Chapter 9: Image Segmentation Image segmentation – partition an image into component parts Contents: (1) Thresholding (2) Edge detection

◎ Thresholoding 。 Single Thresholoding

。 Double Thresholding

。Advantages: (i) Remove unnecessary detail (ii) Bring out hidden detail

○ How to choose a threshold value 。 Histogram method

。 Otsu’s thresholding method Describe the histogram as a probability distribution by

Let t be the determined threshold value Define Find t such that

○ Adaptive Thresholding Divide image into strips Apply Otsu’s method to each strip

。 Rosenfeld’s variable thresholding (i) Divide image into blocks Global thresholding (ii) Compute histograms of block images

For each block image, compute its (1) Smooth histogram h

(2) Fit histogram with mixture of Gaussians let v be the gray level corresponding to the deepest valley of

Compute

(3) Test biomodality

(4) If the bimodality test is past, compute T by (5) For block (x,y) whose threshold value T(x,y) hasn’t yet been determined

(6) Smooth T by (7) Determine thresholding values of image pixels by bilinear interpolation

Variable thresholding Global thresholding

Ramp edge ◎ Edge Detection 。Types of edge: Step edge (jump edge) Roof edge (crease edge) Smooth edge

○ Derivatives

Horizontal filter: , Smooth filter: Prewitt filters 。Consider Horizontal filter: , Smooth filter: Combine Vertical filter: , Smooth filter:

vertical horizontal Edge image Binary image Thinning

。Roberts filter: 。Sobel filter:

◎ Second Derivatives Laplacian:

Invariant under rotation (isotropic filter) Discrete filter: Invariant under rotation (isotropic filter)

Step edge: Ramp edge:

。 Second derivatives are sensitive to noise 。 Other Laplacian masks

○ Zero crossing 0 +, + 0 0 －, － 0 + －, － +

Example: Edge detection by taking zero crossings after a Laplace filtering Marr-Hildreth method Smooth the input image using a Gaussian before Laplace filtering

。 Gaussian smooth + Laplace filtering = Laplacian of Gaussian (LOG): Gaussian: LOG:

Mexican hat: Difference of Gaussian (DOG): ◎ Canny edge detector Features: 1. Precise in edge position (scale space) 2. One-pixel width edges

○ Steps: Let 1. Smoothing and Edge detection (a) Horizontal direction (b) Vertical direction (c) Edge magnitude

(b) Quantize to (a) For each pixel p, 2. Non-maximum suppression 0, 45, 90 or 135 degs. (c) Along p is marked if its edge magnitude is larger than both its two neighbors p is deleted otherwise

3. Hysteresis thresholding For each marked pixel p, (a) If > or (b) If and p is adjacent to an edge pixel p is considered as an edge pixel

◎ Hough Transform

○ Line equation: y = ax + b Parameter space A point on the line Rewrite as Another point on the line

○ Line equation: