphologically filtered to remove some noise

5.3.3 Openings, Holes, and Connectivity

opening can create holes in a connected set that is being opened

=====Fig. 5.25=====

5.3.4 Conditional Dilation

select connected components of image that have nonempty erosion

conditional dilation
, defined iteratively

are points in the regions we want to select

conditional dilation where is the smallest index

=====Fig. 5.26=====

5.4 Generalized Openings and Closings

generalized opening: any increasing, antiextensive, idempotent operation

generalized closing: any increasing, extensive, idempotent operation

=====Oldie 33:18=====

5.5 Gray Scale Morphology

binary dilation, erosion, opening, closing naturally extended to gray scale

extension: uses min or max operation

gray scale dilation: surface of dilation of umbra

gray scale dilation: maximum and a set of addition operations

gray scale erosion: minimum and a set of subtraction operations

5.5.1 Gray Scale Dilation and Erosion

top: top surface of : denoted by
:

umbra of : denoted by

=====Fig. 5.28=====

gray scale dilation: surface of dilation of umbras

dilation of by : denoted by

=====Fig. 5.29=====

=====Fig. 5.30=====

and
, then

=====Fig. 5.31=====

=====lena.im=====

=====lena.im.dil=====

gray scale erosion: surface of binary erosions of one umbra by the other umbra

=====Fig. 5.32=====

and
, then

=====Fig. 5.33=====

=====lena.im.ero=====

=====Fig. 5.34=====

5.5.2 Umbra Homomorphism Theorems

surface and umbra operations: inverses of each other, in a certain sense

surface operation: left inverse of umbra operation

Proposition 5.1

Proposition 5.2

Proposition 5.3

5.5.3 Gray Scale Opening and Closing

gray scale opening of by kernel : denoted by

=====lena.im.open=====

gray scale closing of by kernel : denoted by

=====lena.im.close=====

duality of gray scale dilation, erosion
duality of opening,
closing

=====Fig. 5.37=====

5.6 Openings, Closings, and Medians

median filter: most common nonlinear noise-smoothing filter

median filter: for each pixel, the new value is the median of a window

median filter: robust to outlier pixel values, leaves edges sharp

median root images: images remain unchanged after median filter

5.7 Bounding Second Derivatives

opening or closing a gray scale image: simplifies the image complexity

5.8 Distance Transform and Recursive Morphology

Fig. 5.39 (b) fire burns from outside but burns only downward and right-ward

=====Fig. 5.39=====

5.9 Generalized Distance Transform

5.10 Medial Axis

medial axis transform: medial axis with distance function

5.10.1 Medial Axis and Morphological Skeleton

morphological skeleton of a set by kernel , where

skeleton of given by

=====Fig. 5.40=====

=====Fig. 5.41=====

5.11 Morphological Sampling Theorem

5.11.1 Set-Bounding Relationships

5.11.2 Examples

5.11.3 Distance Relationships

=====Garfield 17:7=====

5.12 Summary

morphological operations: shape extraction, noise cleaning, thickening

morphological operations: thinning, skeletonizing

Project due Nov. 6

Write programs which do binary morphological dilation, erosion, opening,
closing, and hit-and-miss transform on a binary image.

Project due Nov. 13

Write programs which do gray scale morphological dilation, erosion, opening,
and closing on a gray scale image.

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