Abstract: In a state in which a subject is absent, blank data is collected by a self-radioactivity element typified by Lu-176 (S1). In a state in which the subject is present, transmission data is collected by the self-radioactivity element (S2). Emission data is collected by ? rays emitted from the subject injected with a radiopharmaceutical (S3). Absorption-corrected data is calculated based on the blank data and the transmission data (S4 to S7), and the emission data is absorption-corrected using the absorption-corrected data (S8). Although such background data obtained by the self-radioactivity is originally abandoned, the background data is rather used for the absorption-corrected data. Stable absorption correction can be thereby conducted.

Abstract: A method for improving single photon emission computed tomography by controlling acquisition parameters specific to the imaging goals and specific to the individual case under study. Data acquisition is modulated by scanning to adapt to the particular signal to noise characteristics of each object. A preliminary acquisition quickly scans the object of interest. The preliminary data is analyzed to optimize the secondary scan. The secondary scan is then acquired with optimized sampling of the object based on its own particular image characteristics. The system is able to learn, incorporating site specific data into a triaging set.

Abstract: Plural detection boards are stacked and fixed. The detection board has a wiring board, a semiconductor detection device fixed on an upper surface of the wiring board and configured to detect radiation, and a spacer fixed on the upper surface of the wiring board. Each of the detection boards is provided so that the semiconductor detection device and the spacer have a designated positional relationship. In addition, the spacers are stacked and matched in an X-Y plane surface with each other so that the detection boards are fixed by fixing members.

Abstract: A SPECT system which scans over multiple separate scans and individually motion compensates the information obtained from each of these scans. The separate scans may be over different angular extents and may be for different purposes. One of the scans for example may be a scout scan, and the other scans may then be scans which concentrate on areas identified during the scout scan. Alternatively, the scans may all being exactly the same and stitched together after the individual motion compensation. Since each of the scans are shorter, the patient will presumably have moved less during each individual scan, and the amount of motion is hence presumably less.

Abstract: A PET scanner (8) includes a ring of detector modules (10) encircling an imaging region (12). Each of the detector modules includes at least one detector pixel (24,34). Each detector pixel includes a scintillator (20, 30) optically coupled to one or more sensor APDs (54) that are biased in a breakdown region in a Geiger mode. The sensor APDs output a pulse in response to the light from the scintillator corresponding to a single incident radiation photon. A reference APD (26, 36) also biased in a break-down down region in a Geiger mode is optically shielded from light and outputs a temperature dependent signal. At least one temperature compensation circuit (40) adjusts a bias voltage applied to the sensor APDs based on the temperature dependent signal.

Abstract: An apparatus for imaging an eye includes a housing and a system of optical components disposed in the housing. The apparatus is capable of operating in a line scanning laser ophthalmoscope (LSLO) mode and an optical coherence tomography (OCT) mode. The system of optical components can include a first source to provide a first beam of light for the OCT mode and a second source to provide a second beam of light for the LSLO mode. In the OCT mode, a first optic is used that (i) scans, using a first surface of the first optic, the first beam of light along a retina of an eye in a first dimension, and (ii) descans, using the first surface, a first light returning from the eye in the first dimension to a detection system in the OCT mode. In the LSLO mode, the first optic is used where the second beam of light passes through a second surface of the first optic.

Abstract: A system that incorporates teachings of the present disclosure may include, for example, an ionizing radiation sensor having a first scintillator for generating photons from incoming ionizing radiation, an imaging intensifier for amplifying the photons, and an electron-multiplying charge-coupled device (EMCCD) coupled to the imaging intensifier for sensing the amplified photons generated by the imaging intensifier. Additional embodiments are disclosed.

Abstract: An apparatus comprising a radiation source, coincident positron emission detectors configured to detect coincident positron annihilation emissions originating within a coordinate system, and a controller coupled to the radiation source and the coincident positron emission detectors, the controller configured to identify coincident positron annihilation emission paths intersecting one or more volumes in the coordinate system and align the radiation source along an identified coincident positron annihilation emission path.

Abstract: A method and apparatus are provided for correcting primary and secondary emission data. The method includes obtaining an emission data set having primary and secondary emission data representative of primary and secondary emission particles emitting from a region of interest and applying a scatter correction model to the emission data set to derive an estimated scatter vector. The method also includes comparing the emission data set to the estimated scatter vector to identify an amount of secondary emission data in the emission data set and correcting the emission data set based on the amount of secondary emission data identified in the comparing operation.

