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Morphological Image Processing Digital Image Processing MORPHOLOGICAL Hamid R. Rabiee Fall 2015 Image Analysis 2 Morphological Image Processing Edge Detection Keypoint Detection Image Feature Extraction Texture Image Analysis Shape Analysis Color Image Processing Template Matching Image Segmentation Image Analysis 3 Figure 9.1 of Jain book Image Analysis 4 Ultimate aim of image processing applications: Automatic description, interpretation, or understanding the scene For example, an image understanding system should be able to send the report: The field of view contains a dirt road surrounded by grass. Image Analysis 5 Table 9.1 of Jain book Image Analysis 6 original binary erosion dilation opening closing Morphological Analysis Edge Detection Keypoint Detection 7 Morphological Image Processing (MIP) Outline 8 Preview Preliminaries Dilation Erosion Opening Closing Hit-or-miss transformation Outline (cont.) 9 Morphological algorithms Boundary Extraction Hole filling Extraction of connected components Convex hall Thinning Thickening Skeleton Outline (cont.) 10 Gray-scale morphology Dilation and erosion Opening and closing Rank filter, median filter, and majority filter Preview 11 Mathematical morphology a tool for extracting image components that are useful in the representation and description of region shape, such as boundaries, skeletons, and convex hull Can be used to extract attributes and “meaning” from images, unlike pervious image processing tools which their input and output were images. Morphological techniques such as morphological filtering, thinning, and pruning can be used for pre- or post-processing All the concepts are introduced on binary images. Some extensions to gray- scale images are discussed later. Preliminaries 12 The language of mathematical morphology is set theory: In binary images, the sets in question are members of the 2-D integer space 풁ퟐ, where each element of a set is a 2-D vector whose coordinates are the (x, y) coordinates of a white pixel (by convention) in the image White pixels of the image represent foreground (1 pixels) binary images Black pixels of the image are background (0 pixels) background foreground Preliminaries 13 In addition to basic set operations including union, intersection, complement, and difference, we will need: Translation: Reflection: Preliminaries 14 Operations in morphological image processing are shift-invariant Each operation needs a structuring element (SE) SE is a binary image, which it’s origin must be specified Some examples are cross, square, and circle SEs cross square circle Dilation 15 The dilation of set A by structuring element B, is defined as Figure 9.6 of Gonzalez book Dilation 16 Expands the size of foreground objects Smooths object boundaries Closes holes and gaps Original image Dilation with 3 * 3 cross SE Dilation with 7 * 7 cross SE Original image is taken from MIT course slides Dilation 17 Using dilation to fix a text image with broken characters Original image Dilation with 3 * 3 cross SE Dilation with 5 * 5 cross SE Original image is taken from figure 9.7 of Gonzalez book Dilation 18 Dilation is commutative Dilation is associative Note: can be decomposed to structuring elements and This decomposition can cause speed-up Erosion 19 The erosion of set A by structuring element B, is defined as Figure 9.4 of Gonzalez book Erosion 20 Shrinks the size of foreground objects Smooths object boundaries Removes peninsulas, fingers and small objects Original image Erosion with 3 * 3 cross SE Erosion with 7 * 7 cross SE Original image is taken from MIT course slides Erosion 21 Using erosion to clear the thin wires in an image Original image Erosion with a disk of radius 10 Erosion with a disk of radius 5 Erosion with a disk of radius 20 Original image is taken from figure 9.5 of Gonzalez book Relationship between dilation and erosion 22 Duality: Dilation and erosion are duals of each other with respect to set complementation and reflection and Proof Opening 23 The opening of set A by structuring element B, is defined as The geometric interpretation of opening Figure 9.8 of Gonzalez book Closing 24 The closing of set A by structuring element B, is defined as The geometric interpretation of closing is based on opening using duality property Figure 9.9 of Gonzalez book Relationship between opening and closing 25 Duality: Opening and closing are duals of each other with respect to set complementation and reflection and Proof: homework Open filter & close filter 26 Original image Result of opening Result of opening followed by closing : 3 * 3 square SE Figure 9.11 of Gonzalez book Hit-or-miss transformation 27 The hit-or-miss transformation of by ,where is a structuring element pair is defined as Suppose that is enclosed by a small window . The local background of with respect to is defined as the set difference – , and the structuring element B2 is as follows example Hit-or-miss transformation 28 Original image ( to