Class Note for ECE 482 at UA-Comp Visn Dig Image Proc(7)
Class Note for ECE 482 at UA-Comp Visn Dig Image Proc(7)
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Date Created: 02/06/15
Computer Vision amp Digital Image Processing Morphological Image Processing Electrical amp Computer Engineering Dr D J Jackson Lecture 151 Opening and Closing Morphological opening generally Smoothes the contour of an object Breaks narrow isthmuses Eliminates thin protrusions Morphological closing generally Smoothes contour sections Fuses narrow breaks and long thins gulfs Eliminates small holes Fills gaps in a contour Electrical amp Computer Engineering Dr D J Jackson Lecture 152 Opening and Closing The opening of set A by structuring element B denoted A B is defined as AoBAeB B Thus opening is defined as the erosion of A by B followed by a dilation of the result by B The closing of set A by structuring element B denoted A B is defined as AoBA BeB Thus opening is defined as the dilation ofA by B followed by a erosion ofthe result by B Ele nczl a Cnmrluler Enmneennu m n l Jamsnn Lecture 153 Opening Geometric Interpretation Suppose the structuring element B is viewed as a rolling ball The boundary ofA B is established by all points in B that reach the farthest into the boundary ofA as B is rolled about the inside ofthis boundary The opening ofA by B is obtained by taking the union of all translates of B that t into A A B U 5z 31 A t u n emumm M Ele nczl a Cnmrluler Enmneennu m n l Jamsnn Lecture 15 Closing Geometric Interpretation Suppose the structuring element B is viewed as a rolling ball The boundary ofA o B is established by all points in B that reach the closestto the boundary ofA as B is rolled about the outside of this boundary A n B wl BZ m A 9 for any translate of BZ containing w B A Electrical amp Computer Engineering Dr D J Jackson Lecture 155 Opening and Closing Examples FIGURE 910 Mullilloli iuul clumrlll Kiln mulll circle shown ill t uus posilions ll ihlffhe SI us not shaded hL l t l39nr C1Lll39ll Thc Llilrk dul ix the comer 01 he slrurluring claimlit Electrical amp Computer Engineering Dr D J Jackson Lecture 156 Morphological Filtering Morphological operators can be used to construct filters similar in concept to spatial lters If the filtering objective in question is to eliminate noise and distort data of interest as little as possible then a morphological filter consisting of an opening followed by a closing can be used Recall from the de nitions of opening and closing that the more primitive operations of erosion and dilation are used Opening Closing A BAGB B ABA BGB erode dilate dilate erode Electrical amp ComputerEngineering Dr D J Jackson Lecture 157 Morphological Filtering Example e f FIGURE 91 l a Noisy image b Structuring element c Eroded image d Opening of A e Dilation of the opening I Closing of the opening Original image courtesy of the National Institute of Standards and Technology n an in minimum mn Electrical amp ComputerEngineering Dr D J Jackson Lecture 158 HitorMiss Transform Tne murpnmugmax nmunrmss transfurm s a has mm rursnape detectmn The hear men s tu nd me ucatmn ur a Knuwn shape wwtmn a setuf snapes Assume 3 521A cunswsts ur a setuf snapes subsets c o and E u s nesnemunnu the ucatmn at me unne shapes 0 Lemma unger ur eacn shape be ms center at gram mummy 3mm nu HitorMiss Transform Let D be enclosed by a small window W The local background of D with respect to W is de ned as the set difference W D n 7m m mmquot Ewumumoumvuhrammunnn w n4 mum m M HitorMiss Transform The complement ofA AC is needed in the transform operation LetA be eroded by D The erosion ofA by D is the set of locations of the origin of D such that D is completely contained in A Viewed geometrically this is the set ofall locations of the origin of D at which Dfound a match hit in A A quot e lm anl 3 f l Electrical amp Computer Engineering Dr D J Jackson Lecture 1511 HitorMiss Transform Erode the complement of A AG by the local background set W D If we now compute the intersection of the two