Abstract: An imaging system (10) includes at least one radiation detector (20) disposed adjacent a subject receiving aperture (18) to detect and measure at least one of emission and transmission radiation from a subject, the detector (20) at a plurality of projection angles. A processor (64) determines which radiation data belong to a field of view of the radiation detector (20) at each projection angle. An image processor (70, 72) iteratively reconstructs the radiation detected only in the determined field of view into image representations. Truncated data is compensated by supplying the untruncated data from the projections taken at different angles at which the truncated data is untruncated.

Abstract: A region of interest is automatically evaluated. The automatic evaluation is based on assessments of one or more characteristics. The one or more characteristics of the region of interest are assessed in a plurality of image data sets acquired by a respective plurality of imaging modalities. In some embodiments, the evaluation is based on assessments of one or more characteristics for each region of interest derived from a combination of structural and functional image data. In one embodiment, the set of structural image data is a set of CT image data and the set of functional image data is a set of PET image data. The one or more lesions may be detected in the structural and/or functional image data by automated routines or by a visual inspection by a clinician or other reviewer.

Abstract: A host lattice modified GOS scintillating material and a method for using a host lattice modified GOS scintillating material is provided. The host lattice modified GOS scintillating material has a shorter afterglow than conventional GOS scintillating material. In addition, a radiation detector and an imaging device incorporating a host lattice modified GOS scintillating material are provided.

Abstract: A compound comprising a metal chelate linked to a hexose carrier for use as a metallopharmaceutical diagnostic or therapeutic agent is provided. The compound is suitable for imaging by single-photon emission computed tomography, computer assisted tomography, magnetic resonance spectroscopy, magnetic resonance imaging, positron emission tomography, fluorescence imaging or x-ray.

Abstract: An iterative image reconstruction method used with an imaging system that generates projection data, the method comprises: collecting the projection data; choosing a polar or cylindrical image definition comprising a polar or cylindrical grid representation and a number of basis functions positioned according to the polar or cylindrical grid so that the number of basis functions at different radius positions of the polar or cylindrical image grid is a factor of a number of in-plane symmetries between lines of response along which the projection data are measured by the imaging system; obtaining a system probability matrix that relates each of the projection data to each basis function of the polar or cylindrical image definition; restructuring the system probability matrix into a block circulant matrix and converting the system probability matrix in the Fourier domain; storing the projection data into a measurement data vector; providing an initial polar or cylindrical image estimate; for each iteration; reca

Abstract: The present invention provides a laser apparatus capable of improving a scan speed and achieving a scan rate equal to or more than 1 MHz, and an optical tomographic imaging apparatus using the laser apparatus as a light source. The laser apparatus includes a ring resonator, the ring resonator having a structure in which a first modulator, a normal dispersion region, a second modulator and an anomalous dispersion region are arranged in this order, and in this arrangement, a gain medium is included, and being configured so that modulation with respect to the second modulator can be caused to be phase modulation by periodically superimposing phase modulation on modulation with respect to the first modulator.

Abstract: An imaging method and apparatus, the method comprising collecting detector output data from a radiation detector positioned near a subject provided with a radio-active tracer, and resolving individual signals in the detector output data by (i) determining a signal form of signals present in the data, (ii) making parameter estimates of one or more parameters of the signals, wherein the one or more parameters comprise at least a signal temporal position, and (iii) determining the energy of each of the signals from at least the signal form and the parameter estimates. The acceptable subject to detector distance is reduced or increased, spatial resolution is improved, tracer dose or concentration is reduced, subject radiation exposure is reduced and/or scanning time is reduced.

Abstract: Apparatus and methods for determining a boundary of an object for positron emission tomography (PET) scatter correction are provided. One method includes obtaining positron emission tomography (PET) data and computed tomography (CT) data for an object. The PET data and CT data is acquired from an imaging system. The method further includes determining a PET data boundary of the object based on the PET data and determining a CT data boundary of the object based on the CT data. The method further includes determining a combined boundary for PET scatter correction. The combined boundary encompasses the PET data boundary and the CT data boundary.

Abstract: A tungstate-based scintillating material and a method for using a tungstate-based scintillating material is provided. In addition, a radiation detector and an imaging device incorporating a tungstate-based scintillating material are provided.

