Morphological Compound Operations-Opening and CLosing

Transcription

1 Morphological Compound Operations-Opening and CLosing COMPSCI 375 S1 T 2006, A/P Georgy Gimel farb Revised COMPSCI 373 S1C -2010, Patrice Delmas AP Georgy Gimel'farb 1

2 Set-theoretic Binary Operations Many morphological operations are combinations of erosion, dilation, and simple set-theoretic operations Set-theoretic complement of a binary image: f c (x,y) = 1 if f(x,y) = 0, and f c (x,y) = 0 if f(x,y) = f complement f c AP Georgy Gimel'farb 2

3 Set-theoretic Binary Operations Intersection h = f g of two binary images f and g: h(x,y) = 1 if f(x,y) = 1 AND g(x,y) = 1; h(x,y) = 0 otherwise f g h 15 AP Georgy Gimel'farb 3

4 Set-theoretic Binary Operations Union h = f g of two binary images f and g: h(x,y) = 1 if f(x,y) = 1 OR g(x,y) = 1; h(x,y) = 0 otherwise f g h 16 AP Georgy Gimel'farb 4

5 Opening Opening f s of an image f by a structuring element s is an erosion followed by a dilation: f s = (f s) s From: 17 AP Georgy Gimel'farb 5

6 From: Opening of a Binary Image Structuring element Binary image Opened image Opening is so called because it can open up a gap between objects connected by a thin bridge of pixels. Any regions survived the erosion are restored to their original size by the dilation Idempotent operation: ( f s) s = f s Once an image is opened, next openings with the same structuring element have no further effect 18 AP Georgy Gimel'farb 6

7 From: Opening of a Binary Image Binary image Opening with a 5 5 square Opening with 9 9 square structuring element structuring element 19 AP Georgy Gimel'farb 7

8 Closing Closing f s of an image f by a structuring element s is a dilation followed by an erosion: f s = (f s) s Dilation and erosion should be performed with a rotated by 180 structuring element Typically, the element is symmetrical so that the rotated and initial versions do not differ From: 20 AP Georgy Gimel'farb 8

9 From: Closing of a Binary Image Structuring element Binary image Closed image Closing is so called because it can fill holes in the regions while keeping the initial region sizes Idempotent operation (f s) s = f s: once an image is closed, next closings with the same element have no further effect 21 AP Georgy Gimel'farb 9

10 Closing Vs. Opening Closing is the dual operation of opening (just as opening is the dual operation of closing) Closing of a binary image (dual implementation): Take the complement of that image Perform opening with the structuring element, and Take the complement of the result Opening of a binary image (dual implementation): Take the complement of that image Perform closing with the structuring element, and Take the complement of the result 22 AP Georgy Gimel'farb 10

11 Region Boundary The set difference f (f s) between the original image, f, and the eroded image, f s, forms a boundary with the pixels from f which are absent in the eroded image Structuring element From: Binary image Boundary image 23 AP Georgy Gimel'farb 11

12 Morphological Filtering Compound operations (e.g. opening and closing) act as non-linear filters of shape in a binary image Opening and closing with a disc structuring element smooth corners from the inside and the outside, respectively Details smaller in size than the disc are also filtered out Opening is filtering at a scale of the size of the structuring element Only those portions of the image that fit the structuring element are passed by the filter; smaller structures are blocked and excluded The size of the structuring element is most important in order to eliminate noisy details but not to damage objects If it is too large, the object could be degraded by the operation 24 AP Georgy Gimel'farb 12