Optimum Image Filtering Algorithm Over The Unit Sphere Er. Pradeep Kumar Jaswal

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1 Communication Technolog, Vol, Issue 9, September -03 ISSN(Online) ISSN (Print) Optimum Image Filtering Algorithm Over The Unit Sphere Er. Pradeep Kumar Jaswal Introduction A statistical shape model, which includes a network of deformable curves on the unit sphere as reference framework for seeking geometric features such as high curvature regions and labels such features via a deformation process is confined within a spherical map of the outer surface boundar. The mapping problem of spherical coordinate sstem, which include discontinuities at the poles and non uniform sampling, are overcome b defining the statistical shape variation in terms of projections of landmark points onto corresponding tangent planes of the sphere. The edge of an image describes the boundar between an object and the background. It represents a sudden change in the value of the image intensit function. So an edge separates two regions of different intensities. However all the edges in an image are not due to the change in intensit values, where parameters like poor focus or refraction can result in edge in an image []. The shape of edges in an image depends on different attributes like, lighting conditions, the noise level, tpe of material and the geometrical and optical properties of the object []. Gradient operators of first derivative like Sobel, Prewitt, Roberts and second derivative like Laplacian are used to find the edge in an image [3, 4, 6,, 5]. The efficient edge detection operator is evaluated subjectivel b visuall comparing the output images obtained with certain characteristics [5]. Kewords: statistical shape model, spherical coordinate sstem edge detection, gradient operator, image processing, root-mean-square error. Neurons are spreaded over the human brain is a thin convoluted sheet which are folds oriented outwards and inwards respectivel. It is believed that man cortical neural structures are functionall linked to the brain, although this relationship varies throughout the corte and is not well understood at present. Such assumption would have several applications. First, Neurons have their own natural path within deep brain structures in neurosurgical procedures. Second, it is well known that Neuron epansions are related to the underling connectivit of the brain, since the are influenced b forces eerted b connecting fibers. Therefore, structure analsis of the neurons is important in understanding normal variabilit, as well as in studing developmental disorders or effects of aging. Spatial normalization over the unit sphere is used for the mapping of data to define the coordinate sstem, b removing intersubject morphological differences, thereb allowing for group analsis to be carried out. The threedimensional (3 -D) coordinate sstem has Page 768

2 Communication Technolog, Vol, Issue 9, September -03 ISSN(Online) ISSN (Print) been etensivel used in the brain mapping literature, but surface-based coordinate sstems have also been proposed for studing neuron structure mapping over the brain. (a) Schematic drawing of a brain surface overlapped with neurons. (b) Schematic drawing of neurons curves after mapped to the unit sphere. The unit sphere is the reference space and the parameterized curves on the unit sphere form the training shape. (ASM) search was used to locate and label anatomical features in the images. Manuall etracted points, located on the outer surface, were spatiall normalized via a robust point matching algorithm. Mapping from the unit sphere to the corte readil provides segmentation and labeling of the neurons in three dimensions, which is (a) Schematic our ultimate drawing goal. of We a brain use surface a projection of overlapped each neural with point neurons. onto a respective plane tangent to the sphere, thus (b) Schematic overcoming drawing of limitations neurons curves of after the mapped customar the unit spherical sphere. The coordinates, unit sphere which is the reference include space nonuniform and the parameterized sampling and curves discontinuities the unit sphere at form the the training poles. The spherical model shape. on the unit sphere is applied to find and label neural curves on the spherical maps of datasets outside the training set, referred to as unseen datasets, using a full automated hierarchical deformation scheme. Neural curves on the outer corte are etracted via an inverse spherical mapping procedure. The hierarchical deformation scheme provides accurate registration results and is robust to suboptimal solutions. Neurons are projected onto the unit sphere via a nearl homothetic mapping procedure [5]. Our proposed model captures intersubject variabilit of the shape of the neurons and of the mean curvature along the curves. The statistical shape model for the neural curves is intended for use in automatic labeling and spatial normalization of cortical surfaces etracted from magnetic resonance images 3-D points located over the curves were modeled and an active shape model STATISTICAL SHAPE MODEL FOR CURVES ONTHE UNIT SPHERE Corresponding points on the two surfaces (a) and (b) are compared directl in the embedding space (c), the resulting variabilit is the Page 769