Abstract: A PET apparatus comprises a plurality of detector units in the circumferential direction, wherein the detector unit includes a plurality of unit substrates therein, and wherein the unit substrate includes: a plurality of detectors upon which a ?-ray is incident; and an analog ASIC and digital ASIC for processing a ?-ray detection signal outputted by each of the detectors. The analog ASIC includes two slow systems having mutually different time constants, each of which outputs a pulseheight value. A noise determination part of the digital ASIC determines whether a relevant detection signal is an intended ?-ray detection signal or a noise based on a correlation between the pulseheight values, and a noise counting part counts the number of times of noise determination, and a detector output signal processing control part controls the signal processing with respect to an output signal from a relevant detector based on the count.

Abstract: A technique is provided for forming a multi-layer radiation detector. The technique includes a charge-integrating photodetector layer provided in conjunction with a photon-counting photodetector layer. In one embodiment, a plurality of photon-counting photosensor elements are disposed adjacent to a plurality of charge-integrating photosensor elements of the respective layers. Both sets of elements are connected to readout circuitry and a data acquisition system. The detector arrangement may be used for energy discriminating computed tomography imaging and similar computed tomography systems.

Abstract: A gamma ray detector for detecting a gamma ray emitted from a target of measurement includes: an organic scintillator for detecting Compton electrons resulting from a gamma ray emitted from the target of measurement; an inorganic scintillator for detecting a Compton gamma ray; and photodetector modules for detecting light generation in the corresponding scintillators. Light generation signals from the organic and inorganic scintillators are synchronously measured, and a detection window of a gamma ray is generated. Thus, an inexpensive radiation diagnostic device of an ultra-high S/N ratio and low cost is provided.

Abstract: A detector ring of radiation tomography apparatus according to this invention has a fracture portion having no scintillation counter crystal arranged therein. Moreover, the radiation tomography apparatus according to this invention includes a correlated data complementation section. The correlated data complementation section forms correlated data when assuming that a first scintillation counter crystal actually provided in the detector ring is in the fracture portion, and additionally stores it to a correlated data storing section, thereby complementing correlated data in the fracture portion. As noted above, the correlated data complementation section obtains positional information under assumption that the scintillation counter crystals are in the fracture portion and a corresponding number of coincident events. Consequently, this invention may realize acquisition of faithful detecting efficiencies in the scintillation counter crystals.

Abstract: Methods and systems for performing a patient scan using a three-dimensional (3D) cylindrical Positron Emission Tomography (PET) imaging system are provided. The method includes acquiring a count-rate profile of a brain, repositioning at least one of a detector and the brain based on the count-rate profile and a detector sensitivity profile, and scanning the brain when the acquired count-rate profile substantially matches the detector sensitivity profile.

Abstract: A representative positron emission tomography (PET) calibration system includes a PET scanner having a ring detector, a phantom that is placed at approximately the center of the ring detector, and a time alignment calibration manager that is coupled to the PET scanner. The time alignment calibration manager detects coincidence events from the phantom, calculates position of time of flight events from the ring detector based on the detected coincidence events, and calculates time offsets for the ring detector using a mean value calculation based on the calculated position of the time of flight events.

Abstract: A nuclear medicine (NM) imaging system having attenuation correction (AC) has a stationary portion having an NM rotating portion mounted thereon. At least one gamma camera head is mounted on the NM rotating portion proximate a front side of the stationary portion and is configured to be rotated about a central axis. An AC portion comprises an X-ray source mounted on an AC rotating portion. The AC rotating portion is mounted proximate a back side of the stationary portion. The X-ray source is configured to be rotated about the central axis.

Abstract: A non-invasive imaging system, comprising: an imaging scanner; a signal processing system in communication with the imaging scanner to receive an imaging signal from the imaging scanner; and a data storage unit in communication with the signal processing system, wherein the data storage unit stores template data corresponding to a tissue region of a subject under observation, wherein the signal processing system is adapted to compute, using the imaging signal and the template data, refined template data corresponding to the tissue region, and wherein the refined template data incorporates subpopulation variability information associated with the tissue region such that the signal processing system provides an image of the tissue region in which a substructure is automatically identified taking into account the subpopulation variability information.

