METHOD AND SYSTEM FOR ESTIMATING DENSITY OF COMPOSITE MATERIALS

BACKGROUND

[0001] Embodiments of the present disclosure relate generally to diagnostic imaging, and more particularly to methods and systems for estimating densities of individual constituents of composite materials.

[0002] Non-invasive imaging techniques are widely used for diagnostic imaging in security screening, quality control, and medical imaging systems. Particularly, in medical diagnostic imaging, non-invasive imaging techniques such as computed tomography (CT) and magnetic resonance imaging (MRI) may be used for unobtrusive imaging of underlying tissues and organs.

[0003] Conventionally, MRI provides greater soft-tissue contrast than CT imaging. However, due to ubiquity of the CT systems, faster examination times and superior patient tolerance, CT is more widely used in medical diagnostic procedures. Particularly, CT imaging may be employed for providing quantitative measurements corresponding to individual constituents of a composite biological material. The composite biological material, for example, may include tissues, grey matter, cerebrospinal fluid (CSF), and vascular structures, which in turn, may further include fat, water, iodine, and/or protein as constituents.

[0004] Generally, the quantitative measurements corresponding to the individual constituents provide useful indications that may assist in clinical diagnosis and treatment planning. For example, a quantitative measurement of density of fat in the grey matter may aid in identifying presence and/or severity of infarction in cerebral tissues of a patient. Particularly, directly measuring density of fat in cerebral tissues such as the grey matter or the CSF using CT imaging may obviate a need for biopsies that entail puncturing meninges of the patient. Similarly, a measurement of density of fat in plaque present in arteries may provide indicators that quantify a risk posed by the plaque to the health of the patient.

[0005] Single energy CT imaging, however, may not provide quantitative measurements for individual constituents because single energy CT imaging may only allow density measurements for the composite material as a whole. Accordingly, certain medical procedures employ dual energy CT imaging that may aid in material differentiation of the composite material. To that end, dual energy CT imaging entails acquisition of projection data using X-rays generated at two different energy levels. Varying response of the composite material to X-rays at the two energy levels may provide indications that may be useful in material differentiation and determining elemental composition of the composite material.

[0006] To that end, conventional dual energy CT imaging systems are known to employ material decomposition (MD) procedures that project projection data acquired at the two different energy levels onto two sets of basis functions. Typically, the basis functions include either physical components of an X-ray interaction with matter such as photoelectric effect and Compton scattering, or attenuation coefficients of the individual constituents, such as water and iodine. Alternatively, a Multi-Material Decomposition (MMD) procedure that assumes fixed density and/or known priors may be used for decomposing the composite material into multiple constituents.

[0007] However, quantitative measurement of medically relevant parameters, such as density, corresponding to the individual constituents using conventional techniques is an over parameterized problem, which may not result in an accurate solution. Particularly, for the individual constituents that exhibit spectrally close responses to the X-ray incidence, quantitative measurements using conventional techniques may often be inaccurate. Moreover, these quantitative measurements may be further convoluted by presence of noise in the dual energy CT system, and thus, may not be reliable for use in clinical diagnosis and treatment.

BRIEF DESCRIPTION

[0008] In accordance with certain aspects of the present disclosure, a method for imaging is disclosed. Projection data corresponding to a target using X-rays generated at a plurality of energy levels may be acquired. At least two individual constituents corresponding to the target may be identified. A set of voxels including the at least two individual constituents in at least a pair of basis material decomposition images reconstructed using the acquired projection data may be identified. A prior distribution of densities corresponding to the at least two individual constituents in the identified set of voxels may be determined. At least an upper bound on variation of volume fractions corresponding to the at least two individual constituents in the identified set of voxels may be ascertained. A posterior distribution of the densities corresponding to the at least two individual constituents may then be computed based on the determined prior distribution of densities and/or the ascertained upper bound on variation of volume fractions. Densities corresponding to the at least two individual constituents may be estimated based on the computed posterior distribution.

