【正文】 to the axes respectively. In this case ? ? ? ?jifjifx ,1,1 ????? （ 9） ? ? ? ?jifjify ,11, ????? （ 10） This form of operator is known as the Roberts edge operator and was one of the first set of operators used to detect edges in images (Robert, 1965). The corresponding convolution masks are given by: ????????? 01 101 and ?????? ??? 10 012 An advantage of using a larger mask size is that the errors due to the effects of noise are reduced by local averaging within the neighbourhood of the mask. An advantage of using a mask of odd size is that the operators are centered and can therefore provide an estimate that is based on a center pixel (i,j). One important edge operator of this type is the Sobel edge operator. The Sobel edge operator masks are given as ???????????????001202101x ???????????????121000121y The operator calculates the gradient of the image intensity at each point, giving the direction of the largest possible increase from light to dark and the rate of change in that direction. The result therefore shows how abruptly or smoothly the image changes at that point and therefore how likely it is that part of the image represents an edge, as well as how that the edge is likely to be oriented. In practice, the magnitude (likelihood of an edge) calculation is more reliable and easier to interpret than the direction calculation. Mathematically, the gradient of a twovariable function (the image intensity function) at each image point is a 2D vector with the ponents given by the derivatives in the horizontal and vertical directions. At each image point, the gradient vector points to the direction of largest possible intensity increase, and the length of the gradient vector corresponds to the rate of change in that direction. This implies that the result of the Sobel operator at any image point which is in a region of constant image intensity is a zero vector and at a point on an edge is a vector which points across the edge, from darker to brighter values. The algorithm for developing the Sobel model for edge detection is given below. Pseudocodes for Sobel edge detection method Input: A Sample Image Output: Detected Edges Step 1: Accept the input image Step 2: Apply maskGx,Gy to the input image Step 3: Apply Sobel edge detection algorithm and the gradient Step 4: Masks manipulation of Gx,Gy separately on the input image Step 5: Results bined to find the absolute magnitude of the gradient 2y2x GGG ?? （ 11） Step 6: the absolute magnitude is the output edges Second order derivative operators A maximum of the first derivative will occur at a zero crossing of the second derivative. To get both horizontal and vertical edges, we look at second derivative in both the x and y directions. This is the Laplacian of I where 22222I yIxI ??????? （ 12） The Laplacian is linear and rotationally symmetric. Thus, if one search for the zero crossing of the image that is first smoothed with a Gaussian mask and then the second derivative is calculated。 Feldman, 1968). The different operators in eq. (5) and (6) correspond to convolving the image with the following marks ????????? 00 11x ， ????????? 01 01y When this is done, then: i. The top lefthand corner of the appropriate mask is superimposed over each pixel of the image in turn, ii. A value is calculated for Dx orDy by using the mask coefficients in a weighted sum of the value of pixels (i, j ) and its neighbours, iii. These masks are referred to as convolution masks or sometimes convolution kernels. Instead of finding approximate gradient ponents along the x and y directions, approximation of the gradient ponents could be done along directions at 45176。3 mask. The 37 pixels circular mask used in all feature detection experiments is placed at each point in the image and for each point the brightness of each pixel within the mask is pared with that of nucleus. The parison equation is C? ?0rr ??， = ? ? ? ?? ? ? ??????????trItrI00rI if 0 rI if 1 ?? ?? （ 1） wherer? is the position of the nucleus in the dimensional image, 0r? is the position of any other point within the mask, ??rI? is the brightness of any pixel, t is the brightness in difference threshold and C is the output of the parison. This parison is done for each pixel within the mask where total n of the outputs (c) is given as n??0r? =?r?C? ?0rr ??， （ 2） Sobel Filter Design Most edge detection methods work on the assumption that the edge occurs where there is a discontinuity in the intensity function or a very steep intensity gradient in the image. Using this assumption, if one take the derivative of the intensity value across the image and find points where the derivative is maximum, then the egde could be located. The gradient is a vector, whose ponents measure how rapid pixel value are changing with distance in the x and y direction. Thus, the ponents of the gradient may be found using the following approximation: ? ? ? ? ? ?dx yxfydxxfxx yxf , ??????? （ 3） ? ? ? ? ? ?dy yxfdyyxfyx yxf , ??????? （ 4） where dx anddy measure distance along the x and y directions respectively. In discrete images, one can consider dx and dy in terms of numbers of pixel between two points. dx = dy = 1 (pixel spacing) is the point at which pixel coordinates are(i, j ) thus, ? ? ? ?jifjifx ,1 ???? （ 5） ? ? ? ?jifjify ,1, ???? （ 6） In order to detect the presence of a gradient discontinuity, one could calculate the change in the gradient at (i, j ) .This can be done by finding the following magnitude measure M= 22 yx ???