Abstract: A radiation detector (10, 10?) includes scintillator pixels (30) that each have a radiation-receiving end, a light-output end, and reflective sides extending therebetween. The reflective sides have a reflection characteristic (40, 40?, 41, 44) varying between the radiation-receiving end and the light-output end such that a lateral spread of light emanating from the light-output ends of the scintillator pixels responsive to a scintillation event generated in one of the scintillator pixels depends upon a depth of the scintillation event in the scintillator pixel. A plurality of light detectors (46) optically communicate with the light-output ends of the scintillator pixels to receive light produced by scintillation events.

Abstract: A method for optimizing the scanning trajectory of a radiation detector device, e.g., a SPECT scanning device, about an object generally includes: obtaining object image data using a different imaging modality, e.g., a CT scanning device, determining a maximum object boundary based on the image data, calculating an optimal scan trajectory of the SPECT scanning device relative to the object based on the maximum object boundary, scanning the object with the SPECT scanning device along the optimal scan trajectory to detect gamma photons emanating from the object, from which an image can be reconstructed from the detected gamma photons. Preferably, the SPECT device includes at least two detectors arranged at a pre-selected angle relative to one another and the optimal scan trajectory minimizes the distance between the detectors and the object while maximizing the geometric efficiency of the detectors relative to the object.

Abstract: A mobile compact imaging system that combines both of the imaging system of and optical imaging into a single system which can be located in the operating room (OR) and provides faster feedback to determine if a tumor has been fully resected and if there are adequate surgical margins. While final confirmation is obtained from the pathology lab, such a device can reduce the total time necessary for the procedure and the number of iterations required to achieve satisfactory resection of a tumor with good margins.

Abstract: An imaging system includes interleaved emission detectors and transmission detectors. Emission detectors and transmission detectors can be interleaved along the axis of relative patient motion. Emission detectors and transmission detectors can be interleaved orthogonal to the axis of relative patient motion. Emission detectors can be single photon emission computed tomography detectors and the transmission detectors can be x-ray computed tomography detectors.

Abstract: A method is disclosed for producing an attenuation map for a component of an MR/PET system. In at least one embodiment, the method includes ascertaining attenuation values of the component, producing a basic map from the attenuation values, ascertaining a position of the component relative to an examination volume of the MR/PET system, and producing the attenuation map by correcting the basic map using the ascertained position. This enables the actual position of the components to be taken into account in the attenuation correction.

Abstract: When performing a static image reconstruction of acquired single photon emission computed tomography (SPECT) data for myocardium, dynamic tracer uptake, redistribution, and washout information is generated with reduced or eliminated artifacts by back-projecting ray projections onto a reconstructed myocardial surface. A complete SPECT scan is performed after tracer injection, and a static image of the myocardial surface is reconstructed. The reconstructed image is segmented and a polar plot of it is generated. A contemporaneously acquired subset of the SPECT projection data is then back-projected onto the segmented surface of the polar plot. Contributions from emissions not originating from the myocardium (e.g., from adjacent anatomical structures) are compensated. The resultant image data, which describes tracer distributions across heart segments per projection time, are overlaid on the polar plot and presented to a user.

Abstract: A method of processing positron emission tomography (PET) information obtained from a PET detector having a plurality of detector regions, each detector region having at least one detector module and a corresponding regional collector, the method including the steps of receiving PET event information for a single PET event, the PET event information including energy information and crystal position information of the single PET event; receiving non-detector event information; generating an event list that includes (1) a PET event entry, the PET event entry including a fine time stamp, the energy information, and the crystal position information, and (2) a non-detector event entry that includes the received non-detector event information; and transmitting the generated event list to a computer for off-line processing.

Abstract: A method and apparatus are disclosed for high-sensitivity Single-Photon Emission Computed Tomography (SPECT), and Positron Emission Tomography (PET). The apparatus includes a two-dimensional (2D) gamma detector array that, unlike a conventional SPECT machine, moves to different positions in a three-dimensional (3D) volume space near an emission source and records a data vector g which is a measure of gamma emission field. In particular, the 3D volume space in which emission data g is measured extends substantially along a radial direction r pointing away from the emission source, and unlike a conventional SPECT machine, each photon detector element in the 2D gamma detector array is provided with a very large collimator aperture. Data g is related to the 3D spatial density distribution f of the emission source, noise vector n, and a system matrix H of the SPECT/PET apparatus through the linear system of equations g=Hf+n. This equation is solved for f by a method that reduces the effect of noise.