[0009] In accordance with certain other aspects of the present disclosure, an imaging system is presented. The imaging system includes a data acquisition system configured to acquire projection data corresponding to a target using X-rays generated at a plurality of energy levels. Further, the system includes an image processing unit operationally coupled to the data acquisition system. The image processing unit is configured to identify at least two individual constituents corresponding to the target. Additionally, the image processing unit is configured to identify a set of voxels including the at least two individual constituents in at least a pair of basis material decomposition images reconstructed using the acquired projection data. Further, the image processing unit is configured to determine a prior distribution of densities corresponding to the at least two individual constituents in the identified set of voxels. Moreover, the image processing unit is configured to ascertain at least an upper bound on variation of volume fractions corresponding to the at least two individual constituents in the identified set of voxels. The image processing unit is configured to compute a posterior distribution of the densities corresponding to the at least two individual constituents based on the determined prior distribution of densities and/or the ascertained upper bound on variation of volume fractions. The image processing unit is configured to estimate densities corresponding to the at least two individual constituents based on the computed posterior distribution.

[0010] In accordance with certain other aspects of the present disclosure, a computed tomography system is presented. The computed tomography system includes least one radiation source configured to generate X-rays at a plurality of energy levels. Further, the computed tomography system includes an image processing unit operationally coupled to the detector assembly. The image processing unit is configured to identify at least two individual constituents corresponding to the target. Additionally, the image processing unit is configured to identify a set of voxels corresponding to the at least two individual constituents in at least a pair of basis material decomposition images reconstructed using the acquired projection data. Further, the image processing unit is configured to determine a prior distribution of densities corresponding to the at least two individual constituents in the identified set of voxels. Moreover, the image processing unit is configured to ascertain at least an upper bound on variation of volume fractions corresponding to the at least two individual constituents in the identified set of voxels. The image processing unit is also configured to compute a posterior distribution of the densities corresponding to the at least two individual constituents based on the determined prior distribution of densities and/or the ascertained upper bound on variation of volume fractions. The image processing unit may be configured to estimate densities corresponding to the at least two individual constituents based on the computed posterior distribution.

DRAWINGS

[0011] These and other features, aspects, and advantages of the present disclosure will become better understood when the following detailed description is read with reference to the accompanying drawings in which like characters represent like parts throughout the drawings, wherein:

[0012] FIG. 1 is a schematic representation of an exemplary imaging system, in accordance with aspects of the present disclosure;

[0013] FIG. 2 is another block schematic diagram of the exemplary imaging system of FIG. 1, in accordance with aspects of the present disclosure;

[0014] FIG. 3 is a flow diagram illustrating an exemplary method for imaging, in accordance with aspects of the present disclosure;

[0015] FIG. 4 is a diagrammatical representation depicting an exemplary image corresponding to adipose tissue in the neck of a patient, in accordance with aspects of the present disclosure;

[0016] FIG. 5 is a graphical representation depicting exemplary values of individual densities of fat and blood estimated using the method described with reference to FIG. 3, in accordance with aspects of the present disclosure; and

[0017] FIG. 6 is a graphical representation depicting exemplary volume fractions corresponding to a small neighborhood of voxels determined using the method described with reference to FIG. 3, in accordance with aspects of the present disclosure.

DETAILED DESCRIPTION

[0018] The following description presents methods and systems for determining quantitative measurements corresponding to individual constituents of a composite material with greater accuracy. Particularly, embodiments of the present disclosure provide methods and systems for estimating density of the individual constituents of a binary composite material that exhibit spectrally close response to X-ray interactions.

[0019] Although the following description describes embodiments for estimating the density of the individual constituents of the composite material in the context of medical diagnostic imaging, the present disclosure may be implemented in various other imaging systems and applications. Some of these systems may include an X-ray system, a single or multiple source imaging system, a single or multiple detector imaging system, a photon counting and/or energy discriminating detector imaging system. Some other such systems may include a positron emission tomography (PET) scanner, a PET-CT scanner, a single photon emission computed tomography (SPECT) scanner, a SPECT-CT scanner, an X-ray tomosynthesis system, an MR-CT scanner and/or any imaging system that uses an X-ray tube.

[0020] Further, in addition to medical diagnostic imaging, embodiments of the present disclosure may be employed in other non-invasive imaging contexts, such as baggage screening, package screening and/or industrial nondestructive evaluation of manufactured parts. An exemplary environment that is suitable for practising various implementations of the present disclosure will be discussed in the following sections with reference to FIGs. 1 and 2.

