Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation

28/10/2015
Auteurs : Jesús Angulo
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14305

Résumé

Stochastic watershed is an image segmentation technique based on mathematical morphology which produces a probability density function of image contours. Estimated probabilities depend mainly on local distances between pixels. This paper introduces a variant of stochastic watershed where the probabilities of contours are computed from a gaussian model of image regions. In this framework, the basic ingredient is the distance between pairs of regions, hence a distance between normal distributions. Hence several alternatives of statistical distances for normal distributions are compared, namely Bhattacharyya distance, Hellinger metric distance and Wasserstein metric distance.

Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation

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Stochastic watershed is an image segmentation technique based on mathematical morphology which produces a probability density function of image contours. Estimated probabilities depend mainly on local distances between pixels. This paper introduces a variant of stochastic watershed where the probabilities of contours are computed from a gaussian model of image regions. In this framework, the basic ingredient is the distance between pairs of regions, hence a distance between normal distributions. Hence several alternatives of statistical distances for normal distributions are compared, namely Bhattacharyya distance, Hellinger metric distance and Wasserstein metric distance.

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Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Jesús Angulo jesus.angulo@mines-paristech.fr ; http://cmm.ensmp.fr/∼angulo MINES ParisTech, PSL-Research University, CMM-Centre de Morphologie Mathématique GSI'2015 - 2nd Conference on Geometric Science of Information Ecole Polytechnique, Paris-Saclay (France) - October 28th-30th 2015 1 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Motivation: Unsupervised segmentation of generic images Custard: Color image Large homogenous areas, well contrasted objects as well as textured zones and fuzzy boundaries 2 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Motivation: Unsupervised segmentation of generic images Custard: its color gradient image Large homogenous areas, well contrasted objects as well as textured zones and fuzzy boundaries 3 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Motivation: Unsupervised segmentation of generic images Custard: pdf of contours using stochastic watershed Using watershed based techniques large homogeneous areas are oversegmented and textured zones are not always well contoured 4 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Motivation: Unsupervised segmentation of generic images Custard: h-dynamics watershed cut from SW pdf, h = 0.1 Using watershed based techniques large homogeneous areas are oversegmented and textured zones are not always well contoured 5 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Motivation: Unsupervised segmentation of generic images Custard: h-dynamics watershed cut from SW pdf, h = 0.3 Using watershed based techniques large homogeneous areas are oversegmented and textured zones are not always well contoured 6 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Context and goal 7 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Context and goal Context: Segmentation approaches based on statistical modeling of pixels and regions, e.g, mean shift and statistical region merging Hierarchical contour detection and segmentation, e.g., machine learned edge detection, watershed transform Stochastic watershed (SW): to estimate a probability density function of contours 7 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Context and goal Context: Segmentation approaches based on statistical modeling of pixels and regions, e.g, mean shift and statistical region merging Hierarchical contour detection and segmentation, e.g., machine learned edge detection, watershed transform Stochastic watershed (SW): to estimate a probability density function of contours Goal: Take into account regional information in the probability estimation by SW by means of a statistical gaussian model ⇒ More perceptual strength function of contours 7 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Plan 1 Stochastic Watershed using MonteCarlo Simulations 2 Multivariate Gaussian Model of Regions in SW 3 Perspectives 8 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Stochastic Watershed using MonteCarlo Simulations 1 Stochastic Watershed using MonteCarlo Simulations 2 Multivariate Gaussian Model of Regions in SW 3 Perspectives 9 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Stochastic Watershed using MonteCarlo Simulations Regionalized Poisson points Uniform random germs Generate realizations of a Poisson point process with a constant intensity θ (i.e., average number of points per unit area) Random number of points N(D) falling in a domain D (bounded Borel set), with area |D|, follows a Poisson distribution with parameter θ|D|, i.e., Pr{N(D) = n} = e−θ|D| (−θ|D|) n n! Conditionally to the fact that N(D) = n, the n points are independently and uniformly distributed over D, and the average number of points in D is θ|D| (i.e., the mean and variance of a Poisson distribution is its parameter 10 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Stochastic Watershed using MonteCarlo Simulations Regionalized Poisson points Regionalized random germs Let us suppose that the density θ is not constant; but considered as measurable positive-valued function, dened in Rd . For simplicity, let us write θ(D) = θ(x)dx Number of points falling in a Borel set B according to a regionalized density function θ follows a Poisson distribution of parameter θ(D), i.e., Pr{N(D) = n} = e−θ(D) (−θ(D)) n n! If N(D) = n, the n are independently distributed over D with the probability density function: θ(x) = θ(x)/θ(D) 11 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Stochastic Watershed using MonteCarlo Simulations Regionalized Poisson points Generate N random germs in an image m : E → {0, 1} according to density θ(x) using inverse transform sampling 1 Initialization: m(xi ) = 0 ∀xi ∈ E; P = Card(E) 2 Compute cumulative distribution function: cdf (xi ) = k≤i θ(xk ) P k=1 θ(xk ) 3 for j = 1 to N 4 rj ∼ U(1, P) 5 Find the value sj such that rj ≤ cdf (xsj ) . 6 m(xsj ) = 1 12 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Stochastic Watershed using MonteCarlo Simulations Stochastic watershed paradigm Spreading random germs as markers on the watershed segmentation. This arbitrary choice is stochastically balanced by the use of a given number M of realizations, in order to lter out non signicant uctuations Each piece of contour may then be assigned the number of times it appears during the various simulations in order to estimate a probability density function (pdf) of contours In the case of uniformly distributed random germs, large regions will be sampled more frequently than smaller regions and will be selected more often Image gradient as density for regionalization of random germs involves sampling high contrasted image areas: probability of selecting a contour will oer a trade-o between strength of the contours and size of the adjacent regions 13 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Stochastic Watershed using MonteCarlo Simulations Probability density of contours using MonteCarlo simulations of watershed Let {mrkn(x)}M n=1 be a series of M realizations of N spatially distributed random markers according to its gradient image g Each realization of random germs considered as the marker image for a watershed segmentation of gradient image g in order to obtain the binary image: WS(g, mrkn)(x) = 1 if x ∈ Watershed lines 0 if x /∈ Watershed lines Probability density function of contours is computed by the kernel density estimation method: pdf (x) = 1 M M n=1 WS(g, mrkn)(x) ∗ Kσ(x). where the smoothing kernel Kσ(x) is a spatial Gaussian function of width σ 14 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Stochastic Watershed using MonteCarlo Simulations Probability density of contours using MonteCarlo simulations of watershed Color image f (x) Color gradient g(x) 15 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Stochastic Watershed using MonteCarlo Simulations Probability density of contours using MonteCarlo simulations of watershed {mrkn(x)}M n=1: M realizations of N regionalized Poisson points of θ(x) = g(x) · · · {WS(g, mrkn)}1≤n≤M : Watershed segmentations · · · 16 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Stochastic Watershed using MonteCarlo Simulations Probability density of contours using MonteCarlo simulations of watershed Color image f (x) Color gradient g(x) Density of contours pdf (x) Segmented with h = 0.1 17 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Stochastic Watershed using MonteCarlo Simulations Probability density of contours using MonteCarlo simulations of watershed Color image f (x) Color gradient g(x) Density of contours pdf (x) Segmented with h = 0.3 18 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW 1 Stochastic Watershed using MonteCarlo Simulations 2 Multivariate Gaussian Model of Regions in SW 3 Perspectives 19 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Watershed transform ⇒ Tessellation Tessellation τ of E from watershed WS(x): (Finite) family of disjoint open sets (or classes, or regions) τ = {Rr }1≤r≤N , with i = j ⇒ Ri ∩ Rj = ∅ such that E = ∪r Rr WS(x) ⇔ WS(x) = E \ ∪r Rr = ∪li,j Boundary between regions Ri and Rj (1 ≤ i, j ≤ N, i = j): Irregular arc segment li,j = ∂Ri ∩ ∂Rj 20 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Color regions as multivariate normal distributions The color image values restricted to each region of the partition, Pi = f (Ri ), can be modeled by dierent statistical distributions Here we focuss on a multivariate normal model Pi ∼ N(µi , Σi ), of mean µi and covariance matrix Σi Dierent (statistical) distances are dened in the space of N(µi , Σi ) Boundary li,j will be weighted with a function depending on the distance between N(µi , Σi ) and N(µj , Σj ) 21 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Distances for multivariate normal distributions Bhattacharyya distance DB (P1, P2) It measures the similarity of two discrete or continuous probability distributions P1 and P2 by computing the amount of overlap between the two statistical populations: DB (P1, P2) = − log P1(x)P2(x)dx For multivariate normal distributions DB (P1, P2) = 1 8 (µ1−µ2)T Σ−1 (µ1−µ2)+ 1 2 log det Σ √ det Σ1 det Σ2 , where Σ = Σ1 + Σ2 2 Note that the rst term in the Bhattacharyya distance is related to the Mahalanobis distance, both are the same when the covariance of both distributions is the same 22 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Distances for multivariate normal distributions Hellinger metric distance DH (P1, P2) 0 ≤ DB ≤ ∞ and it is symmetric DB (P1, P2), but DB does not obey the triangle inequality and therefore it is not a metric Bhattacharyya distance can be metrized by transforming it into to the following Hellinger metric distance DH (P1, P2) = 1 − exp (−DB (P1, P2)), For multivariate normal distributions DH (P1, P2) = 1 − det Σ √ det Σ1 det Σ2 −1/2 e(− 1 4 (µ1−µ2)T (Σ1+Σ2)−1(µ1−µ2)) Hellinger distance is an α-divergence, which corresponds to the case α = 0 and it is the solely being a metric distance. Hellinger distance can be related to measure theory and asymptotic statistics 23 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Distances for multivariate normal distributions Wasserstein metric distance DW (P1, P2) Wasserstein metric is a distance function dened between probability measures µ and ν on Rn is based on the notion optimal transport: W2(µ, ν) = inf E( X − Y 2 )1/2 , where the inmum runs over all random vectors (X, Y ) ∈ Rn × Rn with X ∼ µ and Y ∼ ν For the case of discrete distributions, it corresponds to the well-known earth mover's distance For two multivariate normal distributions: DW (P1, P2) = µ1 − µ2 2 + Tr (Σ1 + Σ2 − 2Σ1,2), where Σ1,2 = Σ 1/2 1 Σ2Σ 1/2 1 1/2 . In particular, in the commutative case Σ1Σ2 = Σ2Σ1 one has DW (P1, P2)2 = µ1 − µ2 2 + Σ 1/2 1 − Σ 1/2 2 2 F . 24 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Probability density function MonteCarlo estimation 25 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Probability density function MonteCarlo estimation To assign to each piece of contour li,j between regions Ri and Rj the normalized statistical distance between the color gaussian distributions Pi and Pj : πi,j = D(Pi , Pj ) lk,l ∈WS D(Pk , Pl ) , where D(Pi , Pj ) is any of the distances discussed above 25 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Probability density function MonteCarlo estimation To assign to each piece of contour li,j between regions Ri and Rj the normalized statistical distance between the color gaussian distributions Pi and Pj : πi,j = D(Pi , Pj ) lk,l ∈WS D(Pk , Pl ) , where D(Pi , Pj ) is any of the distances discussed above For any realization n of SW, denoted WS(x, n), one can compute an image of weighted contours: Pr(x, n) = πi,j if x ∈ ln i,j 0 if x /∈ ln i,j 25 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Probability density function MonteCarlo estimation To assign to each piece of contour li,j between regions Ri and Rj the normalized statistical distance between the color gaussian distributions Pi and Pj : πi,j = D(Pi , Pj ) lk,l ∈WS D(Pk , Pl ) , where D(Pi , Pj ) is any of the distances discussed above For any realization n of SW, denoted WS(x, n), one can compute an image of weighted contours: Pr(x, n) = πi,j if x ∈ ln i,j 0 if x /∈ ln i,j Integrating across the M realizations, the MonteCarlo estimate of the probability density function of contours: pdf (x) = 1 M M n=1 Pr(x, n) ∗ Kσ(x) 25 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Probability density function MonteCarlo estimation {WS(g, mrkn)}1≤n≤M : Watershed segmentations · · · {Pr(x, n)}1≤n≤M : Weighted contours (Bhattacharyya distance) · · · 26 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Probability density function MonteCarlo estimation Bhattacharyya distance Color image f (x) Color gradient g(x) Density of contours pdf (x) Segmented with h = 0.02 27 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Probability density function MonteCarlo estimation Distance of means 1 Bhattacharyya distance Hellinger distance Wasserstein distance 1F. López-Mir, V. Naranjo, S. Morales, J. Angulo. Probability Density Function of Object Contours Using Regional Regularized Stochastic Watershed. In IEEE ICIP'14. 28 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Probability density function MonteCarlo estimation Distance of means 2 Bhattacharyya distance Hellinger distance Wasserstein distance 2F. López-Mir, V. Naranjo, S. Morales, J. Angulo. Probability Density Function of Object Contours Using Regional Regularized Stochastic Watershed. In IEEE ICIP'14. 29 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Probability density function MonteCarlo Robust estimation Distance of means Bhattacharyya distance Hellinger distance Wasserstein distance 30 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Probability density function MonteCarlo Robust estimation Distance of means Bhattacharyya distance Hellinger distance Wasserstein distance 31 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Multivariate Gaussian Model of Regions in SW Comparison with SRM Statistical Region Merging (SRM) 3 depending on scale parameter Q Sum of contours from nine Q 256, 128, 64, 32, 16, 8, 4, 2, 1 Segmentation for Q = 128 Segmentation for Q = 32 3R. Nock, F. Nielsen. Statistical Region Merging. IEEE Trans. on PAMI, 26(11):14521458, 2004. 32 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Perspectives 1 Stochastic Watershed using MonteCarlo Simulations 2 Multivariate Gaussian Model of Regions in SW 3 Perspectives 33 / 34 Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation Perspectives Perspectives In addition to the color, each pixel x described also by its structure tensor T(x) ∈ SPD(2): Each region Ri : N(0, Σi ), Σi = |Ri |−1 x∈Ri T(x) Each region Ri : The histogram of structure tensors {T(x)}x∈Ri From to color to multi/hyper-spectral images: High-dimensional covariance matrix estimated locally in regions Supervised segmentation: Distance learning from training images of annotated contours MATLAB code available. 34 / 34