Morphological Operations 1 Outline • Basic concepts: • Erode and dilate • Open and close. • GltGranulometry • Hit and miss transform • Thinning and thickening • Skeletonization and the medial axis transform • Introduction to gray level morphology. 2 Pixel connectivity • We need to define which pixels areWarning: neighbors. • Are the Pdarkilixe ls are sampl es, pix els in thisnot arraarray squares . connected? 4 Pixel connectivity • We need to define which pixels are neighbors. • Are the dark pixels in this array connected? 5 Pixel Neighborhoods 4-neighborhood 8-neighborhood 6 Pixel paths • A 4-connected path between pixels p1 and pn is a set of pixels {p1, p2, …, pn} such that pi is a 4-neighbor of pi+1, i=1,,,…,n-1. • In an 8-connectedhd path, pi is an 8-neihbighbor of pi+1. 7 Connected regions • A region is 4-connected if it contains a 4- connected path between any two of its pixels. •A regiiion is 8-connectedifid if it contai ns an 8- connected path between any two of its pixels. 8 Connected regions • Now what can we say about the dark pixels in this array? • What about the light pixels? 9 Connected components labelling • Labels each connected component of a binary image with a separate number. 1 1 1 1 1 1 1 1 1 112213111 111221111 111111111 141155111 111155511 661111551 761188111 761188111 10 Foreground labelling • Only extract the connected components of the foreground 1 1 1 1 1 1 1 1 1 110010111 111001111 111111111 101100111 111100011 001111001 201100111 201100111 11 What Are Morphological Operators? • Local pixel transformations for processing region shapes • Most often used on binary images • Logical transformations based on comparison of pixel neighbourhoods with a pattern. 18 Simple Operations - Examples • Eight-neighbour erode – a.k.a. Minkowski subtraction • Erase any foreground pixel that has one eight-connected neighbour that is background. 19 8-neighbour erode Threshold Erode ×1 Erode ×2 Erode ×5 20 8-neighbour dilate • Eight-neighbour dilate • a.k.a. Minkowski addition • Paint any background pixel that has one eight-connected neighbour that is foreground. 21 8-neighbour dilate Dilate ×1 Dilate ×2 Dilate ×5 22 Why? • Smooth region boundaries for shape analysis. • Remove noise and artefacts from an imperfect segmentation. • Match particular pixel configurations in an image for simple object recognition. 23 Structuring Elements • Morphological operations take two arguments: • A binaryyg image •A structuring element. • Compare the structuring element to the neighbourhood of each pixel. • This determines the output of the morppghological op eration. 24 Structuring elements • The structuring element is also a binary array. • A structuring element has an origin . 1 1 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 0 0 1 0 0 25 Binary images as sets • We can think of the binary image and the structuring element as sets containing the pixels with value 1. 1 1 1 0 0 0 1 1 0 0 I = {(1,1), (2,1), (3,1), 0 0 0 0 0 (2,2), (3,2), (4,4)} 0 0 0 1 0 0 0 0 0 0 (Matlab) S = SPARSE(X) converts a sparse or full matrix to sparse form by squeezing out any zero elements26 Some sets notation • Union and • Difference iiintersection: I1 \ I2 = {x : x ∈ I1 and x ∉ I2} I1 ∪ I2 = {}x : x ∈ I1 or x ∈ I2 I1 ∩ I2 = {}x : x ∈ I1 and x ∈ I2 • We use φ for the • Complement empty set . I C = {}x : x ∉ I 27 More sets notation •The symmetrical set of S with respect to point o is 0 1 1 1 0 ( 1 1 1 1 1 S = {o − x : x ∈ S} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 0 28 Fitting, Hitting and Missing • S fits I at x if {y : y = x + s,s ∈ S} ⊂ I • S hits I at x if {y : y = x + s, s ∈ S} ∩ I ≠ φ • S misses I at x if {y : y = x + s, s ∈ S} ∩ I = φ 29 Fitting, Hitting and Missing