[0021] FIG. 1 illustrates an exemplary CT system 100 configured to provide quantitative measurements corresponding to individual constituents of a composite material such as biological tissues. The composite material, for example, may also include non-biological materials such as inanimate objects, manufactured parts and/or foreign objects such as dental implants, stents, and/or contrast agents present within the body. To that end, in one embodiment, the CT system 100 may include a gantry 102. The gantry 102 may further include at least one X-ray radiation source 104 that may be configured to project a beam of X-ray radiation 106 towards a detector array 108 positioned on the opposite side of the gantry 102. In certain embodiments, one or more radiation sources may be employed to project a plurality of X-rays 106 for acquiring projection data at different energy levels.

[0022] In certain embodiments, the CT system 100 may include an image processing unit 110 configured to generate separate images of different basis materials such as bone and water using projection data acquired in response to X- rays 106 generated at the different energy levels. Particularly, in one embodiment, the image processing unit 110 may be configured to generate images that allow clear differentiation between individual constituents of a composite material that exhibit spectrally close responses to the X-rays 106.

[0023] As used herein, the term "composite material" may correspond to a material that includes two or more individual constituents. Further, the term "spectrally close responses" and/or "spectrally similar responses" refers to corresponding spectral responses of two or more individual constituents to the incident X-rays 106 that have substantially similar X-ray mass attenuation coefficients, where the X-rays 106 are of substantially similar wavelengths. By way of example, in one embodiment, a difference between mass attenuation coefficient of the spectrally similar individual constituents at the plurality of energy levels may be lesser than noise in the CT system 100.

[0024] In certain embodiments, the CT system 100 may be configured to provide quantitative measurements corresponding to the individual constituents, thus identified, for use in diagnosing and/or prescribing treatment for a patient 112. By way of example, the CT system 100 may be configured to discriminate between fat and soft tissues in cerebral tissues of interest. Additionally, the CT system 100 may also be configured to provide quantitative measurements of certain medically relevant parameters corresponding to fat and soft tissues in the cerebral tissues. The quantitative measurements, for example, corresponding to density and volume fractions of fat and soft tissues in the cerebral tissues may allow identification of presence and/or severity of cerebral infarction. The quantitative measurements, thus, may aid a clinician in diagnosing and/or prescribing appropriate treatment for the patient 112. Another exemplary embodiment of an imaging system configured to discriminate between, and provide quantitative measurements of the medically relevant parameters corresponding to the individual constituents of a composite material will be described in greater detail with reference to FIG. 2.

[0025] FIG. 2 illustrates an imaging system 200 that is similar to the CT system 100 of FIG. 1. In accordance with aspects of the present disclosure, the system 200 may be configured to provide quantitative measurements corresponding to individual constituents of a composite material with greater accuracy. For discussion purposes, embodiments of the system 200 are described with reference to a dual energy imaging system configured to acquire projection data at two different energy or peak kilo-voltage (kVp) levels. Particularly, embodiments of the system 200 are described with reference to the dual energy imaging system configured to estimate densities of individual constituents that exhibit spectrally close responses to the X-rays 106 (see FIG. 1).

[0026] Although embodiments of the present disclosure are described with reference to the dual energy imaging system, in certain embodiments, the system 200 may be configured to operate at more than two energy levels for estimating the densities of the individual constituents. Additionally, the system 200 may also be configured to estimate quantitative measurements of other parameters corresponding to the individual constituents, such as volume fractions corresponding to the individual constituents and/or one or more estimates corresponding to local noise variance, with greater accuracy.

[0027] To that end, in certain embodiments, the system 200 may include a radiation source, such as the radiation source 104 of FIG. 1, configured to project the X-rays 106 having a first energy level and a second energy level for imaging a region of interest (ROI) of a subject 204. The subject 204, for example, may include the patient 112 of FIG. 1, a machine, or baggage that correspond to a composite material. Generally, well-separated frequencies may be selected as the first and second energy levels to prevent possible overlap between spectral responses of individual constituents of the composite material. Accordingly, in an exemplary implementation, the first energy level may correspond to 140 kVp and the second energy level may correspond to 80 kVp.

[0028] In one embodiment, the system 200 may implement dual energy imaging by sequentially scanning the ROI at the first and second energy levels. Alternatively, the system 200 may employ two radiation sources mounted orthogonal to each other on the gantry 102 to project the X-rays 106 at the first and second energy levels. In certain embodiments, the system 200 may be configured to rapidly switch voltage of the X-rays 106 within a single scan to allow for dual energy imaging. In certain other embodiments, the system 200 may employ energy discriminating (ED) detectors, layered detectors and/or photon counting (PC) detectors for dual energy imaging.

[0029] Accordingly, in one embodiment, the detector array 108 may include a plurality of arrayed, ED and/or PC detector elements 202. The detector elements 202 may be configured to acquire the projection data at the first energy level and the second energy level simultaneously or in rapid succession. Use of the ED and/or PC detector elements 202 may enable the system 200 to rapidly acquire projection data via ED and/or PC modes. The rapid acquisition of the projection data may allow the system 200 to swiftly reconstruct images with a large energy separation. In one example, the large energy separation may correspond to about 90 kilo electron Volt (keV). Generation of the images with a large energy separation, in turn, may simplify estimation of medically relevant parameters corresponding to the individual constituents of the composite material.

[0030] In certain embodiments, the system 200 may be configured to traverse different angular positions around the subject 204 for generating desired images of the ROI. To that end, the gantry 102 and the components mounted thereon may be configured to rotate about a center of rotation 206 for acquiring projection data at the different energy levels. Alternatively, in embodiments where a projection angle relative to the subject 204 varies as a function of time, the mounted components may be configured to move along a general curve rather than along a segment of a circle.

[0031] Accordingly, in one embodiment, the rotation of the gantry 102 and the operation of the X-ray radiation source 104 may be controlled by a control mechanism 208 in the system 200. In one embodiment, the control mechanism 208 may include an X-ray controller 210 configured to provide power and timing signals to the radiation source 104. For example, the X-ray controller 210 may be configured to provide control signals for projecting the X-rays 106 having a first energy level and a second energy level simultaneously or in rapid succession. Additionally, the control mechanism 208 may also include a gantry motor controller 212 configured to control a rotational speed and/or position of the gantry 102 based on imaging requirements.

[0032] Moreover, in certain embodiments, the control mechanism 208 may further include a data acquisition system (DAS) 214. In one embodiment, the DAS 214 may be configured to sample analog data received from the detector elements 202 and convert the analog data to digital signals for subsequent processing. The data sampled and digitized by the DAS 214 may be transmitted to a computing device 216. The computing device 216 may store the data in a storage device 218. The storage device 218, for example, may include a hard disk drive, a floppy disk drive, a compact disk-read/write (CD-R/W) drive, a Digital Versatile Disc (DVD) drive, a flash drive, and/or a solid-state storage device.

[0033] Additionally, the computing device 216 may be configured to provide appropriate commands and parameters to one or more of the DAS 214, the X-ray controller 210 and the gantry motor controller 212 for controlling system operations such as data acquisition and/or processing. In certain embodiments, the computing device 216 may be configured to control system operations based on operator input. To that end, the computing device 216 may be operatively coupled to a console 220, which may include a keyboard (not shown) or a touchscreen to allow an operator to specify commands and/or scanning parameters. Additionally, the system 200 may also include a display 222 that may be configured to allow the operator to observe object images and/or specify commands and scanning parameters for performing a desired imaging task.

[0034] In one embodiment, for example, the computing device 216 may be configured to use the operator-specified and/or system defined commands and parameters to operate a table motor controller 224. The table motor controller 224, in turn, may be configured to control a motorized table 226. Particularly, the table motor controller 224 may be configured to move the table 226 for appropriately positioning the subject 204 in the gantry 102. Additionally, the table motor controller 224 may be configured to position the subject 204 so as to allow the detector elements 202 to acquire corresponding projection data from the ROI at the different energy levels.

[0035] As previously noted, the DAS 214 may be configured to sample an digitize the projection data acquired by the detector elements 202. Subsequently, an image reconstructor 228 may be configured to reconstruct basis pair material images corresponding to the individual constituents using the sampled and digitized X-ray data. As used herein, the term "basis pair material images" may also be referred to as basis materially decomposed (MD) images. Typically, different basis materials corresponding to the subject 204 have different absorption characteristics for low energy X-rays and high energy X-rays. Accordingly, the image reconstructor 228 may be configured to distinguish between an absorption of X-rays caused by a first basis material, for example fat, and an absorption caused by a second basis material, for example water, based on the received data. Subsequently, the image reconstructor 228 may be configured to decompose the projection data to generate fat and water basis pair MD images.

[0036] Alternatively, the image reconstructor 228 may be configured to generate monochromatic images at desired energy levels. Use of monochromatic images may reduce beam hardening effects in the generated images, thus improving the clinician's evaluation of pathology of the ROI. The image reconstructor 228 may be configured to either store the MD and/or monochromatic images in the storage device 218. Additionally, the image reconstructor 228 may also be configured or transmit the MD and/or monochromatic images to the computing device 216 for generating medically useful information.

[0037] By way of example, the computing device 216 may be configured to generate density maps from the MD and/or monochromatic images corresponding to the individual constituents for assessing structural characteristics of the ROI. In certain embodiments, the computing device 216 may also be configured to transmit the MD and/or monochromatic images, the density maps, and the medically useful information to the display 222. The display 222 may be configured to allow the operator and/or clinician to evaluate the imaged ROI and provide a pathological assessment of the ROI based on the medically useful information.

[0038] It may be noted that conventional MD techniques allow for estimation of effective density peff of a composite material that corresponds to a product of volume fraction a with true density p of the composite material. Conventional MD techniques, however, fail to provide any measurements corresponding to densities p, and p2, and volume fraction a corresponding to individual constituents, for example, of a binary composite material. Moreover, the conventional MD techniques are often afflicted with system noise, and thus, may not allow clear distinction between spectrally close constituents in the composite material.

[0039] Unlike conventional MD techniques that estimate effective density pejf, embodiments of the present disclosure, allow for estimation of densities p] and p2, and volume fraction a corresponding to the individual constituents, where peff = apx + (1 - a)p2 (1)

[0040] Subsequently, the operator and/or the clinician may use the estimated densities and volume fractions to discriminate between spectrally close individual constituents, for example, using Bayesian estimation. To that end, in one embodiment, the operator may use the operator console 220 to identify the individual constituents of the composite material of interest in the ROI from the MD and/or monochromatic images. Alternatively, a dedicated image processing unit 230, similar to the image processing unit 110 of FIG. 1, may be configured to identify the individual constituents based on a priori information, which correlates the composite material corresponding to the ROI with known constituents. It may be noted that in one embodiment, the a priori information may include information that is stored in an electronic library. Accordingly, in one embodiment, the image processing unit 230 may be operatively coupled to the operator console 220, the computing device 216, the storage device 218, and/or the display 222 through a communication link 232.

[0041] In certain further embodiments, the image processing unit 230 may be configured to employ spectral filtering to automatically identify a set of voxels that includes the ROI from the MD and/or monochromatic images. Further, the image processing unit 230 may be configured to use the identified set of voxels for estimating the densities of the individual constituents. To that end, the image processing unit 230 may be configured to obtain prior distribution values corresponding to each individual constituent in each voxel in the identified set of voxels. In one embodiment, for example, the image processing unit 230 may be configured to obtain the prior distribution values corresponding to densities of each individual constituent of the composite material. Furthermore, in one embodiment, the prior distribution may correspond to a Gaussian distribution with a corresponding mean and standard deviation. Moreover, the prior distribution may be obtained from previously determined information that may be available in the storage device 218. The previously determined information, in turn, may be obtained, for example, from published literature and/or from the operator.

[0042] Generally, in biological tissues, densities of the individual constituents may be constant within a small neighborhood of voxels. In one embodiment, the small neighborhood may include a group of about 26 voxels corresponding to an image slice in an image. Accordingly, aspects of the present disclosure allow for an assumption of constancy in the densities of the individual constituents within the small neighborhood. Volume fractions corresponding to the individual constituents in the small neighborhood, however, may exhibit slight variations.

[0043] Accuracy of the MD and/or monochromatic image derived information for medical use may depend upon accurate quantitative measurements of densities and volume fractions corresponding to the individual constituents. The quantitative measurements, however, may also suffer from noise inherently present in the system 200. The present disclosure allows for accurate estimation of the noise in the quantitative measurements at a particular voxel by applying a constraint on variation of the volume fractions within the small neighborhood.

