09/11/2017. Morphological image processing. Morphological image processing. Morphological image processing. Morphological image processing (binary)

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1 Towards image analysis Goal: Describe the contents of an image, distinguishing meaningful information from irrelevant one. Perform suitable transformations of images so as to make explicit particular shape information. Mathematical Morphology provides mathematical tools based on set theory for investigating and manipulating geometric structures in binary images. Image objects are represented as sets. The latter may be derived from graylevel images by segmentation methods (e.g. using thresholds or edges) [next chapter] Some morphological operators, extended to graylevel images, can become useful preprocessing tools Copyright notice: Most images in this package are Gonzalez and Woods,, Prentice-Hall Morphological Operators Erosion and dilation are the most elementary operators of mathematical morphology. More complicated morphological operators can be designed by means of combined erosions and dilations. Some History George Matheron (1975) Random Sets and Integral Geometry, John Wiley. Jean Serra (1982) Image Analysis and Mathematical Morphology, Academic Press. Petros Maragos (1985) A Unified Theory of Translation- Invariant Systems with Applications to Morphological Analysis and Coding of Images, Doctoral Thesis, Georgia Tech. Sets of (binary or graylevel) pixels are defined; standard set operations are defined (union, intersection, complement...). Two more operations are defined: E.g.: A binary object A is present in an image. The SE B runs over the input image. The output set is formed by the positions in which B is completely contained in A (erosion)... [The image needs to be padded to allow for all SE positions] Structuring elements (SE) (sort of probes ) operate on the pixel sets A SE (in gray) can have any shape, can be connected or not. To be used for MM it may need to be padded to form a rectangle. Black dot = origin of SE... or, e.g., B and A overlap by at least one element (dilation)

2 EROSION Set of all points z such that B, translated by z, is contained in A A B = {z (B) z A} = {z (B) z A C = } EXAMPLE DILATION Set of all points z such that B, flipped and translated by z, has a non-empty intersection with A A B = {z (B ) z A } = {z ( (B ) z A ) A EXAMPLE: fix broken characters using dilation note: differently from a linear filter, a binary image is output NOTE:the flipping of the structuring element is included in analogy to convolution. Not all Authors perform it

3 EXAMPLE: Eliminate small objects by erosion+dilation note 1: white objects on black background (opposite wrt prev. slides) note 2: the final dilation will NOT yield in general the exact shape of the original objects OPENING = erosion followed by dilation Smoothes vertices, breaks narrow isthmuses, eliminates small islands and sharp peaks A B = A B B CLOSING = dilation followed by erosion Smoothes vertices, fuses narrow breaks and long thin gulfs, eliminates small holes A B = A B B OPENING shift the SE inside the object border OPENING erode dilate CLOSING shift the SE outside the object border CLOSING dilate erode For clarity, object A is not shaded in the first picture

4 A property: Erosion and Dilation Opening and Closing Opening closing Note: both white and black errors (e.g. due to poor thresholding) are eliminated are dual operators wrt set complementation and reflection: C C ( A B) A Bˆ C C ( A B) A Bˆ BOUNDARY EXTRACTION β(a) = A A B HOLE FILLING X P while X X X 0 end X ( X F k k X A k k 1 k 1 do B) A C Note: use to get an 8-connected border (use a 5x5 SE to get a 2- pixel-wide border) The dilation would fill the whole area were it not for the intersection with A C We call it Conditional dilation

5 EXTRACTION OF CONNECTED COMPONENTS X P 0 while X end X X ( X k k k 1 k 1 do B) A It is again a conditional dilation, with A SE B provides 8- connected contour SE in previous slide for 4-connected cont. HIT-OR-MISS TRANSFORM We search for the location of D Enclose D in a small window W and perform the operations shown (erosion of A with D, erosion of A C with W-D, intersection of the results) (The local background of D wrt W is W-D) (The origin of an object is its center of gravity) Hit-or-Miss transform Let B be the set composed of D and its background. The match of B in A is A B = A D A C (W D ) I.e.: D finds a match in A ("hit") and W-D finds a match ("miss") in A C Example D+background: gray = object D white = background = don't care Concept is that two objects are disjoint iff each has at least a 1-px-wide background around it NOTE: when looking for just a pattern of 1s in an image, not caring about the background, hit-or-miss becomes standard erosion CONVEX HULL [Smallest] convex set containing A For each direction: hit-ormiss; add A; repeat until stable. Join 4 resulting sets = don't care X i 0 = A X i k = (X k-1 B i ) A i =1,2,3,4; k =1,2,3... C (A ) = i=1,..4 X i K To get the smallest convex set containing A, limit growth to the min and max coordinates of A

6 THINNING THICKENING Thinning A B = A - (A B) = A (A B) C A B = A (A B) If B is a set of rotated simple SEs, {B i }: A B = ((...((A B 1 ) B 2 )... ) B N ) A B = ((...((A B 1 ) B 2 )... ) B N ) Thickening = (background thinning complement postprocessing) SKELETONIZATION Maximum disk: largest disk included in A, touching the boundary of A at two or more different places Skeleton: set of the centers of the maximum disks

