Bayesian Defect Signal Analysis for Nondestructive Evaluation of Materials

Bayesian Defect SignalAnalysis for Nondestructive

Evaluation of Materials†

Aleksandar DogandzicElectrical and Computer Engineering

Iowa State University

†Joint work with Benhong Zhang (Ph.D. student), supported by the NSF I-U Cooperative ResearchProgram, CNDE, Iowa State University.

My Research Topics

General Area: Statistical Signal Processing.

Applications include:

• Sensor networks,

• Nondestructive evaluation and testing (NDE/NDT) of materials,

• Wireless communications,

• Radar.

Background: What is NDE?

In nondestructive evaluation (NDE) of materials, noninvasive measurementtechniques are used to determine if various properties of materials,components, or structures have defects that might lessen their abilityto perform their intended function.

Background: What is NDE?

In nondestructive evaluation (NDE) of materials, noninvasive measurementtechniques are used to determine if various properties of materials,components, or structures have defects that might lessen their abilityto perform their intended function.

Typical goals of NDE inspections:

• determining if the wing of an airliner has corrosion around the rivets or

• testing materials for flaws that might cause failure (before the materialis used to make parts or components).

Our Goal in This Talk

• Goal: Find elliptically-shaped regions with elevated signal levels in noisyimages.

• Application: Automatic identification of defects(e.g. cracks, corrosion, porosity, inclusions etc).

Accounting for both the defect-signal amplitude (signal level) andarea greatly improves the detectability performance (compared with thetraditional NDE methods which ignore the spatial extent of the defectsignals).

Approach

Statistical Modeling Approach: Hierarchical Bayesian.

Computational Approach: Markov chain Monte Carlo, forsimulating from the posterior distributions of the desired parameters=⇒ Intuitively, generate a bunch of “proposals” for the defect regionsand signals; if a proposal matches the data well, generate more “similar”proposals.

Outline

• Measurement Model

− Parametric model for defect location and shape,− Measurement-error (noise) model,− Defect-signal (reflectivity) model,− Prior specifications for the location, shape, and defect-signal

distribution parameters (model parameters).

• Bayesian Analysis

− Simulating the model parameters φ,− Simulating the signals θi.

• Numerical Examples.

Terminology

Abbreviations:

• i.i.d. ≡ independent, identically distributed,

• pdf ≡ probability density function,

• cdf ≡ cumulative distribution function,

• MMSE ≡ minimum mean-square error,

• MCMC ≡ Markov Chain Monte Carlo.

Notation

• “T” ≡ a transpose,

• πφ(φ) ≡ prior pdf of φ,

• p(φ |y) ≡ conditional pdf of φ given y,

• R(z) ≡ defect region,

• Rc(z) ≡ noise-only region, i.e. region outside R(z).

•iA(x) =

{1, x ∈ A,0, otherwise

≡ indicator function.

Notation (cont.)

• N (y;µ, σ) ≡ Gaussian pdf of a random variable y with mean µ andstandard deviation σ:

N (y; µ, σ) =1√

2πσ2· exp

[− (y − µ)2

2σ2

],

• Nt(y ; µ, σ) ≡ truncated-Gaussian pdf (to non-negative values) withparameters µ and σ:

Nt(y ; µ, σ) =N (y ; µ, σ)

Φ(µ/σ)· i[0,∞)(y)

where Φ(·) denotes the cdf of the standard normal random variable.

Outline

• Measurement Model

− Parametric model for defect location and shape,

− Measurement-error (noise) model,− Defect-signal (reflectivity) model,− Prior specifications for the location, shape, and defect-signal

distribution parameters (model parameters).

• Bayesian Analysis

− Simulating the model parameters φ,− Simulating the signals θi.

• Numerical Examples.

Parametric Model for Defect Location and Shape

Potential defect-signal region modeled as an ellipse R(z):

R(z) = {r : (r − r0)TΣ−1R (r − r0) ≤ 1}

where

• r = [x1, x2]T ≡ location in Cartesian coordinates

• z = [rT0 , d, A, ϕ]T ≡ vector of the (unknown) location and shape

parameters, and

ΣR =[

cos ϕ − sinϕsinϕ cos ϕ

]·[

d2 00 A2/(d2π2)

]·[

cos ϕ − sinϕsinϕ cos ϕ

]T

.

Parametric Model for Defect Location and Shape (cont.)

Ellipse location and shape parameters:

• r0 =[

x0,1

x0,2

]≡ center of the ellipse in Cartesian coordinates (m);

• d > 0 ≡ axis parameter (m);

• A ≡ area of the ellipse (m2);

• ϕ ∈ [−π/4, π/4] ≡ ellipse orientation parameter (rad).

Parametric Model for Defect Location and Shape (cont.)

Ellipse location and shape parameters:

• r0 =[

x0,1

x0,2

]≡ center of the ellipse in Cartesian coordinates (m);

• d > 0 ≡ axis parameter (m);

• A ≡ area of the ellipse (m2);

• ϕ ∈ [−π/4, π/4] ≡ ellipse orientation parameter (rad).

Note: d and A/(dπ) are the axes of this ellipse.

Outline

• Measurement Model

− Parametric model for defect location and shape,

− Measurement-error (noise) model,

− Defect-signal (reflectivity) model,− Prior specifications for the location, shape, and defect-signal

distribution parameters (model parameters).

• Bayesian Analysis

− Simulating the model parameters φ,− Simulating the signals θi.

• Numerical Examples.

Measurement-Error (Noise) Model

Assume that we have collected measurements yi at locations si, i =1, 2, . . . , Ntot within the region of interest, where Ntot denotes thetotal number of measurements in this region. We adopt the followingmeasurement-error (noise) model:

• If yi is collected over the defect region [i.e. si ∈ R(z)], then

yi = θi︸︷︷︸signal

+ ei︸︷︷︸additive noise

where θi and ei denote the defect signal (related to its reflectivity) andnoise at location si, respectively;

Measurement-Error (Noise) Model (cont.)

• If yi is collected outside the defect region [i.e. si ∈ Rc(z)], then

yi = 0︸︷︷︸signal

+ ei︸︷︷︸additive noise

implying that the signals θi are zero in the noise-only region;

Measurement-Error (Noise) Model (cont.)

• If yi is collected outside the defect region [i.e. si ∈ Rc(z)], then

yi = 0︸︷︷︸signal

+ ei︸︷︷︸additive noise

implying that the signals θi are zero in the noise-only region;

• We model the additive noise samples ei, i = 1, 2, . . . , Ntot as zero-meani.i.d. Gaussian random variables with known variance σ2 (which can beeasily estimated from the noise-only data).

Measurement-Error (Noise) Model (cont.)

Therefore,p(yi | θi) = N (yi ; θi, σ)

where θi = 0 for si ∈ Rc(z).

Outline

• Measurement Model

− Parametric model for defect location and shape,− Measurement-error (noise) model,

− Defect-signal (reflectivity) model,

− Prior specifications for the location, shape, and defect-signaldistribution parameters (model parameters).

• Bayesian Analysis

− Simulating the model parameters φ,− Simulating the signals θi.

• Numerical Examples.

Defect-Signal (Reflectivity) Model

The signals within the defect region [i.e. si ∈ R(z)] are i.i.d. truncatedGaussian with unknown defect-signal distribution parameters µ and τ .Therefore, the joint pdf of the defect signals conditional on z (location andshape), µ, and τ (defect-signal distribution parameters) is

p({θi, si ∈ R(z)} |z, µ, τ) =∏

i, si∈R(z)

Nt(θi ; µ, τ).

Note: τ is a measure of defect-signal variability;for example, if τ = 0 =⇒ all θi within the defect region are equal to µ.

Defect-Signal (Reflectivity) Model: An Illustration.

Defect-Signal (Reflectivity) Model: An Illustration.

Location, Shape, and Defect-SignalDistribution Parameters

The vector of the location, shape, and defect-signal distribution parameters(model parameters) is

φ =

zµτ

where

• z = [x0,1, x0,2, d, A, ϕ]T≡ location and shape parameters and

• µ, τ ≡ defect-signal distribution parameters.

Outline

• Measurement Model

− Parametric model for defect location and shape,− Measurement-error (noise) model,− Defect-signal (reflectivity) model,

− Prior specifications for the location, shape, and defect-signal

distribution parameters (model parameters).

• Bayesian Analysis

− Simulating the model parameters φ,− Simulating the signals θi.

• Numerical Examples.

Prior Specifications for the Model Parameters

πφ(φ) = πx0,1(x0,1) · πx0,2(x0,2) · πd(d) · πA(A) · πϕ(ϕ) · πµ(µ) · πτ(τ)

where

πx0,1(x0,1) = uniform(x0,1,MIN, x0,1,MAX)

πx0,2(x0,2) = uniform(x0,2,MIN, x0,2,MAX)

πd(d) = uniform(dMIN, dMAX)

πA(A) = uniform(AMIN, AMAX), πϕ(ϕ) = uniform(ϕMIN, ϕMAX)

πµ(µ) = uniform(0, µMAX), πτ(τ) = uniform(0, τMAX).

Outline

• Measurement Model

− Parametric model for defect location and shape,− Measurement-error (noise) model,− Defect-signal (reflectivity) model,− Prior specifications for the location, shape, and defect-signal

distribution parameters (model parameters).

• Bayesian Analysis

− Simulating the model parameters φ,

− Simulating the signals θi.

• Numerical Examples.

Simulating the Model Parameters φ

Sample from the posterior pdf

p(φ |y)

where y = [y1, y2, . . . , yNtot]T denotes the vector of all observations.

Comments: Obtaining a closed-form expression for p(φ |y) is impossible,but we can determine it up to a multiplicative constant =⇒ sufficient forapplying MCMC techniques and simulating φs from p(φ |y)!

Finding p(φ |y)(up to a Multiplicative Constant)

Define the vector of random signals θ = [θ1, θ2, . . . , θNtot]T .

Idea. Integrate out the random signals θ from the joint posterior pdfp(φ, θ |y):

p(φ |y) =∫

p(φ, θ |y) dθ =p(φ, θ |y)p(θ |φ, y)

.

Finding p(φ |y)(up to a Multiplicative Constant)

Define the vector of random signals θ = [θ1, θ2, . . . , θNtot]T .

Idea. Integrate out the random signals θ from the joint posterior pdfp(φ, θ |y):

p(φ |y) =∫

p(φ, θ |y) dθ =p(φ, θ |y)p(θ |φ, y)

.

Note: A Bayesian trick for “integrating” θ out without actually performingthe integration!

Computing p(φ,θ |y)

p(φ, θ |y) ∝ πφ(φ) · p(θ |φ) · p(y |θ)

∝ πφ(φ) ·∏

i, si∈R(z)

Nt(θi;µ, τ)

·∏

i,si∈R(z)

N (yi; θi, σ) ·∏

j,sj∈Rc(z)

N (yj; 0, σ)

∝ πφ(φ) ·[ ∏

i, si∈R(z)

Nt(θi;µ, τ) · N (yi; θi, σ)N (yi; 0, σ)

].

Computing p(θ |φ,y)

p(θ |φ, y) ∝∏

i, si∈R(z)

exp[− (θi − µ)2

2τ2− (yi − θi)2

2σ2

]· i[0,∞)(θi)

=∏

i, si∈R(z)

Nt

(θi ; θi(µ, τ),

( 1τ2

+1σ2

)−1/2)

where

θi(µ, τ) =τ2 yi + σ2 µ

τ2 + σ2.

Finally p(φ |y)!

p(φ |y) =p(φ, θ |y)p(θ |φ, y)

∝ πφ(φ) ·

l(y |φ)︷ ︸︸ ︷∏i, si∈R(z)

Nt(θi;µ, τ)

Nt

(θi; θi(µ, τ), (1/τ2 + 1/σ2)−1/2

) · N (yi; θi, σ)N (yi; 0, σ)

.

Here, l(y |φ) ≡ likelihood function of φ.

Simplifying l(y |φ)

l(y |φ) must not depend on θ =⇒ (arbitrarily) set θi = µ in l(y |φ) on theprevious page, yielding

l(y |φ) =∏

i, si∈R(z)

Nt(µ ; µ, τ)

Nt

(µ ; θi(µ, τ), (1/τ2 + 1/σ2)−1/2

) · N (yi;µ, σ)N (yi; 0, σ)

.

Log Likelihood

ln l(y |φ) = −N(z)2

ln(1 +

τ2

σ2

)+

∑i, si∈R(z)

{ln[Φ(θi(µ, τ) ·

√1τ2 + 1

σ2

)Φ(µ/τ)

]+

y2i

2 σ2− (yi − µ)2

2 (τ2 + σ2)

}.

where

N(z) =∑

i,si∈R(z)

1 ≡ number of measurements collected over R(z).

Log Likelihood: Comments

ln l(y |φ) = −N(z)2

ln(1 +

τ2

σ2

)+

∑i, si∈R(z)

{ln[Φ(θi(µ, τ) ·

√1τ2 + 1

σ2

)Φ(µ/τ)

]+

y2i

2 σ2− (yi − µ)2

2 (τ2 + σ2)

}.

If we set ln[Φ(eθi(µ,τ)·

q1

τ2+ 1σ2

)Φ(µ/τ)

]to zero in the above expression (i.e. allow

θi to be negative), we obtain the normalized log likelihood in [1] and ourapproach reduces to that in [1].

[1] A. Dogandzic and B. Zhang, “Bayesian NDE defect signal analysis,” IEEE Trans. Signal Processing, vol.55, pp. 372–378, Jan. 2007.

Outline

• Measurement Model

− Parametric model for defect location and shape,− Measurement-error (noise) model,− Defect-signal (reflectivity) model,− Prior specifications for the location, shape, and defect-signal

distribution parameters (model parameters).

• Bayesian Analysis

− Simulating the model parameters φ,

− Simulating the signals θi.

• Numerical Examples.

Background: Gibbs Sampler

For simplicity, consider sampling scalar random variables φ and u fromtheir joint pdf p(φ, u). We may know this joint pdf up to a multiplicativeconstant only, say p(φ, u) ∝ h(φ, u). Suppose that

• we cannot sample directly from p(φ, u) ∝ h(φ, u) but

• we can sample from the conditional pdfs

p(φ |u) ∝ h(φ, u) and p(u |φ) ∝ h(φ, u).

Note: In many practical applications, p(φ |u) and p(u |φ) are “standard”(and hence easy to sample from) or, if nonstandard, can be simulated usingvon Neumann’s rejection method.

Gibbs Sampler

Do the following:

(a) Start at some φ(0) and u(0);

(b) For t = 1, 2, . . .

Step 1: Sample u(t) from h(φ(t−1), ·) andStep 2: Sample φ(t) from h(·, u(t))

[i.e. create (φ(t), u(t)) using (φ(t−1), u(t−1)) =⇒ Markov chain!].

Under appropriate circumstances, (φ(1), u(1)), (φ(2), u(2)), . . . , (φ(T ), u(T ))can be used to approximate properties of p(φ, u) ∝ h(φ, u)!

Gibbs Sampler

Do the following:

(a) Start at some φ(0) and u(0);

(b) For t = 1, 2, . . .

Step 1: Sample u(t) from h(φ(t−1), ·) andStep 2: Sample φ(t) from h(·, u(t))

[i.e. create (φ(t), u(t)) using (φ(t−1), u(t−1)) =⇒ Markov chain!].

Under appropriate circumstances, (φ(1), u(1)), (φ(2), u(2)), . . . , (φ(T ), u(T ))can be used to approximate properties of p(φ, u) ∝ h(φ, u)!

The draws (φ(t), u(t)) are not i.i.d. but we do not care (much)!

Gibbs Sampler: Comment

Gibbs sampler is named after J.W. Gibbs, a 19th century American physicistand mathematician and one of the founders of modern thermodynamics andstatistical mechanics.

Gibbs Sampler: Comment

Gibbs sampler is named after J.W. Gibbs, a 19th century American physicistand mathematician and one of the founders of modern thermodynamics andstatistical mechanics.

But, Gibbs did not invent the Gibbs sampler. A more descriptive name hasbeen proposed: successive substitution sampling. Yet, the name “Gibbssampler” has won.

Gibbs Sampler: Comment (cont.)

The “Gibbs sampler” is yet another example of Stigler’s Law of Eponymy,which states that

No scientific discovery is named after the person(s) who thought of it.

Gibbs Sampler: Comment (cont.)

The “Gibbs sampler” is yet another example of Stigler’s Law of Eponymy,which states that

No scientific discovery is named after the person(s) who thought of it.

Interestingly, Stigler’s Law of Eponymy is not due to Stigler [2], meaningthat it is an example of itself!

[2] S. Stigler, Statistics on the Table: The History of Statistical Concepts and Methods, Cambridge, MA:Harvard University Press, 1999.

Background: (Univariate) Slice Sampler

Consider now sampling a random variable φ from a nonstandard p(φ) ∝h(φ).

(Seemingly Counter-Intuitive!) Idea:

• Invent a convenient bivariate distribution for, say, φ and u, with marginalpdf for φ specified by h(φ).

• Then, use Gibbs sampling to make

(φ(0), u(0)), (φ(1), u(1)), (φ(2), u(2)), . . . , (φ(T ), u(T )).

Background: (Univariate) Slice Sampler

Consider now sampling a random variable φ from a nonstandard p(φ) ∝h(φ).

(Seemingly Counter-Intuitive!) Idea:

• Invent a convenient bivariate distribution for, say, φ and u, with marginalpdf for φ specified by h(φ).

• Then, use Gibbs sampling to make

(φ(0), u(0)), (φ(1), u(1)), (φ(2), u(2)), . . . , (φ(T ), u(T )).

Create an auxiliary variable u just for convenience!

(Univariate) Slice Sampler

(Univariate) Slice Sampler

“Invent” a joint distribution for φ and u by declaring it to be uniform on

:

p(φ, u) ={

1c, 0 < u < h(φ)0, otherwise

∝ i(0,h(φ))(u).

With this joint pdf, P [φ ≤ 13] =∫ 13

−∞h(φ)

c dφ.

With this joint pdf, P [φ ≤ 13] =∫ 13

−∞h(φ)

c dφ.

The marginal pdf of φ is indeed specified by h(φ) =⇒ if we figureout how to do Gibbs sampling, we know how to generate a φ from h(φ).

Gibbs Sampler is Easy in This Case!

p(u |φ) = uniform(0, h(φ)

)p(φ |u) = uniform on {φ |h(φ) > u}︸ ︷︷ ︸

“slice”

.

Step 1: Given φ(t−1), Sample u(t) ∼ uniform(0, h(φ(t−1))

)

Step 1: Given φ(t−1), Sample u(t) ∼ uniform(0, h(φ(t−1))

)

Step 2: Given u(t), Sample φ(t) Uniform from slice(t)

If we can algebraically solve h(φ) = u(t), our task is easy. What if not?

Step 2 Implementation Using the Rejection MethodWhen we have band bounds on φ, say φMIN ≤ φ ≤ φMAX

generate i.i.d. values φ from uniform(φMIN, φMAX) until we produce a φ inthe slice [i.e. h(φ) > u(t)], which we then accept as φ(t).

Back to Our Problem:Slice Sampling from p(φ |y) ∝ πφ(φ) l(y |φ)

Create an auxiliary variable u so that

p(φ, u |y) ∝ πφ(φ) · i(0, l(y |φ))

(u)

implying

p(u |φ,y) = uniform(0, l(y |φ)

)p(φ |u, y) ∝ πφ(φ) · i

(0, l(y |φ))(u).

Slice Sampling from p(φ |y) ∝ πφ(φ) l(y |φ)

We can easily sample from p(φ , u|y) using Gibbs, whose one cycle is givenby the following two steps:

Step 1: Draw a u(t) from uniform(0, l(y |φ(t−1))

)and

Step 2: Draw a φ(t) from its prior pdf πφ(φ) subject to the indicator

restriction l(y |φ(t)) ≥ u(t).

Comments

• If a vector φ satisfies the indicator restriction:

l(y |φ) ≥ u(t), we say that it is in the slice.

• Step 2 is called “getting a point in the slice,” corresponding to samplingfrom

p(φ |u(t),y) ∝ πφ(φ) · i(0, l(y |φ))

(u(t)).

• A “naive” rejection method for getting a point in the slice: keep drawingφs i.i.d. from πφ(φ) until we get a φ that is in the slice. This may takeforever since φ is a 7-D vector!

Shrinkage Sampling for Getting a Point in the Slice [3]

Recall: The parameter space of φ = [x0,1, x0,2, d, A, ϕ, µ, τ ]T is ahyperrectangle with x0,1 ∈ (x0,1,MIN, x0,1,MAX), x0,2 ∈ (x0,2,MIN, x0,2,MAX),d ∈ (dMIN, dMAX), A ∈ (AMIN, AMAX), φ ∈ (−π/4, π/4), µ ∈ (0, µMAX), andτ ∈ (0, τMAX).

Shrinkage Sampling for Getting a Point in the Slice [3]

Recall: The parameter space of φ = [x0,1, x0,2, d, A, ϕ, µ, τ ]T is ahyperrectangle with x0,1 ∈ (x0,1,MIN, x0,1,MAX), x0,2 ∈ (x0,2,MIN, x0,2,MAX),d ∈ (dMIN, dMAX), A ∈ (AMIN, AMAX), φ ∈ (−π/4, π/4), µ ∈ (0, µMAX), andτ ∈ (0, τMAX).

Shrink the hyperrectangle from which the φs are sampled in such away that the previous value φ(t−1) remains within the hyperrectangle [3].

Note: φ(t−1) is always in the slice since u(t) ∼ uniform(0, l(y |φ(t−1))

)!

[3] R.M. Neal, “Slice sampling,” Ann. Statist., vol. 31, pp. 705–741, June 2003.

Shrinkage Sampling from p(φ |u(t),y)

We first define the initial (largest) hyperrectangle with limits

x0,1,L = x0,1,MIN, x0,1,U = x0,1,MAX,

x0,2,L = x0,2,MIN, x0,2,U = x0,2,MAX,

dL = dMIN, dU = dMAX

AL = AMIN, AU = AMAX

ϕL = ϕMIN, ϕU = ϕMAX

µL = 0, µU = µMAX

τL = 0, τU = τMAX.

Shrinkage Sampling from p(φ |u(t),y) (cont.)

Obtain φ(t) as follows:

1. Samplex0,1 from uniform(x0,1,L, x0,1,U)

x0,2 from uniform(x0,2,L, x0,2,U)

d from uniform(dL, dU)

A from uniform(AL, AU)

ϕ from uniform(ϕL, ϕU)

µ from uniform(µL, µU)

τ from uniform(τL, τU)

yielding φ = [x0,1, x0,2, d, A, ϕ, µ, τ ]T .

2. Check if φ is within the slice, i.e.

l(y |φ) ≥ u(t). (1)

If (1) holds, return φ(t) = φ and exit the loop.

(Recall that u(t) was obtained by sampling from uniform

(0, l(y |φ(t−1))

).)

3. If (1) does not hold, then shrink the hyperrectangle:

• If x0,1 < x(t−1)0,1 , set x0,1,L = x0,1;

else if x0,1 > x(t−1)0,1 , set x0,1,U = x0,1.

• If x0,2 < x(t−1)0,2 , set x0,2,L = x0,2;

else if x0,2 > x(t−1)0,2 , set x0,2,U = x0,2.

• If d < d(t−1), set dL = d; else if d > d(t−1), set dU = d.• If A < A(t−1), set AL = A; else if A > A(t−1), set AU = A.• If ϕ < ϕ(t−1), set ϕL = ϕ; else if ϕ > ϕ(t−1), set ϕU = ϕ.• If µ < µ(t−1), set µL = µ; else if µ > µ(t−1), set µU = µ.• If τ < τ (t−1), set τL = τ ; else if τ > τ (t−1), set τU = τ .• Go back to 1.

A Practical Modification of the Shrinkage Sampler

Computing l(y |φ) may cause a floating-point underflow =⇒ safer tocompute ln l(y |φ) rather than l(y |φ). We then compute

ω(t) = ln[l(y |φ(t−1))]− ε

where ε is an exponential random variable with mean one. We say that φis in the slice if

ln[l(y |φ)] ≥ ω(t)

which is equivalent to (1).

Shrinkage Slice Sampling: A 1-D Illustration.

Shrinkage Slice Sampling: A 1-D Illustration.

Shrinkage Slice Sampling: A 1-D Illustration.

Shrinkage Slice Sampling: A 1-D Illustration.

Shrinkage Slice Sampling: A 1-D Illustration.

Shrinkage Slice Sampling: A 1-D Illustration.

Outline

• Measurement Model

− Parametric model for defect location and shape,− Measurement-error (noise) model,− Defect-signal (reflectivity) model,− Prior specifications for the location, shape, and defect-signal

distribution parameters (model parameters).

• Bayesian Analysis

− Simulating the model parameters φ,

− Simulating the signals θi.

• Numerical Examples.

Simulating the Random Signals θi

To estimate the random signals θ, we utilize composition sampling fromthe posterior pdf p(θ |y) =

∫p(θ |φ,y) p(φ |y) dφ:

• Draw φ(t) from p(φ |y), as described earlier;

• Draw θ(t) from p(θ |φ(t),y) as follows:

− for i ∈ R(z(t)), draw conditionally independent samples θ(t)i from

p(θ(t)i |φ(t), yi) = Nt

(θi ; θi(µ(t), τ (t)),

[ 1(τ (t))2

+1σ2

]−1),

− for i ∈ Rc(z(t)), set θ(t)i = 0,

yielding θ(t) = [θ(t)1 , θ

(t)2 , . . . , θ

(t)Ntot

]T .

Computing the Mean Signal within R(z)

Define the mean signal within the potential defect region:

θ =1

N(z)·

∑i,si∈R(z)

θi.

Then, θ simulated in the tth draw is as

θ(t)

= [1/N(z(t))] ·∑

i,si∈R(z(t))

θ(t)i .

Computing the Defect Area

Define the potential defect area to be proportional to the number ofmeasurement locations i having signals θi that are within 10 dB from themaximum signal θMAX = maxi,si∈R(z) θi in the potential defect regionR(z).

Then, the area of the defect region simulated in the tth draw is

defect area(t) ∝ number of θis within 10 dB from θ(t)MAX

whereθ(t)MAX = max

i,si∈R(z(t))θ(t)i .

Ranking Potential Defects Using Bayes Factors

Bayes factor for comparing models H0 : µ = 0 (defect absent) versus thealternative H1 : µ > 0 (defect present):

BF =

noise-only distribution, according to H0︷ ︸︸ ︷Ntot∏i=1

N (yi; 0, σ2)

marginal distribution of the data in hand under H1

≡ likelihood ratio test statistic for testing H0 versus H1. Here,

• marginal distribution of the data in hand under H1 ≡ prior-weightedaverage of the likelihood (under H1).

Ranking Potential Defects Using Bayes Factors

Bayes factor (up to a multiplicative constant):

BF =[ ∫

l(y |φ) πφ(φ) dφ]−1

=∫

q(φ)l(y |φ) πφ(φ)

p(φ |y) dφ

where q(φ) is an arbitrary pdf having support within the support of theposterior distribution p(φ |y).

Bayes-Factor Computation

BF ≈ 1T·

t0+T∑t=t0+1

q(φ(t))

l(y |φ(t)) πφ(φ(t)).

A Choice of q(φ)q(φ) = qx0,1(x0,1) · qx0,2(x0,2) · qd(d) · qA(A) · qϕ(ϕ) · qµ(µ) · qτ(τ)

where

qx0,1(x0,1) = uniform(x0,1,MIN(T ), x0,1,MAX(T )

)qx0,2(x0,2) = uniform

(x0,2,MIN(T ), x0,2,MAX(T )

)qd(d) = uniform

(dMIN(T ), dMAX(T )

),

qA(A) = uniform(AMIN(T ), AMAX(T )

)qϕ(ϕ) = uniform

(ϕMIN(T ), ϕMAX(T )

)qµ(µ) = uniform

(µMIN(T ), µMAX(T )

), qτ(τ) = uniform

(τMIN(T ), τMAX(T )

).

A Choice of q(φ) (cont.)

Here

x0,1,MIN(T ) = min{x(t0)0,1 , x

(t0+1)0,1 , . . . , x

(t0+T )0,1 }

x0,1,MAX(T ) = max{x(t0)0,1 , x

(t0+1)0,1 , . . . , x

(t0+T )0,1 }

x0,2,MIN(T ) = min{x(t0)0,2 , x

(t0+1)0,2 , . . . , x

(t0+T )0,2 }

x0,2,MAX(T ) = max{x(t0)0,2 , x

(t0+1)0,2 , . . . , x

(t0+T )0,2 }

dMIN(T ) = min{d(t0), d(t0+1), . . . , d(t0+T )}

dMAX(T ) = max{d(t0), d(t0+1), . . . , d(t0+T )}· · ·

Bayesian Analysis: Summary

• We developed MCMC methods for sampling from the posteriordistributions of

− the model parameters φ and− random signals θ = [θ1, θ2, . . . , θNtot]

T .

• We utilize these samples to construct

− minimum mean-square error (MMSE) estimates and− Bayesian confidence regions (credible sets)

for φ and θ.

Estimation of φ and θ

Once we have collected enough samples, we estimate the posterior meansof φ and θ simply by averaging the last T draws:

E [φ |y] ≈ φ =1T

t0+T∑t=t0+1

φ(t), E [θ |y] ≈ θ =1T

t0+T∑t=t0+1

θ(t)

where t0 defines the burn-in period. Here, φ and θ are the (approximate)MMSE estimates of φ and θ.

Note: The proposed MCMC algorithms are automatic, i.e. theirimplementation does not require preliminary runs and additional tuning.

Outline

• Measurement Model

− Parametric model for defect location and shape,− Measurement-error (noise) model,− Defect-signal (reflectivity) model,− Prior specifications for the location, shape, and defect-signal

distribution parameters (model parameters).

• Bayesian Analysis

− Simulating the model parameters φ,− Simulating the signals θi.

• Numerical Examples.

Numerical Example: Experimental UT Data

• Ultrasonic (UT) C-scan data from an inspection of a cylindrical Ti 6-4billet.

• The sample, developed as a part of the work of the Engine TitaniumConsortium, contains 17 # 2 flat bottom holes at 3.2” depth.

• The ultrasonic data were collected in a single experiment by movinga probe along the axial direction and scanning the billet along thecircumferential direction at each axial position.

• The vertical coordinate is proportional to rotation angle and the horizontalcoordinate to axial position.

Ultrasonic C-scan data with 17 defects.

Experimental UT Data Example

Prior Specifications:

µMAX = max{y1, y2, . . . , yNtot}τMAX = 7 σ

dMIN = 1, dMAX = 10,

AMIN = 20, AMAX = 400

ϕMIN = −π/4, ϕMAX = π/4, (full range of ϕs)

x0,i,MIN,x0,i,MAX, i = 1, 2 selected to span the entire region that is being analyzed.

Experimental UT Data Example (cont.)

• Before analyzing the data, we divided the C-scan image into 3 regions.

• In each region, we subtracted row means from the measurements withinthe same row.

• The noise level in Region 2 is lower than the corresponding noise levelsin Regions 1 and 3. Indeed, the sample estimates of the noise varianceσ2 in Regions 1,2, and 3 are:

σ2 = 11.92, 10.32, and 12.02 (respectively).

The underlying non-stationarity of the noise is due to the billetmanufacturing process.

• In the following, we analyze each region separately assuming known noisevariances σ2, given above.

Regions 1,2, and 3.

Experimental UT Data Example (cont.)

We now describe our analysis of Region 1, where we ran seven Markovchains. We perform sequential identification of potential defects, as follows:

• Run 10, 000 cycles of the Gibbs sampler and utilized the last T = 2, 000samples to estimate the posterior pdfs p(φ |y) and p(θ |y).

• Get the approximate MMSE estimates of θi by averaging the T draws:

θi

∣∣chain 1 ≈

1T

t0+T∑t=t0+1

θ(t)i , i = 1, 2, . . . , N.

• Subtract the first chain’s MMSE estimates θi

∣∣∣chain 1

from the

measurements yi, i = 1, 2, . . . , N , effectively removing the first potentialdefect region from the data. Then run the second Markov chain usingthe filtered data:

yi

∣∣chain 2 = yi − θi

∣∣chain 1

compute the approximate MMSE estimates θi

∣∣chain 2 of the second

potential defect signal (using the second Markov chain), subtract themout, yielding

yi

∣∣chain 3 = yi

∣∣chain 2 − θi

∣∣chain 2

and continue this procedure until reaching the desired number of chains.

• We have applied the above sequential scheme to Regions 2 and 3 as well.

-

-

=

Our Sequential Identification in Action.

-

=

Our Sequential Identification in Action.

-

=

Our Sequential Identification in Action.

Our Sequential Identification in Action

Can you guess where the “erased” defects were

located?

Defect estimation results: Approximate MMSE estimates of the defectsignals for the chains having the largest average log posterior pdfs.

Bayesian Confidence Regions (Credible Sets)

90% Bayesian confidence regions for the normalized mean signals θ/σ anddefect areas

([defect area,

θ

σ]−[ defect area,

θ

σ])

C−1([ defect area

θσ

]−

[defect areab

θσ

])≤ ξ

(2)computed for all 24 potential defects in the three regions, where

• defect area and θ denote the MMSE estimates of the defect area and θ,

• C is the sample covariance matrix of the posterior samples

[defect area(t), θ(t)

/σ]T :

C =1T

t0+T∑t=t0+1

([defect area(t)

θ(t)

σ

]−

[defect areab

θσ

]T )

·([defect area(t),

θ(t)

σ]− [ defect area,

θ

σ])

and

• ξ is a constant chosen (for each chain) so that 90% of the samples

[defect area(t), θ(t)

/σ]T , t = t0, . . . , T satisfy (2).

Approximate 90% credible sets for the normalized mean signals θ/σand areas A of all potential defects in the three regions and a possibleclassification boundary for separating defects from non-defects.

Ranking Potential Defects Using Bayes Factors

Logarithms of the estimated Bayes factors (up to an additive constant)for all 24 potential defects in the three regions of the inspected billet;non-defects are marked in red.

Simulated Example: Low-SNR Regime

• Low signal-to-noise ratio (SNR) scenario using simulated data. Inparticular,

− we added i.i.d. zero-mean Gaussian noise with variance σ2 = 2502 tothe defect signals θi coming from one of the flat bottom holes fromthe previous example.

Simulated Example: Low-SNR Regime (cont.)

Simulated Example: Low-SNR Regime (cont.)

Simulated Example: Low-SNR Regime (cont.)

(a) Signals θi, (b) simulated noisy observations yi, and (c) MMSE estimates

θi, for i = 1, 2, . . . , Ntot.

Summary of Accomplishments

Developed

• a parametric model that describes defect shape, location, and reflectivity,

• a hierarchical Bayesian framework and MCMC algorithms for estimatingthese parameters assuming a single defect,

• a sequential method for identifying multiple potential defect regions andestimating their parameters, and

• a simple classification scheme for separating defects from non-defectsusing estimated mean signals and areas of the potential defects.

Our approach provides Bayesian confidence regions (credible sets) for theestimated parameters =⇒ important in NDE applications.

Summary of Accomplishments (cont.)

The results of this work will be published in

A. Dogandzic and B. Zhang, “Markov chain Monte Carlo defectidentification in NDE images,” to appear in Proc. Annu. Rev. ProgressQuantitative Nondestructive Evaluation (QNDE 2006), Portland, OR, Aug.2006.

and, for the simpler case of Gaussian priors on θi within the defect region,

A. Dogandzic and B. Zhang, “Bayesian NDE defect signal analysis,” IEEETrans. Signal Processing, vol. 55, pp. 372–378, Jan. 2007.

Future Work

Developing Bayesian model-based methods that incorporate

• the forward model and

• realistic defect and noise models

into flaw detection, estimation (sizing), and system design.