[0044] To that end, in one embodiment, the image processing unit 230 may be configured to determine an upper bound on variation of volume fractions using previously known information and/or operator input. Further, the image processing unit 230 may be configured to compute a posterior distribution of densities, for example, using a Markov Chain Model to provide a determinate system of equations for density estimation. In accordance with aspects of the present disclosure, the image processing unit 230 may be configured to employ a Gibb's sampler. Particularly, the image processing unit 230 may customize the Gibb's sampler to mitigate wide variations in volume fractions in the target ROI while simulating the Markov chain. The customized Gibb's sampler, in turn, may aid in computation of the density estimates for each individual constituent at every voxel. .

[0045] Particularly, in one embodiment, the image processing unit 230 may be configured to adapt the Gibb's sampler such that a stationary distribution of the Markov chain provides an accurate estimate of the posterior distribution of the densities corresponding to the individual constituents. Accuracy of the density estimates may be further enhanced by mitigating ill effects of noise on the density estimates by computing a probability distribution of volume fractions corresponding to the individual constituents in the identified set of voxels. Although the present embodiment describes use of the Gibb's sampler, alternative embodiments may employ other techniques, such as, the Metropolis Hasting procedure for computing density estimates for each individual constituent.

[0046] Furthermore, as previously noted, the densities corresponding to the individual constituents, particularly spectrally close materials such as fat and water, may provide useful indications regarding the pathology of the target ROI. An exemplary method for providing accurate quantitative measurements of the individual constituents of a composite material for use in medical procedures will be described in greater detail with reference to FIG. 3.

[0047] FIG. 3 illustrates a flow chart 300 depicting an exemplary imaging method for estimating effective densities of individual constituents of a composite material in a target ROI. The exemplary method may be described in a general context of computer executable instructions on a computing system or a processor. Generally, computer executable instructions may include routines, programs, objects, components, data structures, procedures, modules, functions, and the like that perform particular functions or implement particular abstract data types. The exemplary method may also be practised in a distributed computing environment where optimization functions are performed by remote processing devices that are linked through a communication network. In the distributed computing environment, the computer executable instructions may be located in both local and remote computer storage media, including memory storage devices.

[0048] Further, in FIG. 3, the exemplary method is illustrated as a collection of blocks in a logical flow chart, which represents operations that may be implemented in hardware, software, or combinations thereof. The various operations are depicted in the blocks to illustrate the functions that are performed generally, for example, during data acquisition, determining prior and posterior distribution of quantitative measurements and density estimation phases of the exemplary method. In the context of software, the blocks represent computer instructions that, when executed by one or more processing subsystems, perform the recited operations of an imaging system such as the imaging system 200 of FIG. 2.

[0049] The order in which the exemplary method is described is not intended to be construed as a limitation, and any number of the described blocks may be combined in any order to implement the exemplary method disclosed herein, or an equivalent alternative method. Additionally, certain blocks may be deleted from the exemplary method without departing from the spirit and scope of the subject matter described herein. For discussion purposes, the exemplary method will be described with reference to the elements of FIGs. 1-2.

[0050] As previously noted, quantitative measurements of medically relevant parameters corresponding to individual constituents of a composite material aid in clinical diagnosis and treatment planning. However, quantitative measurement of characteristics, such as density, corresponding to individual constituents using conventional MD techniques may not be accurately estimated as such estimation corresponds to an over-parameterized problem. Particularly, in a noise free system, at every voxel there are two unknown density parameters and an unknown volume fraction parameter for a two constituent material mixture model. However, with dual energy CT, only two measurements may be available at any voxel. Accordingly, estimating densities and volume fractions corresponding to individual constituents of even a simple two material mixture model is over-parameterized.

[0051] Accordingly, embodiments of the present method allow for accurate estimation of densities of the individual constituents, the volume fractions, and standard deviation of voxel level measurement noise in imaging system. Particularly, the embodiments described herein allow for estimation of the densities, the volume fractions, and the standard deviation by setting a probabilistic prior distribution on a corresponding density space and the volume fractions. Particularly, embodiments of the present method allow discrimination between individual constituents that include spectrally close materials even in the presence of noise. Accurate discrimination between spectrally close materials, in turn, may aid in accurate assessment of pathology of the target ROl.

