MATHEMATICAL MORPHOLOGY I.INTRODUCTION II.BINARY MORPHOLOGY III.GREY-LEVEL MORPHOLOGY

INTRODUCTION Mathematical morphology Self-sufficient framework for image processing and analysis, created at the École des Mines (Fontainebleau) in 70’s by Jean Serra, Georges Mathéron, from studies in science materials Conceptually simple operations combined to define others more and more complex and powerful Simple because operations often have geometrical meaning Powerful for image analysis

INTRODUCTION Binary and grey-level images seen as sets X XcXc f (x,y) X = { (x, y, z), z  f (x,y) }

Operations defined as interaction of images with a special set, the structuring element INTRODUCTION

MATHEMATICAL MORPHOLOGY I.INTRODUCTION II.BINARY MORPHOLOGY III.GREY-LEVEL MORPHOLOGY

BINARY MORPHOLOGY 1.Erosion and dilation 2.Common structuring elements 3.Opening, closing 4.Properties 5.Hit-or-miss 6.Thinning, thickenning 7.Other useful transforms : i.Contour ii.Convex-hull iii.Skeleton iv.Geodesic influence zones

X B BINARY MORPHOLOGY No necessarily compact nor filled A special set : the structuring element Origin at center in this case, but not necessarily centered nor symmetric Notation x y

Dilation : x = (x 1,x 2 ) such that if we center B on them, then the so translated B intersects X. X B difference BINARY MORPHOLOGY

Dilation : x = (x 1,x 2 ) 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

BINARY MORPHOLOGY Minkowski’s sum of sets : l l

BINARY MORPHOLOGY Dilation : l Dilation

l BINARY MORPHOLOGY Dilation is not the Minkowski’s sum

l l bbbb l BINARY MORPHOLOGY

Dilation with other structuring elements BINARY MORPHOLOGY

Dilation with other structuring elements

Erosion : x = (x 1,x 2 ) such that if we center B on them, then the so translated B is contained in X. BINARY MORPHOLOGY difference

BINARY MORPHOLOGY 2) Better : from Minkowski’s substraction of sets Erosion : x = (x 1,x 2 ) 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

BINARY MORPHOLOGY

Erosion with other structuring elements BINARY MORPHOLOGY

Did not belong to X BINARY MORPHOLOGY Erosion with other structuring elements

BINARY MORPHOLOGY Common structuring elements shapes = origin x y Note : circledisk segments 1 pixel wide points

BINARY MORPHOLOGY Problem :

BINARY MORPHOLOGY

Problem : BINARY MORPHOLOGY d/2 d <d/2

BINARY MORPHOLOGY Implementation : very low computational cost 0 1 (or >0) Logical or

0 1 BINARY MORPHOLOGY Implementation : very low computational cost Logical and

Opening : BINARY MORPHOLOGY Supresses : small islands ithsmus (narrow unions) narrow caps difference also

Opening with other structuring elements BINARY MORPHOLOGY

Closing : BINARY MORPHOLOGY Supresses : small lakes (holes) channels (narrow separations) narrow bays also

BINARY MORPHOLOGY Closing with other structuring elements

BINARY MORPHOLOGY Application : shape smoothing and noise filtering

BINARY MORPHOLOGY Application : segmentation of microstructures (Matlab Help) original negated threshold disk radius 6 closing opening and with threshold

BINARY MORPHOLOGY Properties all of them are increasing : opening and closing are idempotent :

dilation and closing are extensive erosion and opening are anti-extensive : BINARY MORPHOLOGY

duality of erosion-dilation, opening-closing,...

structuring elements decomposition BINARY MORPHOLOGY operations with big structuring elements can be done by a succession of operations with small s.e’s

Hit-or-miss : BINARY MORPHOLOGY “Hit” part (white) “Miss” part (black) Bi-phase structuring element

Looks for pixel configurations : BINARY MORPHOLOGY doesn’t matter background foreground

BINARY MORPHOLOGY isolated points at 4 connectivity

Thinning : BINARY MORPHOLOGY Thickenning : Depending on the structuring elements (actually, series of 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 BINARY MORPHOLOGY 1 iteration =

BINARY MORPHOLOGY 1st iteration 2nd iteration3rd iteration: idempotence

BINARY MORPHOLOGY doesn’t matter background foreground What does the following sequence ?

BINARY MORPHOLOGY 1.Erosion and dilation 2.Common structuring elements 3.Opening, closing 4.Properties 5.Hit-or-miss 6.Thinning, thickenning 7.Other useful transforms : i.Contour ii.Convex-hull iii.Skeleton iv.Geodesic influence zones

i. Contours of binary regions : BINARY MORPHOLOGY

4-connectivity 8-connectivity contour 8-connectivity 4-connectivity contour Important for perimeter computation.

ii. Convex hull : union of thickenings, each up to idempotence BINARY MORPHOLOGY

iii. Skeleton : BINARY MORPHOLOGY Maximal disk : disk centered at x, D x, such that D x  X and no other D y contains it. Skeleton : union of centers of maximal disks.

BINARY MORPHOLOGY 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

BINARY MORPHOLOGY 1st iteration

BINARY MORPHOLOGY result of 1st iteration 2nd iteration reaches idempotence

BINARY MORPHOLOGY 20 iterations thinning 40 iterations thickening

BINARY MORPHOLOGY Application : skeletonization for OCR by graph matching

BINARY MORPHOLOGY and 3 rotations Hit-or-Miss

iv. Geodesic zones of influence : BINARY MORPHOLOGY X set of n connex components {X i }, i=1..n. The zone of influence of X i, Z(X i ), is the set of points closer to some point of X i than to a point of any other component. Also, Voronoi partition. Dual to skeleton.

BINARY MORPHOLOGY threrosion 7x7 GZIopening 5x5 and

BINARY MORPHOLOGY

threrosion 7x7 dist