Abstract: An emission tomography detector module and an emission tomography scanner are disclosed. In at least one embodiment, the emission tomography detector modules includes a scintillator to capture an photon, the scintillator emitting a scintillating light on capturing the photon; a first type of solid-state photodetector to detect the scintillating light; and a second type of solid-state photodetector to detect the scintillating light, wherein the first type of solid-state photodetector and the second type of solid-state photodetector are different with respect to a detecting property.

Abstract: A method is disclosed for determination of positron-emission measurement information about a body area which is affected by at least one periodic movement process of an examination object during the course of positron-emission tomography. In at least one embodiment, the method includes a positron-emission measurement being carried out in the body area to be examined of the examination object in order to determine functional positron-emission measurement information, and recording, at the same time as the positron-emission measurement, anatomical measurement information about the body area to be examined is recorded, restricted to one recording plane, for at least one measurement time period, using an anatomical imaging method with high time resolution, in particular using a computed-tomography method.

Abstract: A positron emission tomography detector module that includes an array of optically isolated crystal elements and photomultiplier tubes that receive light emitted from the array of crystal elements and are arranged to cover the array of crystal elements. The photomultiplier tubes include photomultiplier tubes having two different sizes arranged in various patterns that minimize the number of edges. The axial extent of each detector module is at least three times longer than the other axis of the detector module.

Abstract: The method calculates a scatter estimate for scatter correction of detection data from a subject in a positron emission tomographic scanner. The detection data represent the scattered events and unscattered events of annihilation photons emitted in the subject. The method uses the following steps: determine an estimate of the unscattered events; determine a simulation of the scattered events; determine a value of a scaling factor by fitting the sum of the estimate of the unscattered events and a product of the scaling factor and the simulation of the scattered events to the detection data; and determine the scatter estimate as the product of the scaling factor having said value and the simulation of the scattered events.

Abstract: In a time-of-flight positron emission tomography (TOF-PET) imaging method, three-dimensional time-of-flight line-of-response (TOF-LOR) data are acquired. Each TOF-LOR corresponds to a line-of-response with time-of-flight spatial localization. The TOF-LOR data are slice-binned into a plurality of two-dimensional TOF-LOR data sets based on the time-of-flight spatial localization. At least some of the slice-binned TOF-LOR data correspond to lines of response that are oblique to the two-dimensional data sets. The TOF-LOR data are coarsely angularly rebinned to a plurality of coarse angular bins each having an angular span of at least about 10°. The coarsely angularly binned TOF-LOR data are reconstructed to produce the image slice.

Abstract: The design of a compact, handheld, solid-state and high-sensitivity imaging probe and a micro imager system is reported. These instruments can be used as a dedicated tool for detecting and locating sentinel lymph nodes and also for detecting and imaging radioactive material. The reported device will use solid state pixel detectors and custom low-noise frontend/readout integrated circuits. The detector will be designed to have excellent image quality and high spatial resolution. The imaging probes have two different embodiments, which are comprised of a pixelated detector array and a highly integrated readout system, which uses a custom multi-channel mixed signal integrated circuit. The instrument usually includes a collimator in front of the detector array so that the incident photons can be imaged. The data is transferred to an intelligent display system. A hyperspectral image can also be produced and displayed. These devices are designed to be portable for easy use.

Abstract: A brain imager includes a compact ring-like static PET imager mounted in a helmet-like structure. When attached to a patient's head, the helmet-like brain imager maintains the relative head-to-imager geometry fixed through the whole imaging procedure. The brain imaging helmet contains radiation sensors and minimal front-end electronics. A flexible mechanical suspension/harness system supports the weight of the helmet thereby allowing for patient to have limited movements of the head during imaging scans. The compact ring-like PET imager enables very high resolution imaging of neurological brain functions, cancer, and effects of trauma using a rather simple mobile scanner with limited space needs for use and storage.

Abstract: The invention described herein provides systems and methods for multi-modal imaging with light and a second form of imaging. Light imaging involves the capture of low intensity light from a light-emitting object. A camera obtains a two-dimensional spatial distribution of the light emitted from the surface of the subject. Software operated by a computer in communication with the camera may then convert two-dimensional spatial distribution data from one or more images into a three-dimensional spatial representation. The second imaging mode may include any imaging technique that compliments light imaging. Examples include magnetic resonance imaging (MRI) and computer topography (CT). An object handling system moves the object to be imaged between the light imaging system and the second imaging system, and is configured to interface with each system.

