Chapter 10, Part I

 Segmentation subdivides an image into its constituent regions or objects.  Image segmentation methods are generally based on two properties. ◦ Discontinuity: abrupt changes in intensity, such as edges. ◦ Similarity: finding similar regions similar according to a set of predefined criteria, such as thresholding, region growing, region splitting and merging.

 Three types of gray-level discontinuities: points, lines and edges.  The response of the mask at any point in the image is given by R=w 1 z 1 +w 2 z 2 +…+w 9 z 9 where z i is the gray level of the pixel associated with mask coefficient w i.

Line Detection If the first mask was moved around an image, it would respond more strongly to lines oriented horizontally.

 Let R i denote the response the i-th mask generates.  If |R i | > |R j |, for all j≠i, that point is said to be more likely associated with a line in the direction of R i. ◦ A threshold value T may be used to sift the response on each point for differentiating points on edges.  Alternatively, we may use this scheme to detect lines in a specific direction.

Line Detection in a Specific Direction

 An edge is a set of pixels that lie on the boundary of two regions.  Edge detection is a process to measure the grey-level transitions ◦ Ideal edge: a clear transition in grey level. ◦ ramp edge: edges that are blurred.  First derivative and second derivative can be used to detect edges.  “Zero-crossing” property of second derivative is used to identify the location of edge.

The slope of the ramp is inversely proportional to the degree of blurring in the edges. Basic Formulation

First derivative: –Constant for points in the ramp. –The thickness of edge = the length of ramp. Second derivative: –Producing two values for every edge. –Zero-crossing point is near the midpoint of edge. Zero-crossing

Derivatives are sensitive to noise, especially second derivative. Fairly little noise can result great difficulties on edge detection using derivatives. Image smoothing is requires.

 A 2-D gradient defined as:  The magnitude of this vector is  f = mag(  f) = [G x 2 + G y 2 ] 1/2  The direction is  (x, y) = tan -1 (G x /G y ) ◦ The direction of an edge at (x, y) is perpendicular to the gradient.

 Robert cross-gradient operators: G x =(z 9 –z 5 ) and G y =(z 8 –z 6 ) ◦ It does not have a clear center.  Prewitt 3x3 operators G x =(z 7 +z 8 +z 9 )–(z 1 +z 2 +z 3 ) and G y = (z 3 +z 6 +z 9 )–(z 1 +z 4 +z 37 )  Sobel operators G x =(z 7 +2z 8 +z 9 )–(z 1 +2z 2 +z 3 ) and G y = (z 3 +2z 6 +z 9 )–(z 1 +2z 4 +z 37 )  The gradient is  f   G x  +  G y  ◦ computationally attractive

The price of gradient operator is that the original Prewitt and Sobel masks are not isotropic (invariant to rotation).

Masks for Detecting Diagonal Edges.

Example 10.4

The Laplacian Digital approximation for 3 x 3 region is  2 f = 4z 5 - (z 2 +z 4 +z 6 +z 8 ), or  2 f = 8z 5 - (z 2 +z 4 +z 6 +z 8 +z 1 +z 3 +z 7 +z 9 )

 The Laplacian is generally not used for edge detection since ◦ Sensitive to noise. ◦ The effect of double edges. ◦ Unable to detect edge direction.  The main usage: ◦ Using zero-crossing property for edge locating. ◦ Using it to determine whether a pixel is on the dark or light size of an edge.

 Gaussian: h(r)=-e -r 2 /2  2 where r 2 =x 2 +y 2 ◦ Convolving the Gaussian with the image will blur the image.  The second derivative of h(r) is ◦ the Lapalcian of a Gaussian (LoG)  The Laplacian provides an image with zero crossings for edge location.  The Gaussian is used to supress noise.

Plots of a Log Operator Certain aspects of human vision can be models mathematically in this form.