Planogram rebinning with the frequency-distance relationship

Champley, Kyle; Defrise, Michel; Clackdoyle, Rolf; Raylman, Raymond R.; Kinahan, Paul E.

doi:10.1109/tmi.2008.923950

Cited by 3 publications

(12 citation statements)

References 11 publications

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“…When the TOF profile h = 1, (7) becomes the non-TOF planogram formulation [5, 6, 7]. Similar to (4), we can use f̂ to represent the Fourier transform p̂ in (5) as…”

Section: Tof-pet Data Formationmentioning

confidence: 99%

“…In real imaging applications, the object f always has finite support. We follow Champley et al [6] and assume that the support of f is inside a cylindrical set S f :…”

Section: Tof-pet Data Formationmentioning

confidence: 99%

“…The full implementations of all the FORCEs are beyond the scope of the paper. For 3D non-TOF planograms, a different rebinning based on the frequency-distance relationship was implemented in [6, 7]. Here, we perform numerical simulations with a generic 2D dual-panel PET system ( r 2 = 0 and υ = 0) with timing resolution of 300 ps FWHM and crystal size of 1.2 mm.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…The non-TOF planogram is a 3D extension of the 2D linogram [22, 21], all the LORs passing through a fixed point inside the field of view (FOV) lie on a two-dimensional plane in the four-dimensional space of the non-TOF planograms [5]. For 3D non-TOF planograms, the Fourier rebinning using the frequency-distance relationship was developed by Champley et al [6, 7]. The Fourier rebinning based on John's equation in the local detector coordinates was developed by Defrise and Liu [17], Kao et al [32].…”

Section: Introductionmentioning

confidence: 99%

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Fourier rebinning and consistency equations for time-of-flight PET planograms

Defrise

Matej

et al. 2016

Inverse Problems

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Due to the unique geometry, dual-panel PET scanners have many advantages in dedicated breast imaging and on-board imaging applications since the compact scanners can be combined with other imaging and treatment modalities. The major challenges of dual-panel PET imaging are the limited-angle problem and data truncation, which can cause artifacts due to incomplete data sampling. The time-of-flight (TOF) information can be a promising solution to reduce these artifacts. The TOF planogram is the native data format for dual-panel TOF PET scanners, and the non-TOF planogram is the 3D extension of linogram. The TOF planograms is five-dimensional while the objects are three-dimensional, and there are two degrees of redundancy. In this paper, we derive consistency equations and Fourier-based rebinning algorithms to provide a complete understanding of the rich structure of the fully 3D TOF planograms. We first derive two consistency equations and John's equation for 3D TOF planograms. By taking the Fourier transforms, we obtain two Fourier consistency equations and the Fourier-John equation, which are the duals of the consistency equations and John's equation, respectively. We then solve the Fourier consistency equations and Fourier-John equation using the method of characteristics. The two degrees of entangled redundancy of the 3D TOF data can be explicitly elicited and exploited by the solutions along the characteristic curves. As the special cases of the general solutions, we obtain Fourier rebinning and consistency equations (FORCEs), and thus we obtain a complete scheme to convert among different types of PET planograms: 3D TOF, 3D non-TOF, 2D TOF and 2D non-TOF planograms. The FORCEs can be used as Fourier-based rebinning algorithms for TOF-PET data reduction, inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. As a byproduct, we show the two consistency equations are necessary and sufficient for 3D TOF planograms. Finally, we give numerical examples of implementation of a fast 2D TOF planogram projector and Fourier-based rebinning for a 2D TOF planograms using the FORCEs to show the efficacy of the Fourier-based solutions.

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“…When the TOF profile h = 1, (7) becomes the non-TOF planogram formulation [5, 6, 7]. Similar to (4), we can use f̂ to represent the Fourier transform p̂ in (5) as…”

Section: Tof-pet Data Formationmentioning

confidence: 99%

“…In real imaging applications, the object f always has finite support. We follow Champley et al [6] and assume that the support of f is inside a cylindrical set S f :…”

Section: Tof-pet Data Formationmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fourier rebinning and consistency equations for time-of-flight PET planograms

Defrise

Matej

et al. 2016

Inverse Problems

Self Cite

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“…The PFDR (Champley et al 2008) and Kao (Kao et al 2004) algorithms are the only rebinning algorithms that have been developed specifically for PET scanners with parallel planar detectors. Several rebinning algorithms have been developed for cylindrical PET scanners such as Fourier rebinning (FORE) (Defrise et al 1997), exact Fourier rebinning (FOREX) (Defrise et al 1997) and Fourier rebinning based on John’s equation (FORE-J) (Defrise and Liu 1999).…”

Section: Introductionmentioning

confidence: 99%

Advancements to the planogram frequency–distance rebinning algorithm

Champley¹,

Raylman²,

Kinahan³

2010

Inverse Problems

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In this paper we consider the task of image reconstruction in positron emission tomography (PET) with the planogram frequency–distance rebinning (PFDR) algorithm. The PFDR algorithm is a rebinning algorithm for PET systems with panel detectors. The algorithm is derived in the planogram coordinate system which is a native data format for PET systems with panel detectors. A rebinning algorithm averages over the redundant four-dimensional set of PET data to produce a three-dimensional set of data. Images can be reconstructed from this rebinned three-dimensional set of data. This process enables one to reconstruct PET images more quickly than reconstructing directly from the four-dimensional PET data. The PFDR algorithm is an approximate rebinning algorithm. We show that implementing the PFDR algorithm followed by the (ramp) filtered backprojection (FBP) algorithm in linogram coordinates from multiple views reconstructs a filtered version of our image. We develop an explicit formula for this filter which can be used to achieve exact reconstruction by means of a modified FBP algorithm applied to the stack of rebinned linograms and can also be used to quantify the errors introduced by the PFDR algorithm. This filter is similar to the filter in the planogram filtered backprojection algorithm derived by Brasse et al. The planogram filtered backprojection and exact reconstruction with the PFDR algorithm require complete projections which can be completed with a reprojection algorithm. The PFDR algorithm is similar to the rebinning algorithm developed by Kao et al. By expressing the PFDR algorithm in detector coordinates, we provide a comparative analysis between the two algorithms. Numerical experiments using both simulated data and measured data from a positron emission mammography/tomography (PEM/PET) system are performed. Images are reconstructed by PFDR+FBP (PFDR followed by 2D FBP reconstruction), PFDRX (PFDR followed by the modified FBP algorithm for exact reconstruction) and planogram filtered backprojection image reconstruction algorithms. We show that the PFDRX algorithm produces images that are nearly as accurate as images reconstructed with the planogram filtered backprojection algorithm and more accurate than images reconstructed with the PFDR+FBP algorithm. Both the PFDR+FBP and PFDRX algorithms provide a dramatic improvement in computation time over the planogram filtered backprojection algorithm.

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Data Consistency for Linograms and Planograms

Clackdoyle

2018

IEEE Trans. Radiat. Plasma Med. Sci.

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Planogram rebinning with the frequency-distance relationship

Cited by 3 publications

References 11 publications

Fourier rebinning and consistency equations for time-of-flight PET planograms

Fourier rebinning and consistency equations for time-of-flight PET planograms

Advancements to the planogram frequency–distance rebinning algorithm

Data Consistency for Linograms and Planograms

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