The Generalized NEQ and Detectability Index for Tomosynthesis and Cone-Beam CT: From Cascaded Systems Analysis to Human Observers

Abstract

Purpose: In the early development of new imaging modalities - such as tomosynthesis and cone-beam CT (CBCT) - an accurate predictive model for imaging performance is particularly valuable in identifying the physical factors that govern image quality and guiding system optimization. In this work, a task-based cascaded systems model for detectability index is proposed that describes not only the signal and noise propagation in the 2D (projection) and 3D (reconstruction) imaging chain but also the influence of background anatomical noise. The extent to which generalized detectability index provides a valid metric for imaging performance was assessed through direct comparison to human observer experiments.

Methods: Detectability index (d') was generalized to include anatomical background noise in the same manner as the generalized noise-equivalent quanta (NEQ) proposed by Barrett et al. (Proc. SPIE Med. Imaging, Vol. 1090, 1989). Anatomical background noise was measured from a custom phantom designed to present power-law spectral density comparable to various anatomical sites (e.g., breast and lung). Theoretical calculations of d' as a function of the source-detector orbital extent (θ_tot) was obtained from a 3D cascaded systems analysis model for tomosynthesis and cone-beam CT (CBCT). Four model observers were considered in the calculation of d': prewhitening (PW), non-prewhitening (NPW), prewhitening with eye filter and internal noise (PWE), and non-prewhitening with eye filter and internal noise (NPWE). Human observer performance was measured from 9AFC tests for a variety of idealized imaging tasks presented within a clutter phantom. Theoretical results (d') were converted to area under the ROC curve (A_z ) and compared directly to human observer performance as a function of imaging task and orbital extent.

Results: Theoretical results demonstrated reasonable correspondence with human observer response for all tasks across the continuum in θ_tot ranging from low-angle tomosynthesis (θ_tot ~10°) to CBCT (θ_tot ~180°). Both theoretical and experimental A_z were found to increase with acquisition angle, consistent with increased rejection of out-of-plane clutter for larger tomosynthesis angle. Of the four theoretical model observers considered, the prewhitening models tended to overestimate real observer performance, while the non-prewhitening models demonstrated reasonable agreement.

Conclusions: Generalized detectability index was shown to provide a meaningful metric for imaging performance, helping to bridge the gap between real observer performance and prevalent Fourier-based metrics based in first principles of spatial-frequency-dependent NEQ and imaging task.

Keywords: anatomical clutter; anatomical noise; cascaded systems analysis; cone-beam CT; detectability index; noise-equivalent quanta; noise-power spectrum.

Figure. 1

Cascaded systems analysis for 3D tomosynthesis and CBCT. Stages 0–7 model signal and noise transfer through various physical processes in the detector, while Stages 8 to 13 represent mathematical processes of 3D reconstruction by filtered backprojection as detailed in previous work: 0.) incident x-ray quanta; 1.) interaction of x-rays in detector; 2.) generation of secondary quanta; 3.) spread of secondary quanta; 4.) coupling of quanta to detector apertures; 5.) integration by detector aperture; 6.) sampling of detector pixels; 7.) readout with additive noise; 8.) log normalization; 9.) ramp filter; 10.) apodization filter; 11.) interpolation; 12.) 3D backprojection; and 13.) 3D sampling.

Figure 2

Hypotheses and task functions. Task 1: Detection of a small sphere in clutter. Task 2: Detection of a large sphere in clutter. Task 3. Discrimination of an encapsulated sphere vs. a uniform sphere.

Figure 3

(a) Experimental bench showing components and coordinate systems for CBCT and tomosynthesis. (b) Experimental phantom presenting power-law background designed using principles of fractal self-similarity. Six stimuli were inserted to the central coronal slice.

Figure 4

(a) Observer performing a 9AFC test in a darkened reading room. (b) Nine images were shown on a 3×3 grid as displayed by Matlab-based OPTEx software in a randomized order. The example shows the encapsulated sphere task, with the stimulus highlighted in the lower-left corner.

Figure 5

Theoretical and experimental imaging performance (A_z) plotted as a function of θ_tot for 3 imaging tasks. Reasonable agreement between theory and experiment is observed, with prewhitening observer models (PW and PWE) appearing to overestimate human response, while non-prewhitening observer models (NPW and NPWE) demonstrating closer agreement.