3 Communication Technolog, Vol, Issue 9, September -03 ISSN(Online) ISSN (Print) composite of the intrinsic variabilit and the variabilit introduced b the embedding of the surfaces. A statistical shape model (SSM) is a statistical representation of a shape obtained from training data. SSMs have been applied to image segmentation with great success, especiall in medical image applications, where boundaries between regions are weak in images and epert knowledge is necessar. Tpicall, a shape is represented as an ordered set of landmark points. We restrict our stud to space curves that reside on a surface embedded in three dimensions. We could view a set of space curves as a 3- D shape. However, in that case, the total variabilit of a space curve would be a composite of its own intrinsic variabilit, i.e. its position on the surface and of the variabilit caused b the embedding of the surface in three dimensions. The variabilit component can be eliminated b unfolding the surface and mapping its points to points on the unit sphere. We have chosen the unit sphere as the reference domain on which neural curves are projected in order to make our analsis of of the embeded cortical surface. The SSM consists of two parts: A point distribution model that encodes shape variabilit and On the unit sphere, there eists a natural coordinate sstem, namel the spherical coordinate sstem. However, we cannot use the spherical coordinates of our landmark points to calculate the shape variation directl, for three reasons. First, the coordinates of a given landmark point in a collection of training samples do not generall follow a normal distribution. This makes it inaccurate to approimate the shape distribution using mean and covariance. Second, because the sphere is not homeomorphic the coordinate has a discontinuit at the poles. The discontinuit in the coordinate sstem makes it impossible to find a smooth mapping to warp the model to the data. Third, points close to poles displa an artificiall high variation in and coordinates, simpl because the parametric curves are ver dense near the poles..0 STASTICAL SHAPE MODELING We can form a coordinate vector for the training samples as follows: A statistical representation of the features sought b the SSM. Assumptions for Constructing the Local Coordinate Sstem over the Unit Sphere Page 770

4 Communication Technolog, Vol, Issue 9, September -03 ISSN(Online) ISSN (Print) (Projection of Eigen vector radiall to the tangent plane passing through the mean point of the landmark) The Eigen vector, together with the mean shape, can be used to approimate an new eample through the following procedure:. Project each landmark point of a new eample to the corresponding tangent plane and calculate the coordinates for newl form matri.. Form a vector of coordinates. 3. Project to the subspace spanned b newl form matri b the eigen vectors. 4. Reconstruct the shape using the mean shape. coordinates of the landmark point 5. The mean and the variance of the curvature of each landmark point on the surface are computed. PROPOSED ALGORITHM Let S denote the corrupted image and for each piel S (i,j) denoted as Si,j a sliding or filtering spherical window of size (L+) X (L+) centered at Si,j is defined as shown in figure. The coordinates of this spherical window are Si,j = {Xi-u, j-v, -L u,v L}. S(i-,j-) S(i-,j) S(i-,j+) S(i,j-) S(i,j) S(i,j+) S(i+,j-) S(i+,j) S(i+,j+) Fig. A 3 3 Filtering window with S(i,j) as center piel. Set the minimum Spherical window size w=3;. Read the piels from the sliding window and store it in S. 3. Compute minimum (Smin), maimum (Sma) and median value (Smed) inside the window. 4. If the center piel in the window S(i,j),is such that Smin<S(i,j)<Sma, then it is considered as uncorrupted piel and retained. Otherwise go to step5. 5. Select the piels in the window such that Smin<Sij<Sma if number of piels is less than then increase the window size b and go to step,else go to step Difference of each piel inside the window with the mean value (Smean) is calculated as and applied to robust influence function. f() = /(σ + ) (7) Where σ is outlier rejection point, is given b, s (8) Where outlier and is s the maimum given b, (9) s N epected Where σ N is the local estimate of the image standard deviation and ζ is a smoothening factor. Here ζ =0.3 is taken for medium smoothening. Page 77

