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Clarkson E, Kupinski M. Effect on null spaces of list-mode imaging systems due to increasing the number of attributes. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2022; 39:959-968. [PMID: 36215457 DOI: 10.1364/josaa.443326] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2021] [Accepted: 03/14/2022] [Indexed: 06/16/2023]
Abstract
There are two types of uncertainty in image reconstructions from list-mode data: statistical and deterministic. One source of statistical uncertainty is the finite number of attributes of the detected particles, which are sampled from a probability distribution on the attribute space. A deterministic source of uncertainty is the effect that null functions of the imaging operator have on reconstructed pixel or voxel values. Quantifying the reduction in this deterministic source of uncertainty when more attributes are measured for each detected particle is the subject of this work. Specifically, upper bounds on an error metric are derived to quantify the error introduced in the reconstruction by the presence of null functions, and these upper bounds are shown to be reduced when the number of attributes is increased. These bounds are illustrated with an example of a two-dimensional single photon emission computed tomography (SPECT) system where the depth of interaction in the scintillation crystal is added to the attribute vector.
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Kuo J, Granstedt J, Villa U, Anastasio MA. Computing a projection operator onto the null space of a linear imaging operator: tutorial. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2022; 39:470-481. [PMID: 35297431 PMCID: PMC10560448 DOI: 10.1364/josaa.443443] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2021] [Accepted: 01/07/2022] [Indexed: 06/14/2023]
Abstract
Many imaging systems can be approximately described by a linear operator that maps an object property to a collection of discrete measurements. However, even in the absence of measurement noise, such operators are generally "blind" to certain components of the object, and hence information is lost in the imaging process. Mathematically, this is explained by the fact that the imaging operator can possess a null space. All objects in the null space, by definition, are mapped to a collection of identically zero measurements and are hence invisible to the imaging system. As such, characterizing the null space of an imaging operator is of fundamental importance when comparing and/or designing imaging systems. A characterization of the null space can also facilitate the design of regularization strategies for image reconstruction methods. Characterizing the null space via an associated projection operator is, in general, a computationally demanding task. In this tutorial, computational procedures for establishing projection operators that map an object to the null space of a discrete-to-discrete imaging operator are surveyed. A new machine-learning-based approach that employs a linear autoencoder is also presented. The procedures are demonstrated by use of biomedical imaging examples, and their computational complexities and memory requirements are compared.
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Affiliation(s)
- Joseph Kuo
- Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
| | - Jason Granstedt
- Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
| | - Umberto Villa
- Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO 63130, USA
| | - Mark A. Anastasio
- Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
- Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
- Department of Bioengineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
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Clarkson E, Kupinski M. Effect on null spaces of list-mode imaging systems due to increasing the size of attribute space. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2021; 38:387-394. [PMID: 33690468 PMCID: PMC8101067 DOI: 10.1364/josaa.403016] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2020] [Accepted: 01/18/2021] [Indexed: 06/12/2023]
Abstract
An upper bound is derived for a figure of merit that quantifies the error in reconstructed pixel or voxel values induced by the presence of null functions for any list-mode system. It is shown that this upper bound decreases as the region in attribute space occupied by the allowable attribute vectors expands. This upper bound allows quantification of the reduction in this error when this type of expansion is implemented. Of course, reconstruction error is also caused by system noise in the data, which has to be treated statistically, but we will not be addressing that problem here. This method is not restricted to pixelized or voxelized reconstructions and can in fact be applied to any region of interest. The upper bound for pixelized reconstructions is demonstrated on a list-mode 2D Radon transform example. The expansion in the attribute space is implemented by doubling the number of views. The results show how the pixel size and number of views both affect the upper bound on reconstruction error from null functions. This reconstruction error can be averaged over all pixels to give a single number or can be plotted as a function on the pixel grid. Both approaches are demonstrated for the example system. In conclusion, this method can be applied to any list-mode system for which the system operator is known and could be used in the design of the systems and reconstruction algorithms.