be more visible) Structuring Structuring element B1 element B2 Morphological algorithms 29 Boundary Extraction Hole filling Extraction of connected components Convex hall Thinning Thickening Skeleton Pruning Boundary extraction 30 The boundary of set can be obtained using erosion Also the we can use dilation to obtain boundary The difference between dilation and erosion results thicker boundary Boundary extraction 31 Original image Original image is taken from MIT course slides Hole filling 32 Hole: background region surrounded by a connected border of foreground pixels Assumption: a point inside each hole region is given, , and is symmetric In each iteration Termination is at iteration step if The set union of and contains all filled holes and their boundaries Figure is taken from Gonzalez book Extraction of connected components 33 Connected component: refer to definition! Assumption: a point on one of the connected components of set is given and . The objective is to find all the elements of this connected component. In each iteration: Termination is at iteration step if is the set containing all the elements of the connected component. Figure is taken from Gonzalez book Convex hall 34 The convex hall of an arbitrary set is the smallest convex set containing .. Assumption: represent four structuring elements. In each iteration: : hit-or-miss transformation Termination is at iteration step if and we define Then convex hall of is Limiting growth of convex hall algorithm along the vertical and horizontal directions Convex hall 35 Limiting growth of convex hall algorithm Figure is taken from Gonzalez book Thinning 36 The thinning of a set by a structuring element is defined in terms of hit-or- miss transformation To thin symmetrically a sequence of structuring elements is used Where is a rotated version of Definition of thinning by a set of structuring elements Thinning 37 Figure is taken from Gonzalez book Thickening 38 Thickening is the morphological dual of thinning and is defined as Definition of thickening by a set of structuring elements Usual procedure in practice Thin the background of a set in question and then complement the result. Afterwards remove the resulting disconnected points. The thinned background forms a boundary for the thickening process which is one of the principal reasons for using this procedure. Thickening 39 Result of Thinning of Remove disconnected points Final result of Thickening of Figure is taken from Gonzalez book Skeleton 40 First we define The skeleton of set A is can be reconstructed using The resulting set may be disconnected Other algorithms are needed if the skeleton must be maximally thin, connected and minimally eroded Skeleton 41 Original set Reconstructed set Morphological skeleton Figure is taken from Gonzalez book Gray-scale morphology 42 The basic operations introduced for binary images are extendable to gray- scale images In this case f(x, y) is a gray-scale image and b(x, y) is a structuring element Structuring elements in gray-scale morphology belong to one of two categories: non-flat and flat Gray-scale SEs are used infrequently in practice Dilation and erosion 43 The dilation with flat structuring element is defined as The erosion with flat structuring element is defined as The dilation with non-flat structuring element is defined as The erosion with non-flat structuring element is defined as Dilation and erosion 44 Example: Flat SE Original image Dilation with 5 * 5 square SE Erosion with 5 * 5square SE Width of lines is not preserved Dilation and erosion 45 As in the binary case, dilation and erosion are duals with respect to function complementation and reflection Where and Simplifying the notation Similarly Opening and closing 46 The expressions for Opening and closing gray-scale images have the same form as their binary counterparts. Opening Closing Duality of opening and closing and Opening and closing 47 Example 1: 2-D image Original image Opening with 5 * 5 square SE Closing with 5 * 5 square SE Width of lines is preserved Opening and closing 48 Example 2: Original 1-D signal 1-D signal Flat SE, pushed up underneath the signal Opening Flat SE, pushed down along the top of the signal Closing Figure 9.36 of Gonzalez book Rank filter 49 Generalization of flat dilation/erosion: instead of using min or max value in window, use the p-th ranked value example Increases robustness against noise Best-known example: median filter for noise reduction Concept useful for both gray-level and binary images All rank filters are commutative with thresholding Concepts are taken from MIT course slides Median filter and majority filter 50 Gray-level images example Median filter: instead of using min or max value in window, use median value Binary images example Majority filter: instead of using min or max value in window, use majority value Concepts are taken from MIT course slides References 51 Gonzalez book: Rafael C. Gonzalez, and Richard E.
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