5 Communication Technolog, Vol, Issue 9, September -03 ISSN(Online) ISSN (Print) Piel is estimated using equation (0) and (), piel( l)* f ( ) S (0) ll f ( ) S () ll EDGE DETECTION USING GRADIENT OPERATORS Different first and second derivative gradient operators are eplored with eperiments to find edge map in a gra scale image. Comparative analsis of various edge detection gradient operators of first derivative like Sobel, Prewitt, Roberts and second derivative like Laplacian and Laplacian of Gaussian are performed. A new derivative filter of first derivative gradient tpe is proposed with aa novel approach to find better edge map in a grascale image. The proposed derivative operator is convolved to the input image for all coordinate points to obtain both the horizontal and vertical gradient components. The magnitude of the gradient component is calculated. The gradient component is normalized and threshold to a level to find the edge map information. Subjective and objective methods are used to evaluate the performance of the proposed operator with other eisting edge detection operators. The subjective method is used b visuall comparing of different edge detected output images with characteristics like contrast value in the image, strength of edge map, and noise content obtained b different derivative filters. The objective method like root-mean-square error is used to find the performance of different derivative filters. Image sharpening is a common technique in the edge detection process with the objective of enhancing edges in an image. The proposed technique called Fuzz Gaussian Filteration. Finall, to validate the efficienc of the edge detection and edge sharpening scheme, different algorithms are proposed and simulation stud has been carried out using MATLAB FIRST & SECOND DERIVATIVE FILTERS The gradient of an image f(, ) at the location (, ) is given b the two dimensional column vector [, ]. f G G f f () The magnitude of the first derivative is used to detect the presence of an edge in the image. The magnitude of this vector is given b [8]: G G mag( f ) () Here f /..and...f / are the rates of change of two dimensional function f(, ) along and ais respectivel. A piel position is declared as an edge position if the value of the gradient eceeds some threshold value, because edge points will have higher piel intensit values than those surrounding it [0, ]. Page 77

6 Communication Technolog, Vol, Issue 9, September -03 ISSN(Online) ISSN (Print) We have used a 3X3 region to denote image points of an input image [8, 9] as follows: W W W 3 W 4 W 5 W 6 W 7 W 8 W 9 G G ( W 7 ( W 3 W 8 W 6 W ) ( W 9 W ) ( W.3. Prewitt Operator 9 W W 4 W ) 3 W ) (4) The Prewitt s operator is given b the equations [8, 4]: 7 Fig.. A 3X3 region of an image... Sobel Operator G G Y W W W W 6 (5) The Sobel operator is given b the equations [8,, ]: W W f (, ), W f (, ), W3 f (, ) W f (, ), 4 W f (, ), 5 W f (, ) (3) 6 W f (, ), 7 f (, ), 8 W9 f (, ) Where, W to W9 are piels values in a sub image as shown in Fig.... Roberts Operator The Roberts operator is given b the equations [8, 3]:.4. Proposed Operator Our proposed operator is given b equations: G G ( W 7 ( W 3 W W ) ( W 8 W W ) ( W W W ) 4 3 W W ) (6).5. Laplace Operator The sign of the second derivative is used to decide whether the edge piel lies on the dark side or light side of an edge [5, 0]. The second derivative at an point in an image is obtained b using the laplacian operator []. The Laplacian for an image function f(, ) of two variables is defined as [, 6]: (7) f f f The Laplacian operator is given b the equation: 7 Page 773