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Affiliation(s)
- Eric Clarkson
- Department of Medical Imaging, University of Arizona, Tucson, Arizona 85721, USA
| | - Meredith Kupinski
- College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA
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Rahman A, Zhu Y, Clarkson E, Kupinski MA, Frey EC, Jha AK. Fisher information analysis of list-mode SPECT emission data for joint estimation of activity and attenuation distribution. INVERSE PROBLEMS 2020; 36:084002. [PMID: 33071423 PMCID: PMC7561050 DOI: 10.1088/1361-6420/ab958b] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
The potential to perform attenuation and scatter compensation (ASC) in single-photon emission computed tomography (SPECT) imaging without a separate transmission scan is highly significant. In this context, attenuation in SPECT is primarily due to Compton scattering, where the probability of Compton scatter is proportional to the attenuation coefficient of the tissue and the energy of the scattered photon and the scattering angle are related. Based on this premise, we investigated whether the SPECT scattered-photon data acquired in list-mode (LM) format and including the energy information can be used to estimate the attenuation map. For this purpose, we propose a Fisher-information-based method that yields the Cramer-Rao bound (CRB) for the task of jointly estimating the activity and attenuation distribution using only the SPECT emission data. In the process, a path-based formalism to process the LM SPECT emission data, including the scattered-photon data, is proposed. The Fisher information method was implemented on NVIDIA graphics processing units (GPU) for acceleration. The method was applied to analyze the information content of SPECT LM emission data, which contains up to first-order scattered events, in a simulated SPECT system with parameters modeling a clinical system using realistic computational studies with 2-D digital synthetic and anthropomorphic phantoms. The method was also applied to LM data containing up to second-order scatter for a synthetic phantom. Experiments with anthropomorphic phantoms simulated myocardial perfusion and dopamine transporter (DaT)-Scan SPECT studies. The results show that the CRB obtained for the attenuation and activity coefficients was typically much lower than the true value of these coefficients. An increase in the number of detected photons yielded lower CRB for both the attenuation and activity coefficients. Further, we observed that systems with better energy resolution yielded a lower CRB for the attenuation coefficient. Overall, the results provide evidence that LM SPECT emission data, including the scattered photons, contains information to jointly estimate the activity and attenuation coefficients.
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Affiliation(s)
- Ashequr Rahman
- Department of Biomedical Engineering, Washington University in St. Louis, St. Louis, MO, USA
- Mallinckrodt Institute of Radiology, Washington University in St. Louis, St. Louis, MO, USA
| | - Yansong Zhu
- Department of Radiology and Radiological Sciences, Johns Hopkins University, Baltimore, MD, USA
- Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD, USA
- Department of Physics & Astronomy, University of British Columbia, Canada
| | - Eric Clarkson
- College of Optical Sciences, University of Arizona, Tucson AZ, USA
| | | | - Eric C Frey
- Department of Radiology and Radiological Sciences, Johns Hopkins University, Baltimore, MD, USA
| | - Abhinav K Jha
- Department of Biomedical Engineering, Washington University in St. Louis, St. Louis, MO, USA
- Mallinckrodt Institute of Radiology, Washington University in St. Louis, St. Louis, MO, USA
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Barrett HH. Is there a role for image science in the brave new world of artificial intelligence? J Med Imaging (Bellingham) 2020; 7:012702. [PMID: 34660841 PMCID: PMC8495496 DOI: 10.1117/1.jmi.7.1.012702] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022] Open
Abstract
Knowledge of the principles of image science is essential to the successful application of artificial intelligence in medical imaging.
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Affiliation(s)
- Harrison H Barrett
- University of Arizona, Wyant College of Optical Sciences, Tucson, Arizona, United States.,University of Arizona, Center for Gamma-Ray Imaging, Department of Medical Imaging, Tucson, Arizona, United States
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Caucci L, Liu Z, Jha AK, Han H, Furenlid LR, Barrett HH. Towards continuous-to-continuous 3D imaging in the real world. Phys Med Biol 2019; 64:185007. [PMID: 31470417 PMCID: PMC7038643 DOI: 10.1088/1361-6560/ab3fb5] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/07/2023]
Abstract
Imaging systems are often modeled as continuous-to-discrete mappings that map the object (i.e. a function of continuous variables such as space, time, energy, wavelength, etc) to a finite set of measurements. When it comes to reconstruction, some discretized version of the object is almost always assumed, leading to a discrete-to-discrete representation of the imaging system. In this paper, we discuss a method for single-photon emission computed tomography (SPECT) imaging that avoids discrete representations of the object or the imaging system, thus allowing reconstruction on an arbitrarily fine set of points.
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Affiliation(s)
- L Caucci
- Department of Medical Imaging, University of Arizona, Tucson, AZ 85724, United States of America. College of Optical Sciences, University of Arizona, Tucson, AZ 85719, United States of America. Author to whom any correspondence should be addressed
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