MORPHOLOGICAL IMAGE ANALYSIS III

Transcription

1 MORPHOLOGICAL IMAGE ANALYSIS III Secondary inary Morphological Operators A set operator Ψ( ) which is: Increasing; i.e., Ψ( ) Ψ( ) Anti-extensive; i.e., Ψ( ) Idempotent; i.e., ΨΨ ( ( )) = Ψ( ) is called an opening. A set operator Ψ( ) which is: Increasing; i.e., Ψ( ) Ψ( ) Extensive; i.e., Ψ( ) Idempotent; i.e., ΨΨ ( ( )) = Ψ( ) is called a closing. If ( Ψ ε, Ψ δ ) is an adjunction, then ΨΨ δ ε ( ) is an opening whereas ΨΨ ε δ ( ) is a closing. If { Ψ i } is a collection of openings, then Ψ( ) = i Ψ i ( ) is an opening as well; i.e., unions of openings is also an opening! If { Ψ i } is a collection of closings, then Ψ( ) = i Ψ i ( ) is a closing as well; i.e., intersection of closings is also a closing! John Goutsias (ebruary, 02) 1

2 inary Structural Opening The (binary) structural opening of an image with a structuring element is given by: = ( 8) It can be shown that: = { + h ( + h) } This formula suggests that the structural opening of a set (shape) by a structuring element is the union of all translated structuring elements + h that fit inside (geometric interpretation of the structural opening). =( 8) 8 + h Opening a shape by a structuring element eliminates all components of which are smaller than, in the sense that they cannot contain any translated replica of! It therefore acts as a shape filter. John Goutsias (ebruary, 02) 2

3 inary Structural Opening Example 2 3 : r = { rb b } John Goutsias (ebruary, 02) 3

4 inary Structural Opening The structural opening is a smoothing filter! The amount and type of smoothing is determined by the shape and size of the structuring element used. It attempts to undo the effect of erosion 8 by applying the associated dilation ( 8). The structural opening approximates a shape from below, since John Goutsias (ebruary, 02) 4

5 inary Structural Closing The (binary) structural closing of an image with a structuring element is given by: ² = ( ) 8 It can be shown that: ² = { u u + h ( + h) } This formula suggests that a point u belongs to the structural closing of a set (shape) by a structuring element if all translated structuring elements + h that contain u intersect (geometric interpretation of the closing). ² John Goutsias (ebruary, 02) 5

6 inary Structural Closing Example ² ²2 ²3 : r = { rb b } John Goutsias (ebruary, 02) 6

7 inary Structural Closing The structural closing is a smoothing filter! The amount and type of smoothing is determined by the shape and size of the structuring element used. It attempts to undo the effect of dilation erosion ( ) 8. by applying the associated The structural closing approximates a shape from above, since ² John Goutsias (ebruary, 02) 7

8 Properties of inary Structural Opening { h}= ( + h) = ( ) + h Translation invariance ( + h) = Increasingness with respect to shape r r = r( ) Homogeneity Anti-extensitivity ( ) = Idempotence c = ( ² ) c Duality Properties of inary Structural Closing ²{ h}= ( + h) ² = ( ² ) + h Translation invariance ² ( + h) = ² ² ² Increasingness with respect to shape r² r = r( ² ) Homogeneity ² Extensitivity ( ² ) ² = ² Idempotence c ² = ( ) c Duality John Goutsias (ebruary, 02) 8

9 Some Advanced Properties of Structural Opening and Closing Aset is called open if =. Aset is called closed if ² =. Property: Aset is open if and only if = G for some set G. Proof: Assume first that is open. Then = = ( 8) = G where G 8. Assume now that = G, for some set G. Then = ( G ) = (( G ) 8 ) = ( G² ) G =. Also,, due to the anti-extensitivity property of the opening; therefore, since and, we get =, which shows that is open. Property: Aset is closed if and only if = G8 for some set G. Property: Let A and be two structuring elements such that A is open. Then, A ( A) = ( ) A= A ² A ² ( ² A) ² = ( ² ) ² A= ² A John Goutsias (ebruary, 02) 9

10 Morphological Shape Smoothing Example 1 Example 2 ( ) ² Example 3 D ( D) ² D John Goutsias (ebruary, 02) 10

11 inary Erosion-Dilation-Opening-Closing Example 1 : Henri Matisse, Nu leu III, ² John Goutsias (ebruary, 02) 11

12 inary Erosion-Dilation-Opening-Closing Example 2 : 8 ² John Goutsias (ebruary, 02) 12

13 inary Erosion-Dilation-Opening-Closing Example 3 : 8 ² John Goutsias (ebruary, 02) 13

14 Remark rom the previous examples it is clear that the choice of the structuring element is very important here. Here is another example 8, : 8, : 8, : Most of the time, the structuring element is chosen by the algorithm designer in a way that makes most sense in practical applications. However, many researchers have identified the need to build-up an approach that allows the automatic specification of the structuring element that best fits a particular application. John Goutsias (ebruary, 02) 14