Image processing. The'image'model'used'here: What'is'an'image? 1 Image representation 2 Image Filtering 3 Morphological transformations

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1 Image processing Content 2 Image representation 2 Image Filtering 3 Morphological transformations 2 2 several'possible'defini/ons' computer'point'of'view':'unsigned'char'table Physicist:'observa/on'of'an'environment'by'an'op/cal' sensor'(2d'digi/zed'signal) Mathema/cian:'the'projec/on'of'a'3D'space'on'a' plane... What'is'an'image? 3 With':' L:'number'of'lines'(height) C:'number'of'columns'(width) M' 2 N p The'image'model'used'here: f : [,L ] [,C ] [,M] p I = f(x, y) p'=''for'a'luminance'image'(grey'level) p'='3'for'a'color'image'(rgb,'hsv,'...) 4 3 4

2 Pixels, neighbourhood, and distances Triangle Square Hexagonal Distance between two pixels Each pixel can be localised by its co-ordinates (x,y) into the image plane. Distances between pixels may be defined A distance measure must have the following properties: d(p, Q) > d(p, Q) =d(q, P ) d(p, Q) apple d(p, R)+d(R, Q) Principal'distances Neighbourhood Manathan'distance'' d (P, Q) = x p x q + y p y q Euclidian'distance d 2 (P, Q) = (x p x q ) 2 +(y p y q ) 2.5 Chessboard'distance d (P, Q) = max( x p x q, y p y q ) V k (p) ={Q :<d(p, Q) apple k}

3 Image'Scaling Image'Scaling In computer graphics, image scaling is the process of resizing a digital image bi-linear interpolation Knn 9 9 Image'Scaling Image'Scaling bi-linear interpolation bi-linear interpolation 2 2

4 Image'Scaling Without With bi-cubic interpolation Color'images Images'are'generaly'modelized'by'a'3'components' vector Addi/f'model'RGB'(Red,' Green,'Blue) Many'color'spaces'exist'(RGB,'HSV,'Lab,'YCrCb,' YUV,'...) Only'two'of'them'are'presented'hereaXer'

5 Addi/f'model'RGB'(Red,' Green,'Blue) Addi/f'model'RGB'(Red,' Green,'Blue) HSV'model:'[Hue,'Satura/on,'Value] HSV'model:'[Hue,'Satura/on,'Value] Value'is'the'grey'level' (luminance) Hue'(wavelength)'is'measured' by'the'angle'along'the'value' axis. Satura*on'is'the'module'of' the'normal'vector'along'the' value'axis chroma/c'cube'observed'from' the'white'color Lateral'view'of'HSV'hexagon

6 HSV'model:'[Hue,'Satura/on,'Value] Constant'brightness HSV'model:'[Hue,'Satura/on,'Value] Constant'satura/on HSV'model:'[Hue,'Satura/on,'Value] Constant'hue HSV'decomposi/on

7 Addi/onal'informa/on'on'color' images Content The Colour Image Processing Handbook Sangwine, Stephen J.; Horne, Robin E.N. (Eds.) 998, 456 p. Hardcover, ISBN Image representation 2 Image filtering 3 Morphological transformations P. Bonton et al. Lasmea ISBN : in French Image Filtering methods'are'divided'into'two'main' categories Global'methods'(the'same'func/on' is'applied'on'all'the'pixels) Local'methods'(the'func/on'applied' to'one'pixel'is'related'to'it' neighbourhood) Image Filtering: Global Methods The'same'func/on'is'applied'on'all'the' pixels H(x) =Card{p : I(p) =x} Histogram:'a' basic'tool'for' global'filtering

8 Image Filtering: Global Methods Histogram:'some'examples Image Filtering: Global Methods Global'transforma/ons filtered image Input image Image Filtering: Global Methods histogram'stretching' Image Filtering: Global Methods histogram'stretching' filtered image 255 min max 255 input image

9 Image Filtering: Global Methods histogram'equaliza/on' Image Filtering: Global Methods histogram'equaliza/on' Image Filtering: Global Methods binary image binariza/on 255 Image Filtering: Global Methods And'many'other'transforma/ons: stretching threshold 255 Input image equaliza/on Area'extrac/on' Inverted'image' Gamma'correc/on

10 Spatial Methods Def:'modify'the'pixels'in'an'image' based'on'some'func/on'of'a'local' neighborhood'of'the'pixels' Spatial Methods Def:'modify'the'pixels'in'an'image' based'on'some'func/on'of'a'local' neighborhood'of'the'pixels' I 2 (x) =f(i (x),v I (x)) I 2 (x) =f(i (x),v I (x)) Spatial Methods Two'categories: linear'based'filters, non^linear'based'filters. Linear'based'filters: The'simplest Replace'each'pixel'by'a'linear'combina/on'of'its' neighbors.' 'The'prescrip/on'for'the'linear'combina/on'is' called'the' convolu/on'kernel

