Class Note for ECE 482 at UA-Comp Visn Dig Image Proc(8)
Class Note for ECE 482 at UA-Comp Visn Dig Image Proc(8)
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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 141 Introduction Morphology a branch of biology concerned with the form and structure of plants and animals Mathematical morphology a tool for extracting image components useful in the representation and description of image shape including Boundaries Skeletons Convex hull We will also look at morphological techniques for Filtering Thinning Pruning Electrical amp Computer Engineering Dr D J Jackson Lecture 142 Preview Language of mathematical morphology is set theory Sets in mathematical morphology represent objects in an image For example the set of all black pixels in a binary image is a complete morphological description ofthe image For binary images sets are members of the 2D integer space Z2 Each element of the set is a tuple 2D vector whose coordinates are the xy coordinates of a black or white depending on convention pixel in the image Grayscale digital images are represented as sets in Z3 Coordinates and grayscale value Higher dimensioned sets could represent attributes such as color time varying components etc Electrical amp Computer Engineering Dr D J Jackson Lecture 14quot Basic Concepts from Set Theory Let A be a set in 22 If a a1 a2 is an element of A then we write a e A o If a is not an element of A we write a e A o A set with no elements is called the null or empty setand is denoted by the symbol 6 o A set is specified by the contents of two braces o For binary images the elements of the sets are the coordinates of pixels representing objects If we write an expression ofthe form Cw w d for d e D we mean that C is the set of elements w such that w is formed by multiplying each of the two coordinates of all the elements of set D by 1 Electrical amp Computer Engineering Dr D J Jackson Lecture 144 Basic Concepts from Set Theory If every element of set A is also an element of another set B then A is a subset of B and we write A g B The union oftwo sets A and B denoted by C A u B is the set of all elements belonging to eitherA B or both The intersection of two sets A and B denoted by D A m B is the set of all elements belonging to both A and B Two sets are disjoint or mutually exclusive if they have no elements in common A n B Q The complement of a set A is the set of elements not in A A0 w w e A The difference of two sets A and B denoted AB is de ned as ABwWEAweBAmBC Electrrczn a Cnmruler Enmneennu Dr D l Jamsnn Lecture 1575 Logical Operations Involving Binary Images t will Principal logic operations AND 139 OR I NOT COMPLEMENT Functionally complete Combined to form any other logic operation Logic operations described have a oneto one correspondence with the set operations intersection union and complement Logic operations are restricted to binary images not the case for general set operations mmmm 3 I new Electrrczn a Cnmruler Enmneennu Dr D l Jamsnn Lecture 1576 Basic Concepts from Set Theory The reflection of set B is defined as A BW W b for b e B 0 The translation of set B by point z Z1 22 is defined as BZccbzforbeB a b c FIGURE 9 113A wt hi il n relict39lmn nml quot m m translation by an Electrical amp Computer Engineering Dr D J Jackson Lecture 147 Structuring Elements 0 Set reflection and translation are used extensively in morphological operations based on structuring elements SE 0 An SE is a small set or subimage used to probe an area of interest for certain properties May be of arbitrary shape and size In practice an SE is generally a regular geometric shape square rectangle diamond etc Generally padded to a rectangular array for image processing FIGURE 92 First i row Examples of re structuring elements Second Lv row Structuring elements converted to rectangular arrays The dots denote the centers of the SEs ha 14 Electrical amp Computer Engineering Dr D J Jackson Lecture 148 Structuring Elements continued Twodimensional or flat structuring elements consist of a matrix of 039s and 139s typically much smaller than the image being processed The center pixel of the structuring element called the origin identifies the pixel of interestthe pixel being processed The pixels in the structuring element containing 139s define the neighborhood of the structuring element Electrical amp Compmer Engineering Dr D J Jackson Lecture 149 Operation with a Structuring Element example A uh cdu FIGURE 93 M A 31 each shaded square is u nicmlxcr ul Ill wll h A drumming elemum ClTlte m padded with backgrnund elememi no form n lecmnyular array and pvmidr background hurdcr 