[0052] To that end, at step 302, projection data corresponding to a target may be acquired using X-rays generated at a plurality of energy levels. Particularly, in one embodiment, a data acquisition system such as the DAS 214 of FIG. 2 may be configured to acquire projection data in response to X-rays produced at a first energy level, Ei, and a second energy level, E2. The projection data may be acquired at the first and second energy levels, for example, using multiple sources and multiple detectors, multiple detector layers, energy discriminating detectors and/or photon counting detectors.

[0053] In one embodiment, the DAS may be configured to acquire a first set of projection data at a low energy level, for example, in approximately one half of a full gantry rotation plus a detector fan-angle. Further, the DAS may be configured to acquire a second set of projection data at a high energy level, for example, in approximately the other half of a full gantry rotation plus a detector fan angle. Scanning the ROI by separating data acquisition into a low energy half and a high energy half, thus, may allow for data acquisition corresponding to a nearly full 360-degree angle plus twice the fan angle without increasing scan time. In alternative embodiments, however, the system may be configured to employ certain other techniques, such as a step and shoot scan, a helical scan, a gated scan and/or alternate biasing of an X-ray tube between the high and low energy levels for acquiring the projection data for dual energy imaging.

[0054] As previously noted, the DAS may be configured to sample and digitize the projection data for use in reconstructing basis pair material images corresponding to the individual constituents of the target ROI. In one example, the target ROI may include CSF. Additionally, the individual constituents of interest and/or the basis materials may include fat and water in the CSF. In certain embodiments, an image processing unit, such as the image processing unit 230 of FIG. 2, may be configured to decompose the digitized data into first and second data sets that correspond to first and second basis materials such as protein and water, respectively.

[0055] In one embodiment, the image processing unit may be configured to process the first and the second data sets to generate fat and water basis pair MD images. Alternatively, the image processing unit may be configured to generate monochromatic images corresponding to desired energy levels. In certain embodiments, the MD and/or monochromatic images may be stored in a storage repository, such as the storage device 218 of FIG. 2.

[0056] Additionally, the MD and/or monochromatic images may be transmitted to a computing device such as the computing device 216 of FIG. 2 for generating medically useful information. The medically relevant information, for example, may include density maps corresponding to the individual constituents for use in assessing structural and/or functional characteristics of the target ROI. In one embodiment, the computing device 216 may be configured to transmit the MD and/or monochromatic images and the medically useful information to a display such as the display 222 of FIG. 2. The display may be configured to allow the operator and/or clinician to evaluate the target ROI based on the MD and/or monochromatic images and/or the medically useful information.

[0057] To that end, at step 304, at least two individual constituents corresponding to the target ROI may be identified. Although, the present embodiment describes only two individual constituents Mi and M2, this is only for clarity of description. In alternative embodiments, however, the present method may be employed to estimate density and volume fractions of more than two individual constituents corresponding to a composite material.

[0058] To that end, in one embodiment, an operator may use an interface to the system such as the operator console 220 of FIG. 2 to identify the individual constituents, for example Mi and M2, corresponding to the target ROI using the MD and/or monochromatic images. Alternatively, the image processing unit may be configured to identify the individual constituents based on stored information, such as information available in an electronic library, which correlates the target ROI with known constituents. The electronic library may store such information in a lookup table that may identify the individual constituents of the target ROI, for example including the CSF, as fat and water.

[0059] In one embodiment, mass attenuation coefficients corresponding to the identified individual constituents Mx and M2 at an energy E may be obtained from previously determined information, for example, information stored in the storage device 218 of FIG. 2. The stored coefficients may in turn be obtained from NIST (National Institute of Standards and Technology) standards or may be measured in a laboratory. The mass attenuation coefficients may be denoted by nM{E,M\) and juM(E,M2), respectively. Further, a matrix ju corresponding to per unit density linear attenuation coefficients of the two individual constituents at the two energy levels, E] and E2, may be defined, for example, using equation (2): jMu(Ei>Mi) //M(£,,M2)1 lliM{E2,Mx) /uM(E2,M2)

