Bayesian inference and mathematical imaging. Part IV: mixture, random fields, and hierarchical models. Dr. Marcelo Pereyra http://www.macs.hw.ac.uk/ ∼ mp71/ Maxwell Institute for Mathematical Sciences, Heriot-Watt University January 2019, CIRM, Marseille. M. Pereyra (MI — HWU) Bayesian mathematical imaging 0 / 55
Skin Skin cancer is the most common form of cancer Skin melanoma kills 14000 in Europe every year Diagnosis and treatment are main public health issues Human Skin Layers (MacNeil, 2007) M. Pereyra (MI — HWU) Bayesian mathematical imaging 1 / 55
Ultrasound imaging Ultrasound imaging US: diagnostics, routine tests, therapy and surgery US imaging of skin: new high frequency 3D ultrasound probes Study skin diseases & improve diagnosis Assess lesion boundaries prior to surgery (measure depth) Dermis view with skin lesion outlined by the red rectangle M. Pereyra (MI — HWU) Bayesian mathematical imaging 2 / 55
Ultrasound imaging Limitations: Manual annotation of 3D images is time-consuming Strong speckle noise ( SNR < 5 . 9 dB ), poor contrast & edges Segmentation is extremely operator-dependant Objective: Automatic and reliable segmentation of skin layers & lesions in 3D. M. Pereyra (MI — HWU) Bayesian mathematical imaging 3 / 55
Outline Statistical model for US signals (Pereyra and Batatia, 2012) 1 Supervised Bayesian US image segmentation (Pereyra et al., 2012b) 2 Unsupervised Bayesian US image segmentation (Pereyra et al., 2012a) 3 Conclusion 4 M. Pereyra (MI — HWU) Bayesian mathematical imaging 4 / 55
Outline Statistical model for US signals (Pereyra and Batatia, 2012) 1 Supervised Bayesian US image segmentation (Pereyra et al., 2012b) 2 Bayesian model Bayesian algorithm Experimental results Unsupervised Bayesian US image segmentation (Pereyra et al., 2012a) 3 Conclusion 4 M. Pereyra (MI — HWU) Bayesian mathematical imaging 5 / 55
Physical Signal Model Point scattering framework (Morse and Ingard, 1987) x n ≜ x ( t n ) = M a i p ( t n − τ i ) ∑ (1) i = 1 r n ≜ r ( t n ) = ∣ M a i [ p ( t n − τ i ) + ˜ p ( t − τ i )]∣ ∑ (2) i = 1 M : total number of point scatterers a i : cross-section of i th scatterer τ i : time of arrival of i th backscattered wave p ( t ) + ˜ p ( t ) : analytic extension of the interrogating pulse p ( t ) Medical ultrasound Imaging: M , a and τ are unknown quantities M. Pereyra (MI — HWU) Bayesian mathematical imaging 6 / 55
Statistical Signal Model Important questions What are the possible statistical distributions of x n and r n ? What information about M , a and τ in f ( x n ) and f ( r n ) ? hola Conventional analytical answer: central limit theorem ( M is very large) (Wagner et al., 1983) M x n = ∑ a i p ( t n − τ i ) ∼ N( 0 ,σ 2 n ) i = 1 r n = ∣ M a i [ p ( t n − τ i ) + ˜ p ( t n − τ i )]∣ ∼ R ayleigh ( σ n ) ∑ i = 1 n ∝ M ⟨ a 2 i ⟩ is the power backscattered by the n th resolution cell σ 2 M. Pereyra (MI — HWU) Bayesian mathematical imaging 7 / 55
Statistical Signal Model For many biol. tissues x n ∼ N( 0 ,σ 2 ) and r n ∼ R ayleigh ( σ ) are poor models, the empirical tails are not well modeled (Shankar et al., 1993; Shankar, 2000, 2003; Raju and Srinivasan, 2002) Figure : Comparison of the empirical envelope pdf obtained from forearm dermis, and the corresponding estimations using the generalized gamma, Weibull and Nakagami distributions M. Pereyra (MI — HWU) Bayesian mathematical imaging 8 / 55
Statistical Signal Model How are these non-Gaussian statistics explained ? M is not large enough to enforce the CLT M is large and a i has very high variance σ 2 n fluctuates strongly within homogenous regions The signal formation model is inaccurate hola What is an appropriate non-Gaussian distribution for x n and r n ? → to answer these questions we study the limit distributions of x n and r n . Limit distribution : domain of attraction (equilibrium point) in the space of probability density functions hola The Gaussian distribution (CLT) - is only a particular case (finite variance). There are infinitely other equilibrium points. M. Pereyra (MI — HWU) Bayesian mathematical imaging 9 / 55
Main results Main results: If f ( x n ) converges as M → ∞ to a non-Gaussian distribution then 1 x n has a symmetric α -stable limiting distribution x n ∼ S α S ( α,γ ) with α ∈ ( 0 , 2 ) and γ ∈ R + 2 The distribution of the scattering cross-section a i is heavy-tailed with the same characteristic exponent α f A ( a i ) ∝ a −( α + 1 ) i 3 r n is the envelope of a complex S α S random variable r n ∼ α R ayleigh ( α,γ ) M. Pereyra (MI — HWU) Bayesian mathematical imaging 10 / 55
Result 1: S α S statistical model If x n converges in distribution as M → ∞ , then it converges to a S α S ( α,γ ) distribution with α ∈ ( 0 , 2 ) and γ ∈ R + hola 1 x n is a sequence of random summands a i p ( t − τ i ) Its limit distribution must be invariant to addition 2 The characteristic function must be closed under exponentiation, only the α -stable family has this property 3 a i p ( t − τ i ) is statistically symmetric, f X ( x n ) converges to a symmetric α -stable distribution with parameters α ∈ ( 0 , 2 ] and γ ∈ R + 4 The case α = 2 corresponds to the Gaussian distribution and x n is known to be not-Gaussian, we conclude that α ∈ ( 0 , 2 ) M. Pereyra (MI — HWU) Bayesian mathematical imaging 11 / 55
Result 2: Power-law scattering cross-section distribution Given that a i ∈ R + and p ( t − τ i ) ∈ [ − P , P ] is bounded, if x n ∼ S α S ( α,γ ) with α < 2, then a i follows a heavy-tailed distribution with exponent α f A ( a i ) ∝ a −( α + 1 ) i Key idea: use necessary conditions for convergence to infer the class of f A ( a i ) . x n is in the domain of attraction of a S α S distribution with α < 2 only if F Z ( z i = a i p ( t n − τ i )) verifies the Doebling & Gnedenko conditions (Samorodnitsky and Taqqu, 2000) C1 F Z ( − z i ) 1 − F Z ( z i ) = C + = 1 lim z i → ∞ C − C2 1 − F Z ( z i ) + F Z ( − z i ) 1 − F Z ( lz i ) + F Z ( − lz i ) = l α , ∀ l > 0 lim z i → ∞ C1 & C2 imply that F Z ( z i ) ∝ ∣ z i ∣ − α + o (∣ z i ∣ − α ) . M. Pereyra (MI — HWU) Bayesian mathematical imaging 12 / 55
Result 2: Power-law scattering cross-section distribution Also, z i is a product of random variables z i = a i u i (Rohatgi, 1976) − P f U ( u i )[ 1 − F A ( z i / u i )] du i if z i < 0 F Z ( z i ) = { ∫ 0 F Z ( 0 − ) + ∫ 0 f U ( u i ) F A ( z i / u i ) du i if z i ≥ 0 P with u i = p ( t n − τ i ) . Then, for z i ≫ 0 0 f U ( u i ) F A ( z i / u i ) du i ≈ cz − α P ∫ i This condition is verified by all power-law distributions f A ( a i ) = L ( a i ) a −( α + 1 ) i where L ( a i ) is a slow varying function (i.e., lim s → ∞ L ( s ) = 1) L ( ks ) M. Pereyra (MI — HWU) Bayesian mathematical imaging 13 / 55
Result 3: α Rayleigh envelope distribution The envelope r n is the amplitude of the analytic extension of x n r n cos ( ϕ n ) = x n + y n , r n = ∣ x n + y n ∣ Assuming that the position of scatterers is purely random x n ∼ S α S ( α,γ ) ⇒ y n ∼ S α S ( α,γ ) , y n ⊥ x n By deriving f ( r n ,ϕ n ) from f ( x n , y n ) and marginalizing w.r.t. ϕ n r n ∼ α R ayleigh ( r n ∣ α,γ ) where 2 α R ayleigh ( r n ∣ α,γ ) ≜ ∫ ∞ R ayleigh ( r n ∣ σ ) S α 2 ( σ 2 ∣ γ cos ( πα 4 ) α , 1 , 0 ) d σ 0 = ∫ ∞ r n λ exp [ − ( γλ ) α ] J 0 ( r n λ ) d λ 0 M. Pereyra (MI — HWU) Bayesian mathematical imaging 14 / 55
Interpretation of α and γ For modeling and physical interpretation purposes the scattering cross-sections can be assumed to follow a Pareto distribution f A ( a i ) ≅ α a α m a −( α + 1 ) i a m is given by a m = lim i F A ( a i ) a i → ∞ a α hola Moreover, γ is a scale or spread parameter √ γ = D ∗ ( α ) α Ma m √ where D ∗ ( α ) = 2 ) , M is the number of scatters and ⟨ p α i ⟩ is 2 π ⟨ p α i ⟩ α Γ ( α ) sin ( πα the α -th fractional moment of p ( t − τ i ) M. Pereyra (MI — HWU) Bayesian mathematical imaging 15 / 55
Experimental validation Figure : Comparison of the empirical envelope pdf obtained from forearm dermis, and the corresponding estimations using the heavy-tailed Rayleigh, generalized gamma, Weibull and Nakagami distribution M. Pereyra (MI — HWU) Bayesian mathematical imaging 16 / 55
Experimental validation Comparison of distributions tails by means of a logarithmic pdfs M. Pereyra (MI — HWU) Bayesian mathematical imaging 17 / 55
Outline Statistical model for US signals (Pereyra and Batatia, 2012) 1 Supervised Bayesian US image segmentation (Pereyra et al., 2012b) 2 Bayesian model Bayesian algorithm Experimental results Unsupervised Bayesian US image segmentation (Pereyra et al., 2012a) 3 Conclusion 4 M. Pereyra (MI — HWU) Bayesian mathematical imaging 18 / 55
Outline Statistical model for US signals (Pereyra and Batatia, 2012) 1 Supervised Bayesian US image segmentation (Pereyra et al., 2012b) 2 Bayesian model Bayesian algorithm Experimental results Unsupervised Bayesian US image segmentation (Pereyra et al., 2012a) 3 Conclusion 4 M. Pereyra (MI — HWU) Bayesian mathematical imaging 19 / 55
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