Image Structuring element 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 30 Erosion • The image E = ISis the erosion of image I by structuring element S. ⎧1 if S fits I at x E(x) = ⎨ ⎩0 otherwise E = {y : y = x + s ∈ I for every s ∈ S} 31 Implementation (naïve) % I is the input image, S is a structuring element % with origin (ox, oy). E is the output image. function E = erode(I, S, ox, oy) [X,Y] = size(I); [SX, SY] = size(S); E = ones(X, Y); for x=1:X; for y=1:Y fi1fj1for i=1:SX; for j=1:SY if(S(i,j)) E(x,y ) = E(x,y ) & ImageEntry(I, x +i-ox, y+j-oy); end end; end end; end 32 Example 1 1 1 1 1 Structuring 1 1 1 1 1 element 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 33 Example 0 1 1 1 0 Structuring 1 1 1 1 1 element 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 34 Example 1 0 0 0 0 Structuring 0 1 0 0 0 element 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 35 Dilation • The image D = I ⊕ S is the dilation of image I by structuring element S. ⎧1 if S hits I at y D(x) = ⎨ ⎩0 otherwi se D = {y : y = x + s ∈ I for any s ∈ S} 36 Alternative Implementation • For another implementation of erosion and dilation, note that: E(x) = min(I(x + s)) s∈S D(x) = max(I(x + s)) s∈S 37 Example 1 1 1 1 1 Structuring 1 1 1 1 1 element 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 38 Example 0 1 1 1 0 Structuring 1 1 1 1 1 element 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 39 Example 1 0 0 0 0 Structuring 0 1 0 0 0 element 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 40 Erosion and dilation • Erosion and dilation are dual operations: (I S)C = I C ⊕ S( • Commutativity and associativity I ⊕ S = S ⊕ I (I ⊕ S) ⊕T = I ⊕ (S ⊕T ) I S ≠ S I (I S) T = I (S ⊕T ) 41 42 Input Output Structuring Element: 43 Opening and Closing •The opening of I by S is I o S = (I S) ⊕ S •The closing of I by S is I • S = (I ⊕ S) S 44 Opening and Closing • Opening and closing are dual transformations: C C ( (I • S) = I o S • Opening and closing are idempotent operations: I o S = (I o S) o S I • S = (I • S) • S 45 Example close 1 1 1 1 1 Structuring 1 1 1 1 1 element 1 1 1 1 1 1 1 1 1 1 open 1 1 1 1 1 46 Example close 0 1 1 1 0 Structuring 1 1 1 1 1 element 1 1 1 1 1 1 1 1 1 1 open 0 1 1 1 0 47 Example close 1 0 0 0 0 Structuring 0 1 0 0 0 element 0 0 1 0 0 0 0 0 1 0 open 0 0 0 0 1 48 Morphological filtering • To remove holes in the foreground and islands in the background, do both opening and closing . • The s ize and sh ape of th e structuri ng el ement determine which features survive. • In the absence of knowledge about the shape of filifeatures to remove, use a circular structuring element. 49 Example Close then open 1 1 1 1 1 Structuring 1 1 1 1 1 element 1 1 1 1 1 1 1 1 1 1 Open then close 1 1 1 1 1 50 Example Close then open 0 1 1 1 0 Structuring 1 1 1 1 1 element 1 1 1 1 1 1 1 1 1 1 Open then close 0 1 1 1 0 51 Example Close then open 1 0 0 0 0 Structuring 0 1 0 0 0 element 0 0 1 0 0 0 0 0 1 0 Open then close 0 0 0 0 1 52 Count the Red Blood Cells 53 Granulometry • Provides a size distribution of distinct regions or “granules” in the image. • We open the image with increasing structuring element size and count the number of regions after each operation . • Creates a “morphological sieve”. 54 Granulometry function gSpec = granulo(I, T, maxRad) % Segment the image I B = (I>T); % Open the image at each structuring element size up to a %i% maximum and count the rema iiiining regions. for x=1:maxRad O = imopen (B,s tre l(‘dis k’,x )); numRegions(x) = max(max(connectedComponents(O))); end gSpec = diff(numRegions); 55 Count the Red Blood Cells 56 Threshold and Label 57 Disc(11) 58 Disc(19) 59 Disc(59) 60 Number of regions smaller than radius Number ofRe Struct.