Biomedical Image Analysis. Segmentation by Thresholding

Transcription

1 Biomedical Image Analysis Segmentation by Thresholding Contents: Thresholding principles Ridler & Calvard s method Ridler TW, Calvard S (1978). Picture thresholding using an iterative selection method, IEEE Transactions Systems, Man, Cybernetics, SMC-8: Optimal thresholding: Otsu s method Nobuyuki Otsu (1979). A threshold selection method from gray-level histograms. IEEE Trans.Sys., Man, Cyber. 9(1): BMIA 14 V. Roth & P. Cattin 212

2 Partitioning by Thresholding Simplest case: grey value distribution has two modes { 1 iff(x, y) > T g(x, y) = 0 iff(x, y) T Multiple thresholding: a iff(x, y) > T 2 g(x, y) = b ift 1 < f(x, y) T 2 c iff(x, y) T 1 BMIA 14 V. Roth & P. Cattin 213

3 Influence of Noise BMIA 14 V. Roth & P. Cattin 214

4 Influence of Illumination BMIA 14 V. Roth & P. Cattin 215

5 Influence Summary Based on the these observations we can state, that the further apart the peaks are, the better the chances of separating the modes the modes broaden with increasing noise illumination/reflectance can skew the histogram BMIA 14 V. Roth & P. Cattin 216

6 Ridler Calvard s Method Given: histogram, sum of background distribution b(z) and forground distribution f(z). z: grey value, B, F : prior probabilities (B + F = 1), Histogram can be written as p(z) = B b(z) + F f(z) Optimal threshold Topt is F f(t ) = B b(t ) BMIA 14 V. Roth & P. Cattin 217

7 Ridler Calvard s Method (3) Assumption: b(z) and f(z) Gaussian: F e (T µ F ) 2 2σ F 2 = B e (T µ 2 B) 2σ B 2 2πσ 2 F 2πσ 2 B After some algebraic manipulations... (σ 2 B σ 2 F )T 2 +2(µ B σ 2 F µ F σ 2 B)T +σ 2 Bµ 2 F σ 2 F µ 2 B+2σ 2 Bσ 2 F ln Bσ F F σ B = 0 BMIA 14 V. Roth & P. Cattin 218

8 Ridler Calvard s Method (4) Assumption: equal standard deviations σ F = σ B = σ: 2(µ B µ F )T (µ B + µ F )(µ B µ F ) + 2σ 2 ln B F = 0 Solving for T : T = µ B + µ F 2 + σ 2 µ B µ F ln F B Assumption: B = F : T = µ B+µ F 2 Ridler-Calvard for bimodal images. BMIA 14 V. Roth & P. Cattin 219

9 Thresholding with Ridler Calvard s Method In practice: µ B and µ F unknown, must be iteratively estimated based on suggested thresholds. 1. Select an initial value for the global threshold T 2. Segment image using T two classes of pixels B and F 3. Compute average intensities µ B, µ F for the pixels in B, F 4. Compute new threshold T = 1 2 (µ B + µ F ) 5. Repeat until T < ɛ Nicely separated classes usually suitable threshold. BMIA 14 V. Roth & P. Cattin 220

10 Example Fingerprint segmentation: threshold 125 was determined using Ridler Calvard s method. BMIA 14 V. Roth & P. Cattin 221

11 Example (2) Retina blood vessel example: threshold 80 was determined using Ridler Calvard s method. green channel of a retina fundus image histogram thresholded image BMIA 14 V. Roth & P. Cattin 222

12 Optimum Global Thresholding (Otsu 1979) Assumptions: normalised histogram of the grey level image is known The image contains only two classes of pixels Within classes: similar pixels, but classes themselves differ Two criteria: Variance within each class should be small The difference in the mean values between the classes should be large BMIA 14 V. Roth & P. Cattin 223

13 Otsu s Method On input: image with N pixels, grey levels in [i = 0,..., L 1]. n i : #(pixels) with grey level i Total number: N = L 1 i=0 n i Probability of level i is p i = n i N The histogram is normalised: p i 0 and p i = 1 BMIA 14 V. Roth & P. Cattin 224

14 Otsu s Method (2) Threshold k divides the pixels into two classes C 0 and C 1 Pixels [0, 1,..., k] belong to class C 0 and [k + 1,..., L 1] to class C 1 Class probabilities: C 0 : ω 0 = k i=0 p i = ω(k) C 1 : ω 1 = L 1 i=k+1 p i = 1 ω(k) Normalised histogram P(pixel class C 0/1 ) = ω 0/1 BMIA 14 V. Roth & P. Cattin 225

15 Mean levels of each class: Otsu s Method (3) C 0 : µ 0 = k i=0 ip i /ω 0 C 1 : µ 1 = L 1 i=k+1 ip i /ω 1 Mean level of the entire image: µ T = L 1 i=0 ip i Normalised histogram BMIA 14 V. Roth & P. Cattin 226

