Prewitt, Sobel and Scharr gradient 5x5 convolution matrices

Transcription

1 Prewitt, Sobel and Scharr gradient x convolution matrices First Draft, February Second Draft, June Guennadi Henry Levkine hlevkin at yahoo.com Vancouver, Canada. Abstract. Prewitt, Sobel and Scharr x gradient operators are very popular for edge detection. This article demonstrates how to get Sobel and Scharr gradient operators analytically. For that was using linear approximation of brightness in window x. Every pixel in window has its own weight. Special weight matrix selection gives Sobel gradient operator. Similar considerations were made for Prewitt and Scharr gradient operators. t will also demonstrate how to build x Sobel and Scharr gradient operators using similar weight matrices. Selection of different weight matrices gives a lot of new smooth and gradient operators. Key words image processing, image filtration, image gradient, edge detection, Prewitt operator, Sobel operator, Scharr operator, local polynomial approximation ntroduction n this article we use dimensional Local Polynomial Approximation LPA method, which will be shortly described here more detailed description is in []. n this method for every pixel we calculate D polynomial, which approximates image intensity in some block around the pixel. A common view for such a polynomial is n i j x, y cij x y i, j Here we consider only linear approximation and use special notation For, x, y * x * y, calculation we use Min Square Fit Method to minimize function where i, j,, wij * i * j i, i, j j w ij Weight of pixel i, j in x window,, - Coefficients of linear approximation i, j Brightness of pixel i, j Gradient of function x, y is defined as a vector with coordinates G x x, y x, y x ; G y x, y x, y y ;

2 n polynomial approximation the components of gradient are x G y G Using different weight matrices we can generate many different convolution kernel matrices for gradient estimation. Here we summarise masks for x and x blocks for gradient calculations. For making of calculation more simple we give table of differences between predicted x,y and measured x,y For calculation of,, we make the sum of squares of elements from this table, multiplied on their weights. Prewitt gradient operators x and x We have to select weight matrix to get x Prewitt gradient operators w This weight matrix is uniform- alee pixels around the center have the same weight. For x blocks estimation of,, coefficients is obvious. Convolution kernels for their calculation are w w 7 9 Convolution matrices for gradient do not depend on parameter w in the center of weight matrix. f we do quantitative analysis of image, and we use gradient operators and 9, for consistency we recommend to use filtration mask 7 or mask from article []. f we process image with -byte brightness for pixel, minimal possible X and Y gradient values equal -7, maximum equals 7 don t forget multiplier /. x Prewitt weight matrix is uniform as in the x case

3 W For x blocks convolution kernel for calculation of is smoothing W W Kernels for gradients are, don t depend from the parameter W in the center of weight matrix. Sobel gradient operators x and x When we calculate kernels for coefficient estimation, all the pixels in the block have the same weight. But it is possible to use different pixel weights higher for pixels close to block center. We consider here weight matrix Sobel weight matrix / / / / w or W This weight matrix is isotropic. t means that weights are dropping as r / in any direction, where r is the distance between center of matrix and element of it.

4 Now we calculate coefficients,, using this weight matrix. t was found using LPA that corresponding kernels are W 7 These kernels are well known and widely used, kernel for is used for noise filtration. Kernels for, are called Sobel s kernels and are used for gradient estimation. We named kernel for as Rosenfeld s smoothing kernel. This approach using pixel weight matrices gives rich opportunities for creating kernels. t is convenient for different tasks in image processing. Let s try to build x weight matrix, which is generalization of x Sobel weight matrix. n this matrix weights are decreasing by r / law, where r is the distance between center of matrix and current element of matrix. This weight matrix looks like w or W Center of it equals to Sobel weight matrix x. We call this matrix as Sobel x weight matrix. After LPA calculation we found that coefficient calculating using convolution matrix W W 9 Coefficients, gradients are calculating using convolution matrices

5 These convolution matrices and represent Sobel x operators for x- and y-gradients.. Scharr gradient operators using x and x masks One of the popular gradient operators is Scharr operator x. t also can be get from linear LPA using special weight matrix. We call it Scharr weight matrix W This matrix is not isotropic weights in diagonal directions drop faster than in vertical and horizontal directions. After minimization of corresponding,, we get convolution matrices for,, w w We introduce simplified Scharr like weight matrix which is also anisotropic because weights in diagonal directions drop faster compared to horizontal and vertical directions

6 / / / / w or W Kernels for,, calculations are w w Comparing with Sobel operators we see that they look similar, but Sobel weight matrix is isotropic and Scharr weight matrix is anisotropic. Here is generalization of simplified Scharr weight matrix to x case / / / / / / / / / / / / / / / / / / / / w or W We found this matrix, based on assumptions - n horizontal and vertical directions the weight is proportional to reciprocal distance from the center - n diagonal directions the weight is proportional to reciprocal distance* / - t the direction between horizontal and diagonal we use assumption that the weight is proportional to reciprocal distance* / After calculation we found that,, equals W W 7

7 9 New x convolution matrices modified Prewitt for gradient calculation Weight matrix here is w or * w Convolution matrix for coefficient is * * w w Convolution matrices for gradient are

8 Comparing of Prewitt, Sobel and Scharr x and x gradient operators For comparison of gradient operators we use classic test image Lenna of the size x from []. For every gradient operator we calculate corresponding -bytes image for x gradient operators and another for x gradient operators. They are presented in Pic. Pic.. After that we compare results for x Prewitt, Sobel and Scharr and results for x Prewitt, Sobel and Scharr. We are working with grey scale Lenna image, but gradient matrices are represented in the color form red color is used for pixels with positive gradient, blue color for pixels with negative gradient. Pic., Pic. and Pic. show us small practically invisible difference between them. t s because of the fact, that Lenna image can be presented in x windows by quadric surfaces and x Prewitt, Sobel, Scharr gradients gives right results in this case. For x gradient operators the situation is different. Lenna image is not good for x quadric surfaces presentation. As a result x Prewitt operator will give us not good but smoothed gradient image. Sobel and Scharr gives better results because of the fact that far pixels in x window have less weights then pixels close to the center. x Sobel and Scharr gradient images are practically the same. 7. Conclusions LPA together with weight matrix selection is useful for building of smoothing and gradient operators for image processing. For every gradient operator we have set of smoothing operators, which using makes consistency in simultaneous use of smoothing and gradient operators. References [] G. Levkine, mage Local Polynomial Approximation LPA and its Applications [] [] J. M. S. Prewitt. Object Enhancement and Extraction in Picture Processing and Psychopictorics, Academic Press, 97 [] Reference to Sobel operator in Duda, Hart Pattern Classification and scene analysis, Wiley & Sons, 97 [] Reference to Scharr operator in Wikipedia article Sobel operator [] W. Pratt Digital mage Processing, Wiley-nterscience, 7

9 [7] M. Sonka, V. Hlavac, R. Boyle mage processing, analysis and Machine Vision, [] Handbook of mage and Video Processing, editor Al Bovik, Elsevier, [9] K. Castleman, Digital mage Processing, Prentice-Hall, 979 Pic. Prewitt x horizon gradient of Lenna

10 Pic. Sobel x horizon gradient of Lenna

11 Pic. Simplified Scharr x horizon gradient of Lenna

12 Pic. Prewitt x horizon gradient of Lenna

13 Pic. Sobel x horizon gradient of Lenna

14 Pic. Simplified Scharr x horizon gradient of Lenna