7 Communication Technolog, Vol, Issue 9, September -03 ISSN(Online) ISSN (Print) f ( W W4 W6 W8 ) 4W 5 (8).6. Laplacian of Gaussian Operator The Laplace operator is ver much sensitive to noise. Hence it is clear that some sort of noise cleaning procedures must be preceded before the Laplacian. For noise smoothing the gaussian filter can be applied. Then the resultant procedure is called Laplacian of Gaussian (LOG). It can be defined as: g"( r, c) g( r, c)* G( r, c) (9) The root-mean-square error [8] between the input and output image is defined as: e rms MN M N 0 0 fˆ(, ) f (, ) (0) Where, denotes the original input image and f ˆ (, ) denotes the output image. Both the images have M rows and N columns and is the root-mean-square error. 3.. Fuzzfication and Defuzzfication process eistence of an edge can not have binar answer. The images have a continuouschange in brightness level as well as in shades. This information will be lost if we take binar answer to this problem. All the edge points constitute as set which is called an edge map. It is not wise to tell whether a certain piel belongs to the edge map or not, rather each of the piels should be assigned some membership towards the set. This process is called Fuzzfication. In this process, we perform Gaussian filtering with different values of sigma. Piels with membership value one definitel belong to the edge map where as piels with membership value equal to zero do not belong to the set. However piels with intermediate membership values ma or ma not belong to the edge map depending upon a prescribed threshold value. Thresholding operation is called a Defuzzfication process. After thresholding we obtain a binar image which is the edge map representation of the image. Though fuzz sets have a realistic approach towards the real world problems, the are seldom implemented directl, because for computer representation the data has to be in binar value. Hence the fuzz set is now transformed into binar set. Around 5 to 0% of the total piels generall belong to the edge map. Piels having membership less than threshold value are assigned value 0 and those having greater membership are assigned value of. The fuzz logic is different from the conventional binar logic. In binar logic the variable can be 0 and, but the fuzz variable can be an values in between o and. In digital images conformation of Page 774

8 Communication Technolog, Vol, Issue 9, September -03 ISSN(Online) ISSN (Print) Input Image Edge Map Fuzz ficatio n b Gaussi Fuzz Image De fuzfication B Thresholding to m - horizontall and 0 to n - verticall for m rows and n columns. 4. Fill the mask w with mask coefficients. 6. The sum of all the coefficients of each mask must be zero. 7. Compute Mask half-width, a = (m )/ // for a mask of odd size Fig.4. Block Diagram of Fuzz Gaussian Filter 4.0 METHODS The first algorithm is used to convolve the mask to the input image for all coordinate points of the image. The second algorithm describes the process of normalization and the third algorithm describes the process used to threshold the image at a particular level to find the edge map of a gra scale image. The fourth algorithm describes the general procedure used to find the edge map for all gradient operators. ALGORITHM 4.. Convolving an image with odd mask. Begin. Read all the piel vales of input image with M rows and N columns where, f(, ) represents the piel value at and co-ordinate..store all the piel vales in an integer matri of dimension M X N. 3. Select the mask w, which is an arra with dimension m X n, indeed from 0 7. Compute Mask half-height, b= (n )/ // for a mask of odd size 8. Create an M X N output image, g with M rows and N columns 9. for all piel coordinates, and, do 0. g(, ) = 0. end for. for = b to N b - do // column piels in the image ecluding Border piels 3. for = a to M a do //row piels in the image ecluding Border 4. Sum = 0 5. for k = -b to b do // For a single piel in the image 6. for j = -a to a do 7. sum = sum + w(k, j) f( + k, + j) 8. end for 9. end for 0. end for. end for. g(, ) = sum End ALGORITHM 4.. Normalization of an image for displa. Begin. Read all the piel vales of input image with M rows and N columns where, f(, ) represents the piel value at and co-ordinate.. Store all the piel vales in an integer matri of dimension M X N which is to Page 775