16 Otsu s Method (4) The class variances are given by C 0 : σ 2 0 = k (i µ 0 ) 2 p i /ω 0 i=0 C 1 : σ 2 1 = L 1 i=k+1 (i µ 1 ) 2 p i /ω 1 Global variance of all pixels: σ 2 G = L 1 (i µ T ) 2 p i i=0 BMIA 14 V. Roth & P. Cattin 227

17 Otsu s Method (5) Between-class variance: σ 2 B = ω 0(µ 0 µ T ) 2 +ω 1 (µ 1 µ T ) 2. Idea: observe between-class variance under variation of k. Optimal threshold k is found by exhaustive search σ 2 B(k ) = max 0 k L 1 σ2 B(k) If the maximum is not unique, obtain k by averaging the thresholds of all maxima. BMIA 14 V. Roth & P. Cattin 228

18 Otsu s Method (6) The normalised metric η at optimum threshold η(k ) is defined by η(k ) = σ2 B (k) σ 2 G Quantitative estimate of separability. It has values in the range 0 η(k ) 1 The lower bound is reached by images with a single, constant intensity. Upper bound: images with two classes with intensities equal to 0 and L 1. BMIA 14 V. Roth & P. Cattin 229

19 A Note on Otsu s Method Between-class variance: σ 2 B = ω 0(µ 0 µ T ) 2 +ω 1 (µ 1 µ T ) 2. Within-class variance: σ 2 W = ω 0σ 2 0+ω 1 σ 2 1 σ 2 G = σ2 W +σ2 B. k maximizes σ 2 B (k) k minimizes 1 σ 2 B (k) k minimizes σg 2 = σ2 σb 2 B (k)+σ2 W (k) (k) σb 2 (k) = 1 + σ2 W (k) σ 2 B (k) k maximizes σ2 B (k) σ 2 W (k) BMIA 14 V. Roth & P. Cattin 230

20 Efficient Otsu Implementation Reformulating and transforming the equations on the previous slides allows for a computationally more efficient (but less intuitive) implementation of the between-class variance σ 2 B(k) = ω 0 (k)ω 1 (k)(µ 0 (k) µ 1 (k)) 2 = [µ Tω 0 (k) µ(k)] 2 ω 0 (k)[1 ω 0 (k)], with µ(k) = k i=0 ip i being the cumulative average intensities up to level k. Advantage: global mean µ T computed only once only µ(k) and ω(k) recomputed for every k. BMIA 14 V. Roth & P. Cattin 231

21 Efficient Otsu Implementation (2) 1. Compute normalised histogram p i 2. Cumulative sums ω(k) = k i=0 p i for k = 0, 1, 2,..., L 3. Cumulative means µ(k) = k i=0 ip i, and global mean µ T = L 1 i=0 ip i 4. Between-class variance σb 2 (k) for k = 0, 1, 2,..., L 1 5. Obtain the Otsu threshold k maximizing σ 2 B (k) 6. Return separability measure η = σ2 B (k ) σ 2 G BMIA 14 V. Roth & P. Cattin 232

22 Multiple Thresholding The thresholding method introduced by Otsu can be easily extended to K classes: Between class variance generalises to σ 2 B = K k=1 ω k (µ k µ T ) 2 with ω k = p i and µ k = 1 ip i ω k i C k i C k The K classes are separated by K 1 thresholds and k1, k2,..., kk 1 maximizes the between-class variance σ 2 B(k 1, k 2,..., k K 1) = max 0<k 1 <k 2 <...k n 1 <L 1 σ2 B(k 1, k 2,..., k K 1 ) BMIA 14 V. Roth & P. Cattin 233

23 Multiple Thresholding (2) BMIA 14 V. Roth & P. Cattin 234

24 Smoothing to Improve Global Thresholding BMIA 14 V. Roth & P. Cattin 235

25 Smoothing to Improve Global Thresholding When the object is small, neither Otsu s method nor image filtering can help... BMIA 14 V. Roth & P. Cattin 236

26 Edges to Improve Global Thresholding Chances for finding good threshold increased if peaks are tall, narrow, symmetric, and separated by deep valleys. Idea: Consider only the pixels on or close to edges histogram depends less on relative size of B and F peaks have about the same height P (B) P (F ) increased symmetry BMIA 14 V. Roth & P. Cattin 237

27 Algorithm Compute edge image of f(x, y) Specify threshold T binary image g T (x, y) Compute histogram of f(x, y) after masking with strong edge pixels Otsu global segmentation of f(x, y) BMIA 14 V. Roth & P. Cattin 238

28 Example with Gradient Edge Detection BMIA 14 V. Roth & P. Cattin 239

29 Example with Laplacian Edge Detection (2) BMIA 14 V. Roth & P. Cattin 240

30 Variable Thresholding Global thresholding is problematic under nonuniform illumination conditions variable thresholding. Basically, any global thresholding method can also be applied locally (blockwise or sliding window). Depending on the window and image size such a local application can be computationally expensive. Specialized variable thresholding methods are often more efficient. Examples: Image Partitioning, Local Image Properties, Moving Averages. BMIA 14 V. Roth & P. Cattin 241