1L1 Slruuluringt ulcmunl l1 ruclunyulnrurr c Sm pruccssunl hy lhc ll llilllll lnf cluman Electrical amp Compmer Engineering Dr D J Jackson Lecture 1410 Structuring Elements Matlab Functions strel Crealemumhuluulcalslruclurmuelemenl syntax s e scxellshspeypaxamecexsl Description 5 e scxellshspeypaxamecexsl erealesaslmelurmeelemenl seerlnewespeemeenv shape The lablellslsalllhesuppu edshapes Dependneun shape met mavlake aeelllenal parameters eeelne svnlax eesenpllens lnalrellewrer eelalls abuul erealme each Wpe urslmelurme element Flm sunctunnll Elements arbllram M dlamund gerludlclme m reclangle M uclagun Mm lm sullct Elements arbllram all Electrical 8 Computer Engineering Dr D J Jackson Lecture 1411 Structuring Elements Matlab Funct39ons re ect Reflect structuring element Syntax SE2 zetleccrsm Description SE2 re ectcSEJ re ects a structurlng elementthrough llS center The effect lsthe same as lfyou rotated the structuran element39s dnmeln1EiE degrees arnund its center ar a 2D structuring element If SE IS an array nfstruclurlng element ublects than reflectASE re ects each element of SE and SE2 hasthe same elze as SE Class Support SE and SE2 are STREL unlects Electrical 8 Computer Engineering Dr D J Jackson Lecture 1412 Structuring Elements Matlab Functions translate Tianslale slmclunng element Syntax 522 trenSJBLELSiINY Descrlpnon 52 iEllEEHSiZyvl lianslalesaslrucluimuelemenl s in ND space VisanNrelemenlvecluicontaining the u sels mm deswedlianslalmn in each mmensmn Class suppm sEand SEzaie STRELDbleclS VisavecluiuiduublEpiecismnvalues Elwcalampcmzmlmm nr n J mtgquot mum Dilation Mth A and B as sets in 22 the dilation ofA by B denoted A 98 is de ned as A Bz zmA This formulation is based on the re ection ofB about its origin and shi ing this re ection by z The dilation ofA by B is the set of all displacements 2 such that B and A overlap by at least one element Therefore another expression for the dilation ofA by Bis A B ZBz AA Set 8 is the structuring element Elwcalampcmzmlmm nr n J mtgquot Lmue1L1A Dilation Example B gbecause Bis symmetric with respect to its origin The dashed line shows the original set A and the solid boundary shows the limit beyond which any further displacements of the origin ofB by 2 would cause the intersection ofB and A to be empty All points inside this boundary constitute the dilation ofA by B The second case shows more dilation vertically than horizontally Electrrczn a Cnmruler Enmneermu Dr D l Jamsnn lecture lus Dilation Application One simple application of dilation is for br idging gaps In the image below the maximum break length is two pixels Although low pass filtering can be used to accomplish the same task this generates a grayscale image that must then be thresholded to produce a resulting binary image letorkHyr mm mmquot mem were wrlnn my mm m erg ralhlr mm flSllRi 97 in Sample ml ul mtrr rcSUlulltm with broken h lruclilrllig elm till murttrl lit tlrl mum wgmcnh cw gruntll Electrrczn a Cnmruler Enmneermu Dr D l Jamsnn lecture 15716 Matlab Dilation Example I imread brokenlines bmp imshowI tit1e origina1 SE stre1 square 3 Jimdilate I SE figure imshowJ tit1e Dilated 1 1 1 SE 1 1 1 1 1 1 Eludncal a cnnumer Englneenng m n J Jacksnn mu m Matlab Dilation Example 4 ngmun 3 IE rgmun 3 am Ede Ed Mew lnse Innis Mndaw eiv Ede Ed Mew lnse Innis indaw eiv Ongmai Diiaied Eludncal a Cnnumer Engineering m o J Jacksnn um 1671a Erosion With A and B as sets in 22 the erosion ofA by B denoted A98 is defined as AeB z Bz g A o In words the erosion of A by B is the set of all points 2 such that B translated by z is contained in A Dilation and erosion are duals of each other with respect to set complementation and reflection Therefore AeB A0 e E and A A 66 B AC 98 Electrical amp Computer Engineering Dr D J Jackson Lecture 1419 Erosion Example u39 A I I ifl B 7 1 e B r1 quot 154 1d l x 134 E 152 d E e 39r 1338 d a b c r k I 314 d e ng am FIGURE 94 a Set A b Square structuring element B c Erosion of A by B shown shaded d Elongated structuring element e Erosion of A by B using this element The dotted border in c and e is the boundary of set A shown only for reference Electrical amp Computer Engineering Dr D J Jackson Lecture 1420 Erosion and Dilation Application 0 One simple application of erosion is for eliminating irrelevant detail in terms of size from a binary image 0 Note In general dilation does not restore fully the eroded objects Erosion of a binary image with a 13x13 size structuring element and subsequent dilation of the result with the same element Electrical amp Compmer Engineering Dr D J Jackson Lecture 1421
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