9 Communication Technolog, Vol, Issue 9, September -03 ISSN(Online) ISSN (Print) be normalized for M rows and N columns. 3. The output image is g(, ) with M rows and N columns 4. Calculate the minimum value for each column of the input matri. 5. Calculate the smallest value among all the minimum column values. 6. Calculate the maimum value for each column of the input matri. 7. Calculate the largest value among all the maimum column values. 8. Calculate range = largest value smallest value 9. for = to N do 0. for = to M do. g(, ) = (f(, ) smallest piel value) * 55 /range.. end for 3. end for End ALGORITHM 4.3. Thresholding an image Begin. Select a gra scale image with M rows and N columns where f(, ) represents the piel value at and coordinates.. Store all the piel values of the image in matri form. 3. Choose a value for the label. 4. for = to N do 5. for = to M do 6. if f(, ) is greater than level then 7. f(, ) = 55, it sets the point to white 8. else 9. f(, ) = 0, it sets the point to black 0. end. end for. end for End ALGORITHM 4.4. Edge detection b derivative operators. Begin. Select a gra scale input image.. Store all the piel values of the image along and coordinates in matri form. 3. Generate the convolution mask for different gradient operators and store it in different matrices. 4. The sum of all the coefficients of each mask must be zero. 5. Each mask along the horizontal and vertical direction is convolved with the input image. 6. The magnitude of the gradient vector is obtained. 7. Finall, the gradient vector is normalized and threshold to a particular level for displa of edge map information. End We have taken 8-bit grascale image of size 56 X 56 and the input image is processed using the different gradient first and second derivative operators like Sobel, Robert, Prewitt and Laplacian to find edge map [5,7, 3, 4]. The proposed mask for horizontal and vertical direction is convolved to the input image and then the magnitude of the gradient vector is obtained [, 7]. Finall it is normalized and threshold to find the edge map information RESULTS Page 776

10 Communication Technolog, Vol, Issue 9, September -03 ISSN(Online) ISSN (Print) Output from Sobel Operator: (a) (b) (c) (d) (a) Original Lena image (b) Horizontal component (c)vertical component (d) output image with edge map (a) Original Lena image (b) Horizontal component (c)vertical component (d) output image with edge map Fig.8.Results of Laplacian Operator 5.5. Output from Laplacian of Gaussian Operator: Lena image source b MathWorks Inc., USA (MATHLab)) Fig.5 Results of Sobel Operator 5. Output of Robert Operator (a) Original Lena image (b) Horizontal component (c)vertical component (d) output image with edge map Fig.9.Results of Laplacian of Gaussian Operator (a) Original Lena image (b) Horizontal component (c)vertical component (d) output image with edge map ( Lena image source b MathWorks Inc., USA (MATHLab)) Fig.6.Results of Roberts Operator 5.3. Output from Prewitt Operator (a) Original Lena image (b) Horizontal component (c)vertical component (d) output image with edge map Fig.7.Results of Prewitt Operator 5.4. Output from Laplacian Operator: 5.6. Output from proposed Operator: (a) Original Lena image (b) Horizontal component (c)vertical component (d) output image with edge map Fig.0.Results of Proposed Operator 5.6. Output from Fuzz Gaussian filter Fig..Edge Sharpening b Fuzz Gaussian filter 6.0. CONCLUSION The proposed operator s performance for edge detection in a nois image is evaluated both subjectivel and objectivel against the first Page 777

11 Communication Technolog, Vol, Issue 9, September -03 ISSN(Online) ISSN (Print) and second order derivative filters.the evaluation of edge detected images show that proposed operator, Sobel and Prewitt operator ehibit better performances respectivel. The Robert and Laplacian have poor performance in terms of contrast, edge map strength and noise content. Prewitt, Sobel and proposed operator have good contrast, edge map strength and low noise content then Laplacian of Gaussian. Prewitt is more acceptable than Roberts, Laplacian and Laplacian of Gaussian, while Sobel and proposed are more acceptable than Prewitt. It also shows that Laplacian is ver much sensitive to noise. The root mean square error of Laplacian and Robert is less than Prewitt, which is less than Sobel and proposed operator. The fuzz Gaussian filter has a good contrast and sharpness which is required to sharpen an image. This paper concludes that the subjective and objective evaluation of edge map shows that proposed, Sobel, Prewitt, Laplacian of Gaussian, Roberts and Laplacian ehibit better performance for edge detection respectivel and the results of the subjective evaluation matches with the results of the objective evaluation. REFERENCES [] E. Argle. Techniques for edge detection, Proc. IEEE, vol. 59, pp , 97. [] H.Chidiac, D.Ziou, Classification of Image Edges, Vision Interface 99, Troise- Rivieres, Canada, 999.pp [3] Hueckel.,M., A local visual operator which recognizes edges and line. J. ACM, vol. 0, no. 4, pp , Oct [4] M.Heath, S. Sarkar, T. Sanocki, and K.W. Bower. A Robust Visual Method for Assessing the Relative. Performance of Edge Detection Algorithms. IEEE Trans. Pattern Analsis and achine Intelligence, vol. 9(), pp , Dec [5] M. Heath, S. Sarkar, T. Sanocki, and K.W. Bower. Comparison of Edge Detectors: A Methodolog and Initial Stud Computer Vision and Image Understanding, vol. 69, no., pp Jan [6] M.C. Shin, D. Goldgof, and K.W. Bower. Comparison of Edge Detector Performance through Use in an Object Recognition Task. Computer Vision and Image Understanding, vol. 84, no., pp , Oct. 00. [7] T. Peli and D. Malah. A Stud of Edge Detection Algorithms. Computer Graphics and Image Processing, vol. 0, pp. -, 98. [8] R. C. Gonzalez, R. E. Woods, Digital Image Processing, nd ed., Upper Saddle River, New Jerse, Prentice-Hall, Inc., 00. [9] Pratt, W.K., Digital Image Processing, 4 th ed., Hoboken, New Jerse, John Wile & Sons, Inc, 007. [0] F.Bergholm. Edge focusing, in Proc. 8th Int. Conf. Pattern Recognition, Paris, France, pp , 986. [] Chanda, B., and Dutta Majumder, D. Digital Image Processing and Analsis, India, Prentice Hall of India, 008. [] Sobel, I.E., Camera Models and Machine Perception, Ph.D. dissertation, Stanford Universit, Palo Alto, Calif, 970. [3] Roberts, L.G., Tippet, J.T., Machine Perception of Three- Dimensional Solids, Cambridge, Mass, MIT Press, 965. [4] Prewitt, J.M.S., Lipkin, B.S., and Rosenfeld, A. Object Enhancement and Etraction., New York, Academic Press, 970. [5] Chanda, B., Chaudhuri, B.B. and Dutta majumder, D., A differentiation/ enhancement edge detector and its properties, IEEE Trans. On Sstem, Man and Cbern., SMC- 5:pp. 6-68, 985. [6] Marr, D.C. and Hildreth, E., Theor of edge detection, Proc. Roal Soc. Lond., vol. B, pp. 87-7, 980. [7] Hueckel, M., An operator which locates edges in digitized pictures, J. Assoc. Comput., vol. 8, pp. 3-5, 97. [8] Page 778

12 Communication Technolog, Vol, Issue 9, September -03 ISSN(Online) ISSN (Print) Forsth, D.A., and Ponce, J., A Modern Approach, India, Prentice Hall of India, 003. [9] Gilat, A., Matlab An Introduction with Applications, New York, John Wile & Sons, Inc, 004. [0] Duda, R.O, Hart, P.E., Pattern Classification and Scene Analsis, New York, Wile-Interscience, 00. [] Image Processing Toolbo, User guide for use with Matlab, the MathWorks Inc., 00. [] Cganek, C., and Siebert, J.P., An Introduction to 3D Computer Vision Techniques and Algorithms, New York, John Wile & Sons, Ltd, 009. [3] Ziou, D. and S. Tabbone, Edge detection techniques an overview. Int. J. Patt. Recog. Image Anal., vol. 8, pp , 998. [4] Yakimovsk Y., "Boundar and object detection in real world image", Journal ACM, vol. 3, pp , 976. [5] Davis, L. S., "Edge detection techniques", Computer Graphics Image Process., vol. 4, pp , Page 779