Challenges for the evaluation of the diagnostic imaging systems with - PowerPoint PPT Presentation

Challenges for the evaluation of the diagnostic imaging systems with nonlinear behavior Lucretiu M. Popescu Division of Imaging and Applied Mathematics Center for Devices and Radiological Health Food and Drug Administration E-mail: lucretiu.popescu@fda.hhs.gov February Fourier Talks University of Maryland, College Park February 21–22, 2013

Motivation • Integral-geometry models used for image reconstruction are replaced by physical and statistical models ◮ PET and SPECT already use iterative reconstruction algorithms with corrections for physical effects ◮ X-ray Computed Tomography (CT) has started the transition to iterative reconstruction algorithms 1

Motivation • Integral-geometry models used for image reconstruction are replaced by physical and statistical models ◮ PET and SPECT already use iterative reconstruction algorithms with corrections for physical effects ◮ X-ray Computed Tomography (CT) has started the transition to iterative reconstruction algorithms • In CT there is a need to reduce the dose while maintaining diagnostic effectiveness 1

CT dose reduction estimation problem • The iterative reconstruction algorithms (IRA) promise improved image quality (IQ) 2

CT dose reduction estimation problem • The iterative reconstruction algorithms (IRA) promise improved image quality (IQ) • Need to determine an IQ metric related with diagnostic performance 2

CT dose reduction estimation problem • The iterative reconstruction algorithms (IRA) promise improved image quality (IQ) • Need to determine an IQ metric related with diagnostic performance • It should be a scalar, generate IQ vs. dose plots and find the equivalence points Device 1 Device 2 Image Quality Dose 2

Traditional CT image reconstruction • Integral-geometry model � g ( y ) = f ( l )d l L ( y ) 3

Traditional CT image reconstruction • Integral-geometry model � g ( y ) = f ( l )d l L ( y ) • X-ray transmission tomography model � � g 0 j � L j µ ( l )d l ⇒ � − g j = g 0 j e µ ( l )d l = log g j L j where g 0 j data recorded without the object g j data recorded with the object 3

Discrete representation • Projection g = H f 4

Discrete representation • Projection g = H f • Reconstruction f = H − 1 g 4

Discrete representation • Projection g = H f • Reconstruction f = H − 1 g • In the presence of noise ˆ n f = H − 1 ( g + ˆ f = f + ˆ n g ) 4

Discrete representation • Projection g = H f • Reconstruction f = H − 1 g • In the presence of noise ˆ n f = H − 1 ( g + ˆ f = f + ˆ n g ) • The image quality is linearly determined by H − 1 and ˆ n g 4

Discrete representation • Projection g = H f • Reconstruction f = H − 1 g • In the presence of noise ˆ n f = H − 1 ( g + ˆ f = f + ˆ n g ) • The image quality is linearly determined by H − 1 and ˆ n g • Noise propagation is independent of the object (system property) n f = H − 1 ˆ ˆ n g 4

X-ray transmission tomography in real world X−ray tube Detector 10 φ (E) 9 8 7 φ (E) [1/KeV] 6 5 4 3 2 1 0 20 30 40 50 60 70 80 90 100 110 • Polychromatic source • Attenuation dependent on energy. Scatter • Energy integrating detectors, nonlinear response • Statistical behavior 5

X-ray transmission tomography physical model � − � L j µ ( l,E )d l ε j ( E ) ξ ( E )d E + Is j g j = I φ j ( E )e where g j the detector signal for projection j I the X-ray source intensity φ j ( E ) the source spectrum L j µ ( l,E )d l attenuation along the projection j − � e ε j ( E ) detector efficiency ξ ( E ) detector response signal; e.g. ξ ( E ) ∝ E Is j scattered photons contribution 6

Iterative reconstruction algorithm • The voxel’s attenuation represented as µ i ( E ) = f i µ 0 ( E ) • Find the extreme value of a cost function g j − g j ) 2 (ˆ � S ( f ) = + βR ( f ) η j g j j βR ( f ) regularization term, R ( f ) = � � ψ ( f i − f k ) i k ∈N i 7

Iterative reconstruction algorithm • The voxel’s attenuation represented as µ i ( E ) = f i µ 0 ( E ) • Find the extreme value of a cost function g j − g j ) 2 (ˆ � S ( f ) = + βR ( f ) η j g j j βR ( f ) regularization term, R ( f ) = � � ψ ( f i − f k ) i k ∈N i • Properties ◮ Nonlinear behavior ◮ Noise strongly dependent on the object ◮ External constraints can be introduced 7

Image quality (IQ) measures • Resolution ◮ identify line or grid patterns ◮ point spread function (PSF) ◮ modulation transfer function, MTF = F [ PSF ] 8

Image quality (IQ) measures • Resolution ◮ identify line or grid patterns ◮ point spread function (PSF) ◮ modulation transfer function, MTF = F [ PSF ] • Noise ◮ pixel variance (no spatial correlations) ◮ noise power spectrum (NPS) 8

Image quality (IQ) measures • Resolution ◮ identify line or grid patterns ◮ point spread function (PSF) ◮ modulation transfer function, MTF = F [ PSF ] • Noise ◮ pixel variance (no spatial correlations) ◮ noise power spectrum (NPS) • For ranking we need to express the IQ as a single number 8

Contrast to noise ratio (CNR) CNR = ROI contrast pixel variance • Does not account for spatial correlations of the noise • Depends on the ROI original contrast • Arbitrary scaling 9

Task based evaluation • A test task relevant for the clinical application 10

Task based evaluation • A test task relevant for the clinical application • Yet simple enough ◮ Can be analytically studied ◮ Convenient to be applied experimentally 10

Task based evaluation • A test task relevant for the clinical application • Yet simple enough ◮ Can be analytically studied ◮ Convenient to be applied experimentally Detection of small, low contrast, signals 10

Detection of a signal at known location • We have ◮ g 1 – signal average ◮ g 0 – background average ◮ K – noise covariance (same for signal and background) 11

Detection of a signal at known location • We have ◮ g 1 – signal average ◮ g 0 – background average ◮ K – noise covariance (same for signal and background) • Likelihood ratio test for a given location ˆ g � Pr (ˆ g | 1) � = ( g 0 − g 1 ) t K − 1 ˆ λ (ˆ g ) = log g Pr (ˆ g | 0) If λ (ˆ g ) > λ th then ˆ g is declared positive 11

Detection of a signal at known location • We have ◮ g 1 – signal average ◮ g 0 – background average ◮ K – noise covariance (same for signal and background) • Likelihood ratio test for a given location ˆ g � Pr (ˆ g | 1) � = ( g 0 − g 1 ) t K − 1 ˆ λ (ˆ g ) = log g Pr (ˆ g | 0) If λ (ˆ g ) > λ th then ˆ g is declared positive • Signal to noise ratio (SNR) { E [ λ ( g 1 )] − E [ λ ( g 0 )] } 2 d 2 = 1 2 { var[ λ ( g 1 )] − var[ λ ( g 0 )] } 11

Interpretation of SNR • At high dose “noise” → 0 ⇒ SNR → ∞ 12

Interpretation of SNR • At high dose “noise” → 0 ⇒ SNR → ∞ • If we compare two modalities, then at high dose ∆ SNR = SNR 2 − SNR 1 can have arbitrary values 12

Interpretation of SNR • At high dose “noise” → 0 ⇒ SNR → ∞ • If we compare two modalities, then at high dose ∆ SNR = SNR 2 − SNR 1 can have arbitrary values • SNR is not suited for direct quantitative comparisons 12

Interpretation of SNR • At high dose “noise” → 0 ⇒ SNR → ∞ • If we compare two modalities, then at high dose ∆ SNR = SNR 2 − SNR 1 can have arbitrary values • SNR is not suited for direct quantitative comparisons • We need to turn SNR into quantity that has a more direct connection with the signal detection performance 12

Relative operating characteristic (ROC) backgr. g ( z ) 0.6 signal f ( z ) Prob. dens. 0.4 0.2 0 -2 -1 0 1 2 3 4 5 Template matching score, z 13

Relative operating characteristic (ROC) backgr. g ( z ) 0.6 signal f ( z ) Prob. dens. 0.4 0.2 0 -2 -1 0 1 2 3 4 5 Template matching score, z 13

Relative operating characteristic (ROC) 1 backgr. g ( z ) 0.6 signal f ( z ) z d f ( z )d z 0.8 Prob. dens. 0.4 � ∞ 0.6 Sensitivity, 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 3 4 5 � ∞ 1 - Specificity, z d g ( z )d z Template matching score, z 13

Relative operating characteristic (ROC) 1 backgr. g ( z ) 0.6 signal f ( z ) z d f ( z )d z 0.8 Prob. dens. 0.4 � ∞ 0.6 Sensitivity, 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 3 4 5 � ∞ 1 - Specificity, z d g ( z )d z Template matching score, z • Area under the ROC curve A = Prob ( signal score > background score ) ∈ (0 . 5 , 1) � d • Relation with SNR: A = 1 � �� 1 + erf 2 2 13

Detection of signals at unknown locations -30 -20 -10 0 10 20 6 6 4 4 2 2 y (cm) y (cm) 0 0 -2 -2 -4 -4 -6 -6 -6 -6 -4 -4 -2 -2 0 0 2 2 4 4 6 6 x (cm) x (cm) 14

Detection of signals at unknown locations • One dimensional random field example 40 ˆ f ( x ) sample f ( x ) ideal 30 20 10 0 -10 -20 -100 -80 -60 -40 -20 0 20 40 60 80 100 x [mm] 15

‘Image’ scanning 40 ˆ f ( x ) sample z ( x ) scan value scan window 30 20 10 0 -10 -20 -100 -80 -60 -40 -20 0 20 40 60 80 100 x [mm] 16