31 Nonuniform Illumination Example Endoscopic image of a placenta with heavy vignetting (lower illumination towards the boarder) BMIA 14 V. Roth & P. Cattin 242

32 Nonuniform Illumination Example (a) Original retina image, (b) green channel, (c) histogram of the green channel, (d) thresholded with Otsu s method T = 92, η = 0.51 and (e) thresholded with T = 51 BMIA 14 V. Roth & P. Cattin 243

33 Image Partitioning Simplest approach: subdivide image into non-overlapping rectangles, chosen small enough to assume uniform illumination. (a) Original pattern with a ramp, (b) Otsu (global), (c) Otsu with 2 2 subdivision, (d) Otsu with 4 4 subdivision Trade-off between subimage size and image structure! BMIA 14 V. Roth & P. Cattin 244

34 Image Partitioning Example (a) Retina (green channel), (b) Otsu, 2 2, (c) 3 3, (d) BMIA 14 V. Roth & P. Cattin 245

35 Variable thresholding: Local Image Properties Idea: compute threshold at every point (x, y) based on specific properties in a neighborhood S xy : T xy = aσ Sxy + bµ Sxy, where σ Sxy is the standard deviation and µ Sxy the mean in S xy, and a, b non-negative constants. The segmented image is then computed as { 1 iff(x, y) > T xy g(x, y) = 0 iff(x, y) T xy The values for a, b are application dependent. BMIA 14 V. Roth & P. Cattin 246

36 Example BMIA 14 V. Roth & P. Cattin 247

37 Moving Averages Efficient method for local thresholding. Averages computed line-by-line: m(k + 1) = 1 k+1 n i=k+2 n z i = m(k) + 1 n (z k+1 z k n+1 ), where z i are the pixels,and n = #(points) used in the average. Initialization: m(1) = z 1 /n (image padded with zeros). BMIA 14 V. Roth & P. Cattin 248

38 Moving Averages: Example BMIA 14 V. Roth & P. Cattin 249

39 Moving Averages: Example BMIA 14 V. Roth & P. Cattin 250

40 Moving Averages: Problems BMIA 14 V. Roth & P. Cattin 251

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42 Image Blurring The main idea of Moving Averages is to blur the input image and to use the blurred image as a local threshold. We can achieve a similar effect by blurring the image with a Gaussian. Advantage: no anisotropy (as the Moving Average). BMIA 14 V. Roth & P. Cattin 253

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44 Multivariable Thresholding Visualize 2D multivariable thresholding with joint histogram: Manual labeling of the Brain MR regions of the different brain tissues can be visualized in the joint histogram. This manual segmentation is a prerequisite to train the different tissues in the joint histogram. BMIA 14 V. Roth & P. Cattin 255

45 Multivariable Thresholding contd. Training dataset where (bg) background, (wm) white matter, (gm) grey matter, and (csf) cerebro spinal fluid BMIA 14 V. Roth & P. Cattin 256

46 Multivariable Thresholding contd. Simplest approach to training the different tissue types in the joint histogram: determine the cluster centers. Each pixel characterized by two intensity values (I 1 and I 2 at the same location (x, y)) represented as vector z. Multivariable thresholding can be viewed as distance computation { 1 if D(z, a)<t ; g(x, y) = 0 otherwise, with z = pixel intensities, a = cluster center of the tissue type of interest and the distance measure D(, ). BMIA 14 V. Roth & P. Cattin 257

47 Multivariable thresholding to detect csf BMIA 14 V. Roth & P. Cattin 258

48 Multivariable Thresholding contd. This approach can be easily extended to detecting multiple types of tissues. 1 if D(z, a) < T a ; 2 if D(z, b) < T b ; g(x, y) = 3 if D(z, c) < T c ; 4 if D(z, d) < T d ; 0 otherwise. BMIA 14 V. Roth & P. Cattin 259

49 Multivariable Thresholding contd. (left) Multivariable thresholding to detect multiple brain tissues; (right) Automatic segmentation result BMIA 14 V. Roth & P. Cattin 260

50 Distance Measures Multivariable Thresholding is based on calculating a distance measure of the form { 1 if D(z, a) < T ; g(x, y) = 0 otherwise. For an n-dimensional vector z the Euclidean distance is: D E (z, a) = z a = [ (z a) T (z a) ] 1/2. The equation D E (z, a) = T describes a hypersphere in the n-dimensional Euclidean space. BMIA 14 V. Roth & P. Cattin 261

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52 Distance Measures (2) A more powerful measure is the Mahalanobis distance: D M (z, a) = [ (z a) T C 1 (z a) ] 1 2, where C is the covariance matrix of the z-vectors. C = (z z)(z z) T, z = 1 N z i. i The equation D M (z, a) = T describes an n-dimensional hyperellipse. BMIA 14 V. Roth & P. Cattin 263

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