Abstract: An electronic storage medium that comprises at least one radiopharmaceutical identity, SPECT measured values of at least one radiopharmaceutical kinetic parameter of a flow rate across a tissue membrane, for the radiopharmaceutical, and a set of instructions for associating the at least one radiopharmaceutical kinetic parameter with a disease signature.

Abstract: An apparatus for imaging an eye includes a housing and a system of optical components disposed in the housing. The apparatus is capable of operating in a line scanning laser opthalmoscope (LSLO) mode and an optical coherence tomography (OCT) mode. The system of optical components can include a first source to provide a first beam of light for the OCT mode and a second source to provide a second beam of light for the LSLO mode. In the OCT mode, a first optic is used that (i) scans, using a first surface of the first optic, the first beam of light along a retina of an eye in a first dimension, and (ii) descans, using the first surface, a first light returning from the eye in the first dimension to a detection system in the OCT mode. In the LSLO mode, the first optic is used where the second beam of light passes through a second surface of the first optic.

Abstract: A light receiver for detecting incident time is installed on the side of a radiation source of a scintillator (including a Cherenkov radiation emitter), and information (energy, incident time, an incident position, etc.) on radiation made incident into the scintillator is obtained by the output of the light receiver. It is, thereby, possible to identify an incident position and others of radiation into the scintillator at high accuracy.

Abstract: A multi-stage process utilizing one or more radiation sensors on a distributed network for the detection and identification of radiation, explosives, and special materials within a shipping container. The sensors are configured as nodes on the network. The system collects radiation data from one or more nodes. The collected radiation data is dynamically adjusted according to at least one of a plurality of background radiation data based on a determined background environment about the container. The collected and adjusted radiation data is compared to one or more stored spectral images representing one or more isotopes to identify one or more isotopes present. The identified one or more isotopes present are corresponded to possible materials or goods that they represent.

Abstract: A positron emission tomography imaging system includes a detector configured to detect radiation emitted from an object to be examined, a data processing apparatus configured to reconstruct image data for positron emission tomography diagnosis from detection data obtained from the detector, and a user terminal configured to obtain desired information by operating the detector and the data processing apparatus. The detector includes a detection part having a plurality of detection elements configured to detect the radiation emitted from the object to be examined for every designated timing, and an event information file generation part configured to generate an event information file based on the detection data obtained from the detector.

Abstract: When generating a 3D image of a subject or patient, a cone beam X-ray source (20a, 20b) is mounted to a rotatable gantry (14) opposite an offset flat panel X-ray detector (22a, 22b). A wedge-shaped attenuation filter (24a, 24b) of suitable material (e.g., aluminum or the like) is adjustably positioned in the cone beam to selectively attenuate the beam as a function of the shape, size, and density of a volume of interest (18) through which X-rays pass in order to maintain X-ray intensity or gain at a relatively constant level within a range of acceptable levels.

Abstract: Methods of nuclear imaging can include, in a pre-scan, detecting radiation emitted from a patient in a first plurality of viewing angles including at least a first viewing angle and a second viewing angle, generating nuclear data from the detected radiation, reconstructing a first nuclear event distribution from the nuclear data, selecting a region of interest, determining a first signal-to-noise ratio of the first nuclear event distribution within the region of interest, selecting a second plurality of viewing angles not including the first viewing angle, reconstructing a second nuclear event distribution from the nuclear data associated with the second plurality of viewing angles, determining a second signal-to-noise ratio of the second nuclear event distribution within the region of interest, determining that the second signal-to-noise ratio is greater than or equal to the first signal-to-noise ratio, and nuclear imaging the patient by detecting nuclear data based on a nuclear imaging process that is based

Abstract: A method and apparatus are provided for reconstructing 3D image. The method may include the steps of: detecting a plurality of line of response (LORs) emitted from an object; transforming the plurality of LORs into first sinogram data; back-projecting the first sinogram data with a plurality of projection angles to produce image data for the object; and projecting the produced image data with the plurality of projection angles to transform the image data into second sinogram data. The back-projecting may include filling pixels of image plane for each of the plurality of projection angles with the first sinogram data and rotating a coordinate axis of the image plane with corresponding projection angle to produce the image data. The projecting may include rotating the image data with corresponding projection angle in an opposite direction before projecting the image data with the plurality of projection angles. The projecting and the back-projecting may use symmetry properties in coordinate space.