7 GEODESIC DILATION AND EROSION In its original sense, a geodesic is the shortest route between two points on the Earth's surface, namely, a segment of a large circle It is a generalization of the notion of a straight line to curved spaces In dilation, structures grow at their boundaries. To constrain this growth within some predefined boundaries we use Geodesic dilation: the minimum between the dilation of the image (marker, F) by an SE (B) and a control image (mask, G) that contains the restraining boundaries Geodesic erosion is similarly defined as the maximum between the erosion of the image... D 1 G (F )= (F B ) G E 1 G (F )= (F B ) G Geodesic dilation and erosion Geodesic dilation and erosion can be iterated until convergence (that always occurs) Reconstruction by dilation, Reconstruction by erosion As we know, erosion + dilation = opening. Erosion + reconstruction by dilation is called Opening by reconstruction Erosion removes small structures, and erodes the boundaries of large structures. Parts of the latter remain present in the eroded image. A reconstruction by geodesic dilation, with the original image for a control image (mask), makes these parts grow back to their original size. The small structures that were completely removed by the erosion will not grow back Opening by reconstruction removes small structures, keeping larger structures intact Opening by reconstruction Closing by reconstruction Dilation + reconstruction by erosion is called Closing by reconstruction

8 Opening by reconstruction Extract characters that contain long vertical strokes BOUNDARY CLEANING Delete objects that touch the boundary of the image (probably incomplete objects) Set marker image as the border of the original image; compute reconstruction by dilation; subtract the result from the image SUMMARY SUMMARY

9 SUMMARY SUMMARY SUMMARY SUMMARY

10 FOREWORD GRAY-LEVEL IMAGES The complement X c of a binary set X is defined as all elements not belonging to X. The complement f c of an image f is defined as f mirrored in a central greyvalue line. This definition will allow us to operate on gray-level images. For an image with a graylevel range {L,... M} the complement is f c (x,y) = L + M f(x,y). Complementing twice yields the original set or function: (X c ) c = X FOREWORD To define a relation between grey-value images and sets, we regard each grey level as a set; then the image f is a stack of sets F with the relation F(c) = {(x,y) f(x,y) c}. We are now able to translate any operators defined only for sets to apply to images: we apply the operator to all of the level sets F(c) that together make up the image Erosion / dilation Erosion and dilation of an image f(x,y) by a structuring element b(x,y): [NOTE: b can be non-flat but is almost always flat] With a flat SE: ( f b)( x, y) min{ f ( x s, y t) ( s, t) b} ( f b)( x, y) max{ f ( x s, y t) ( s, t) bˆ} max{ f ( x s, y t) ( s, t) b} (same as min and max in order statistics filters) Note: like in the binary case, applying specularity in either dilation or erosion permits to define the duality: C ( f b) C f bˆ 448x425 px erosion, SE=disk(r=2) dilation Erosion makes the output image darker; bright details having area smaller than the SE are lessened The opposite for dilation.

11 Opening and closing of an image f(x,y) by a structuring element b(x,y) have the same form as their binary counterpart: f b = f b b f b = f b b Geometric interpretation: View the image as a 3-D surface map f(x,y), and suppose we have a flat SE Opening: move the SE pushing it against the underside of the surface, and take the highest point it reaches in each position (x,y) Closing: move the SE on top of the surface, and take the lowest points it reaches in each position (x,y) Opening / closing 1-D illustration Opening makes the image darker, by clipping small and bright details Closing... [not to be confused with figs. showing binary opening and closing in slide 11] Opening / closing Morphological smoothing: opening closing Simplifies an image opening, radius 3 closing, radius 5 xxxxxxxx Note the decreased size of the small bright (opening) or dark (closing) details Has no appreciable effects on the darker (opening) or brighter (closing) details

12 Morphological gradient: g = f b (f b) Top hat / Bottom hat (White top hat / Black top hat) Th = f f b Bh = f b f Opening (closing) removes small light (dark) structures; then, taking the difference from the original we obtain only the removed components. They are output with a gray value that is relative to local background (see previous slide showing intensity profiles) Using a large SE we remove all structures from an image, then we subtract them from the original and show them relative to their local background remove slow gray-level variations in the background useful for threshold-based image segmentation (see later) Top hat / Bottom hat Granulometry

13 Granulometry Opening has most effects on regions that contain structures having the same size as the SE. Perform openings with SE of increasing size r Compute surface area (= sum of all pixel values) for each output image It decreases with r, since opening reduces the intensity of pixels Plot the differences between adjacent surface area values This shows the distribution of the particles size... if the amount and area of objects are suitable (graylevel or binary) Texture segmentation See MM concepts, examples, and Matlab commands: