Computational Methods Short Course on Image Quality

Transcription

1 Computational Methods Short Course on Image Quality Matthew A. Kupinski What we Will Cover Sources of randomness Computation of linear-observer performance Computation of ideal-observer performance Miscellaneous

2 Sources of Randomness g = Hf + n Every term in the imaging equation may have random variations* Neglecting a source of randomness can cause spurious results Addressing all sources of randomness can be computationally difficult *Random variations in H will not be covered Example Image Pinhole plate Object What size pinhole maximizes detectability?

3 Example Image Pinhole plate Object For a non-random background f, the ideal observer uses a wide-open aperture Object Models

4 Object Models Reduce the object model f down to a finite number of parameters f (ρ) Linear Observers Hotelling observer SNR2 = g K g g Wiener estimator EMSE = tr K θ K g,θ K g K g,θ "

5 Data Covariance If g is an M vector where M is the number of pixels K g is an M M matrix We need to both estimate and invert this very large matrix Data Covariance Let us assume that we have a single 28 by 28 detector The matrix K g has 27 million elements We need at least 282 or 6, images to produce an invertible estimate of K g

6 Data Covariance Without any assumption of independence, we can decompose the covariance matrix In the case of the Hotelling observer Kg = Kn + Kg (4.27) is the noise covariance averaged over objects Kn K g is the covariance of the noise-free background images Data Covariance For photon-counting detectors, the noise covariance K n is a diagonal matrix with elements given by g An estimate of K g can be produced using long-exposure images of random backgrounds The addition of a known diagonal matrix ensures that the overall covariance K g is invertible even with few samples

7 Data Covariance We now have a full-rank estimate of the data covariance We still need to invert this estimate Neumann series Matrix-inversion lemma Channels Neumann Series " K g = σ2 I + A = σ2 I + 2 A σ (4.37) #j " 2A = 2 σ j= σ (4.38) K g

8 Matrix-Inversion Lemma K g = K n + K g = K n + W W (4.46) Where W is a M Ns matrix Kn + W W " # $ = Kn # " " $ " W Kn Kn W I + W Kn W (4.47) Channelized Hotelling Observer v = U g (4.22) SNR2v = v K v v = g U U K g U " U g (4.23)

9 Channelized Hotelling Observer VALIDATION OF THE LAGUERRE-GAUSS CHANNELS (Work of Brandon Gallas) xpansion functions ating Hotelling s: e-gauss functions er expansion is rotationally sym. The Wiener Estimator Hotelling observer Kg = Kn + Kg Wiener estimator noise Kg = Kg bg + K g + K param g

10 Ideal Observers Bayesian ideal observer Λ(g) = pr(g H2 ) = ΛBKE (g b)"b g,h pr(g H ) ΛBKE (g b) = pr(g H2, b) pr(g H, b) (4.8) (4.8) Ideal Observers Λ(g) = pr(g H2 ) = ΛBKE (g b)"b g,h pr(g H ) (4.8) If we could draw samples from pr(b g, H ), then we could approximate the ideal observer by, Ns ΛBKE (g bn ) Λ(g) Ns n= (4.87)

11 MCMC Markov chain Monte Carlo Requires parameterized object model f (ρ) Directed search through parameter space Each iteration only depends on the previous one pr(b g, H ) need only be known to a scale MCMC i i q

12 MCMC.6 LBKE Iteration MCMC 8 4 x

13 MCMC ' 637,-.+//,2(3)4/.5 "& "% 89+,:2; 89+,:2< 89+,:2= "$ "# "# "$ "% "& ' ()*+,-.+//,2(3)4/.5 What about Estimation? Bayesian estimation θ MMSE = θ ΛBKE (g b, θ, θ )"b g,θ ΛBKE (g b, θ, θ ) = " pr(g b, θ) pr(g b, θ ) Maximum-likelihood estimation s= db pr(b g, θ) θ log [pr(g b, θ)] " F = ss g θ

14 No Gold "* "& "% "% # ' ") "$ "( "# "$ "# "' "# "$ "% "& +,-. /2.3. ' "# "# "$ "% ()*+,-./+.+ No Gold "% "$ '()*+,)-./ "' "# "# "# "$ '()*+,)- # "% "& "& '

15 No Gold "%.7 ").5 Estimate 2 34/56/78' "$ "(.4 "#.3 "'.2. "' "# "# "$ "% "& '.2 *+,-./ No Gold "%.7 ").5 Estimate 2 34/56/78' "$ "(.4 "#.3 "'.2. "' "# "# "$ "% *+,-./-2- "& '.2.4

16 No Gold "%.7 ").5 Estimate 2 34/56/78' "$ "(.4 "#.3 "'.2. "' "# "# "$ "% "& '.2 *+,-./ No Gold Model θ pm = am θp + bm + npm θp : Gold standard (4.) θ pm : Estimate for modality m npm : Noise am, bm,σm : Linear model parameters Note: The same population of patients are imaged on all modalities

17 Assumptions Gold standard does not vary from one modality to the next There is an underlying distribution on the gold standard: pr(θ ρ) The noise npm is Gaussian distributed with variance σm Patients are independent Method We can estimate{am,bm, σm, ρ} maximum-likelihood estimation using

18 Simulation Study Estimate.8.8 Estimate 3 Estimate Simulation Study.8 Estimate Estimate 3 Estimate

19 Simulation Study.8.8 Estimate Estimate 2 Estimate Software No-gold comparisons Channelized Hotelling observer MCMC via the Image-Quality Toolkit (coming soon)

20 Concluding Remarks Knowledge of forward model and statistics The more images, the better Many methods for computing task performance are computationally feasible