computed values this give use the location of D If B denotes the set composed of D and its background the match or set of matches of B in A denoted A B is A B A 9D m Ace WD N6 in m Mylar u lA Jl w ll e n r Electrical amp Computer Engineering Dr D J Jackson Lecture 1512 Boundary Extraction The boundary of a set A denoted BA is obtained by first eroding A by B and then performing the set difference between A and its erosion AA A AeB 0 B as always is a suitable structuring element l Shaded elements are 1 s and white elements are 0 s i 11 ma 1 b c I FIGURE 913 a Set A b Structuring element B c A eroded by B d Boundary given by the set difference between A and its erosion Electrical amp Computer Engineering Dr D J Jackson Lecture 1513 Boundary Extraction Although a 3x3 structuring element is commonly used it is not unique A 5x5 structuring element of 1 s would result in a boundary between 2 and 3 pixels thick When the origin of the structuring element B is on the edges of the set part of B may be outside the image The normal treatment ofthis condition is to assume that values outside the borders of the image are 0 Electrical amp Computer Engineering Dr D J Jackson Lecture 1514 Boundary Extraction ab FIGURE 914 a A simple binary image with 15 represented in White b Result of using Eq 951 with the structuring element in Fig 913b Electrical amp computer Engineering Dr D J Jackson Lecture 1515 Region Filling Assume A denotes a set containing a subset whose elements are 8 connected boundary points of a region Beginning with a point p inside the boundary the object is to ll the entire region with 1 s Adopt the convention that all nonboundary background points are labeled 0 and assign a value of 1 to pto begin The following procedure lls the region with is XKXM BnAC k 1 2 3 Where Xup and B is the symmetric structuring element B or o or o The algorithm terminates at step k if XKXM The set Xk u A contains the lled set and its boundary Electrical amp computer Engineering Dr D J Jackson Lecture 1516 Region Filling 39 39 a l d g h FIGURE 915 Hole lling a Set A shown shaded i i b Complement i T T of A 39 c Structuring element B 1 Initial point inside the boundary eh Various steps of Eq 952 i Final result union of a and 01 c f i 3 i l gin l i l39 M i l Electrical amp Computer Engineering Dr D J Jackson Lecture 1517 Region Filling The dilation process in the algorithm would fill the entire area if left unchecked The intersection with A5 at each iteration limits the result to inside the region of interest This is the first example of how a morphological process can be conditioned to meet a desired property This process is called conditional dilation I h L FlGURE Mb i Hum Imagu mu imi m quotmu M Di mt quotyum g uk wining plquot in rm lmlclilliny ilyniilllmile 12mm ii lilllny um mum in Rnull i lllliny in lhIle Electrical amp Computer Engineering Dr D J Jackson Lecture 1518 Extraction of Connected Components Concepts of connectivity and connected components are used Let Y represent a connected component in a set A and assume that a point p of Y is known The following expression yields all the elements of Y XkXH BmA k1 2 3 Where Xop and B is a suitable structuring element If XkXk1 the algorithm has converged and YXk Electrrczn a Cnmruler Enmneerlnu Dr D l Jamsnn Lecture 15719 Extraction of Connected Components I c ll c l g rlcullr v17 lhlk lln cm Cl nlulnll yn mm um wylklllmlllculllllrrclcdt mllyllll Electrrczn a Cnmruler Enmneerlnu Dr D l Jamsnn Lecture 1572 Extraction of Connected Components This process is similar to region filling except that we use A instead of Ac in the process The difference arises because all ofthe elements sought the elements ofthe connected component are labeled 1 The intersection with A at each step eliminates dilations that are centered on elements labeled 0 The shape ofthe structuring element assumes 8 connectivity between pixels Electrrczn a Cnmruler Enmneennu Dr D l Jamsnn Lecture 15721 Extraction of Connected Components 3 H Eluklmtlm llu H H 39lcmctc Uml H 391 quot Lareplmu W