11 Let'W'be'the'kernel'(matrix)'of'size' [^n,n]x[^m,m]' I 2 (x) = X u2w W (u)i (x + u) 4 Some'classic'kernels'(average'operators) W A 9 W 2 A W = 2 4 2A Some'classic'kernels'(average'operators) Average (neighbourhood) 5x5 Classic'kernels'(Gaussian'filter)! i 2 + j 2 W (i, j) =C exp

12 Classic'kernels'(Gaussian'filter) Gaussian (sig = 3 and support = 5x5) Classic'kernels'(Shen^Castan)! i + j W (i, j) =C exp b Gradient'approxima/on'kernels'(Sobel' filter) Gradient'approxima/on'kernels'(Sobel' filter) W Horizontal 2 A 2 Vertical W 2 2 2A W 2 = W T 47 Horizontal 23 Vertical

13 Gradient'approxima/on'kernels' (Laplacian'filter) L(x, y) Approximated'by: W 4 A W 8 A Gradient'approxima/on'kernels' (Laplacian'filter) Linear'filters:'Some'proper/es'(separable' filters) Linear'filters:'Some'proper/es'(separable' filters) I 2 (x, y) =I (x, y) H xy (x, y) If H x,y (x, y) =H x (x) H2 y (y) Then I 2 (x, y) =[I (x, y) H x (x)] H2 y (y) a a b ba = a a 2 ab a ab b 2 aba a 2 ab a

14 Linear'filters:'Some'proper/es'(separable' filters) Exercise:'show'that'the'following'filters' are'separable ^'Sobel'filter' ^'Gaussian'filter 53 Non4linear4filters ^'Mathema/cal'Neighborhood'Operators ^'Calcula/on'within'the'kernel'is'defined'by'non^ linear'mathema/cal'and''sta/s/cal'opera/ons' 'Minimum'' 'Maximum'' 'Median 'Range 'Majority'' 'Standard'devia/on,' Median4filter ^'Robust'Filter ^'Non^linear'opera/on ^'Each'pixel'is'modified'according'to'the' median'value'of'it'neighbourhood' Can be computed using a quick sort algorithm Median4filter Input Mean Median

15 Median4filter Content Image representation 2 Image filtering 3 Morphological transformations MM'was'originally'developed'for'binary' images,'and'was'later'extended'to' grayscale'func/ons'and'images What can we do with MM? born'in'964'from'the'collabora/ve'work' of'georges4matheron'and'jean4serra,'at' the'école&des&mines'de'paris,'france Remove noise separate shapes compare shapes

16 Main'idea:''probe'an'image'with'a'simple,'pre^ defined'shape,'drawing'conclusions'on'how'this' shape'fits'or'misses'the'shapes'in'the'image.' Some structuring elements This'simple'"probe"'is'called'structuring'element,' and'is'itself'a'binary'image'(i.e.,'a'subset'of'the' space'or'grid) Basic'operators:'erosion Basic'operators:'dila/on Erosion of the binary image A by the structuring element B: Before SE After Dilation of the binary image A by the structuring element B: SE Before After A B = {z 2 E B z A} A B = {z 2 E (B s ) z \ A 6=?} translation of B by z symmetric of B

17 Example Example Initial image eroded time Initial image dilated time eroded 2 times eroded 3 times dilated 2 times dilated 3 times Basic'operators:'opening Basic'operators:'closing Before After Before After A B =(A B) B A B =(A B) B

18 Basic'operators:'opening Basic'operators:'closing Initial image opening Initial image closing Basic'operators:'opening+closing Initial image result Some'proper/es A B A A A B A A B B A (A B) =A B A (A B) =A B

19 References J. Serra, Image Analysis and Mathematical Morphology, Academic Press, New-York, 982. Image Processing Exercices histogram stretching (<=I(x)<=9 gray levels) ) compute the original histogram 2) compute the stretched histogram J. Serra (Ed.), Image Analysis and Mathematical Morphology, Part II: Theoretical Advances, Academic Press, London, 988. P. Soille, Morphological Image Analysis, Springer- Verlag, Berlin, Image Processing Exercices Image scaling: (<=I(x)<=9 gray levels) ) give the general bi-linear expression 2) compute the 2x scaled image A Original image B A Interpolated image 75 Image Processing Exercices Filtering Compute the filtered image for : ) W A 9 2) a median filter (3x3 support) Conclude

20 Image Processing Exercices Mathematical Morphology Propose a binary structured element and a set of morphological transformations to remove the noise and close the square Noise 77 77