Gurmrln r v thw llllnraycmll Electrrczn a Cnmruler Enmneennu Dr D l Jamsnn Lecture 15722 Convex Hull A set A is convex if and only if the straight line segment joining any two points ofA lie entirely within A The convex hull H of an arbitrary set S is the smallest convex set containing 8 The set difference H S is called the convex de ciency The convex hull and convex deficiency will be useful for object description We present an algorithm for obtaining the convex hull CA of a set A Electrical amp Computer Engineering Dr D J Jackson Lecture 1523 Convex Hull Let 8quot i1234 represent the four structuring elements shown below The procedure consists of implementing the following XXk1 BioA i1234 and k123 with X3214 xx xxx x x x x x CED XXX XX B2 B3 B4 shaded1 white0 Xdon t care Electrical amp Computer Engineering Dr D J Jackson Lecture 1524 Convex Hull Now let D k where there is convergence in the sense that XI Ir1 The convex hull ofA is CA 00quot In other words the procedure consists of iterativer applying the hitormiss transform to A with 81 When no further changes occur we perform the union with A and call the set D1 Sets D2 D3and D4 are generated in a similar manner The union of the four sets is the convex hull ofA Electrical amp Computer Engineering Dr D J Jackson Lecture 1525 Convex Hull H a r n u w bccl efg 11 FIGURE 919 a Structuring elements b Set A c7f Results of convergence with the structuring elements shown in a g Convex 11u11h Convex hull showing the contribution of each structuring element Electrical amp Computer Engineering Dr D J Jackson Lecture 1526 Convex Hull One obvious shortcoming of the procedure is that the convex hull can grow beyond the minimum dimensions required to guarantee convexity One approach to reduce this effect is to limit the growth of the convex hull such that it does not extend beyond the horizontal and vertical dimensions of the original set noun van Rmm u mumm gum u mt lm nun xlvnnlhm u mt dilulmn Electrical amp Computer Engineering Dr D J Jackson Lecture 1527 Thinning The thinning ofa set A by a structuring element B denoted A B can be de ned in terms ofthe hitormiss transform A 693 A A B AmA Bc A more useful expression for thinning A symmetrically is based on a sequence of structuring elements BB1BZ B3Bquot A BA B1 B2 Bquot Electrical amp Computer Engineering Dr D J Jackson Lecture 1528 Thinnin The process is to thin A by one pass with B1 Thin the result with one pass of 82 and so on until A is thinned with one pass of Bn The entire process is repeated until no further changes occur Each individual thinning pass is performed using A n A B As a postprocessing step the thinned set may be converted to mconnectivity to eliminate multiple paths Electrical at Cnmnuter Engineering Dr D l Jawsnn Lecture 15729 Thinnin Electrical at Cnmnuter Engineering Dr D l Jawsnn Lecture 1573n Thickening Thickening is the morphological dual of thinning and is de ned by the expression AOB A u A B Where B is a structuring element suitable for thickening As with thinning thickening can be de ned as a sequential operation A0B AoB1oB2 JOBquot The structuring elements for thickening have the same form as those forthinning but with all 1 s and 0 s interchanged Electrical amp Computer Engineering Dr D J Jackson Lecture 1531 Thickening A separate algorithm for thickening is not absolutely required In practice we can thin the background ofthe set in question and complement the result In otherwords to thicken set A we form CAC thin C and then form CC Depending on the nature of A this procedure may result in some disconnected points Therefore thickening is commonly followed with a postprocessing step to remove disconnected points Electrical amp Computer Engineering Dr D J Jackson Lecture 1532 Thickening FIGURE 922 a Set A b Complement of A 0 Result of thinning the complement of A d Thickened set obtained by complementing c 6 Final result with no disconnected points Electrical amp Computer Engineering Dr D J Jackson Lecture 1533
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