[0060] Further, the densities of the individual constituents, Mx and M2 may be respectively denoted by px and p2. Additionally, a volume fraction indicative of volume fractions corresponding to Mx and M2 may be denoted by ax and 1 - ax, respectively. Accordingly, quantitative measurements corresponding to linear attenuation of the individual constituents, Mx and M2 may be represented, for example, using a column vector, as indicated by equation (3):
'P\ ° 1 r a ~L \£11 ,,v
Mi.=Mxn x + Pi»°\{Y,})x L(P\,P2,a,a)x n(px,p2)x n{a) (8)

[0072] Further, a Bayes estimate of p,, denoted by p, may be defined, for example, using equation (9): p, = C({Yt})j^xL(x, p2,a,i)■ Particularly, the Gibb's sampler may be used to simulate the Markov chain whose stationary distribution may correspond to a posterior distribution represented by a(pi,p2,(T,a\{Yj}). Once the Markov chain converges to a stationary distribution, an estimate of px and corresponding standard errors SE( px) defined in equation (7) may be estimated based on a mean and a standard deviation of a sequence {p\n}tn>x\, respectively. An example of the density, px and the standard error, SE( px) corresponding to an individual constituent may be defined using equation (10) and equation (11), respectively: where m corresponds to a burn-in length required for the Markov chain to converge to a corresponding stationary distribution.
[0076] Further, the Bayesian estimate of the density px corresponding to an individual constituent and the corresponding standard errors SE( px) may be used in the Gibb's sampler for simulating the Markov chain. Certain exemplary steps for simulating the Markov chain {plw, p2n, crn, an }(n>A using the Gibb's sampler (hereinafter referred to as default Gibb's sampler) is presented in

Gibb's Sampling Procedure 1:

Gibb's Sampling Procedure 1:

Data: Yx, ,Yn

Result: px,p2,a initialization;

[0077] The Gibb's Sampling Procedure 1, thus, allows for a Bayesian estimation of densities of the individual constituents along with corresponding standard deviation. The density and the standard deviation values may be correlated with pathological characteristics to aid in medical diagnosis. An exemplary output of the Gibb's Sampling Procedure 1 for use in deriving medically relevant information may be represented using equation (12): A <-%.&<-%.*<-% 02)

[0078] However, in certain scenarios, the conditional posterior distribution of 7r{aj\{Yj},pi,p2,cr) may simulate «;'s whose variance may be much larger than that expected in a region of nearly homogenous tissue. Large variances in the simulated at 's may result in large standard deviations for the posterior distributions of px and p2 ■ Particularly, the standard deviations may be more pronounced when columns of the ju matrix of equation (4) are substantially similar due to spectrally close constituents that have similar linear attenuations at the two energies.

[0079] In one embodiment, the large variances in the volume fractions may be mitigated by assuming that the prior distribution n(at) on a;does not lie within U[0, 1]. In accordance with aspects of the present disclosure, a fourth proposition may be used that assumes the conditional posterior distribution of 7r\aj\{Yi},Pi,p2,cr) to be a truncated Gaussian distribution having mean ma and * variance an}(n>\) using the modified Gibb's sampler for density estimation is presented in Gibb's Sampling Procedure 2: Gibb's Sampling Procedure 2: Data: Y}, ,Yn
Result: px,p2,o initialization; end

[0081] It may be noted that in accordance with aspects of the present disclosure, the Gibb's Sampling Procedure 2 provides a more accurate Bayesian estimation of the densities of the individual constituents along with corresponding standard deviations. Particularly, the Gibb's Sampling Procedure 2 mitigates the over parameterization problem for density estimation by assuming the conditional posterior distribution of volume fractions to be a truncated Gaussian distribution. The truncated Gaussian prior distribution of the volume fractions enables accurate estimation of the densities of the individual constituents that correspond to spectrally similar materials. An exemplary output of the Gibb's Sampling Procedure 2 for use in deriving accurate density estimates for the individual constituents, which in turn, may provide medically relevant information may be represented, for example, using equation (12).

[0082] Conventional Gibb's Sampling Procedures tend to be sensitive to initialization parameters. However, use of the Gibb's Sampling Procedures 1 and 2 minimizes sensitivity to the initialization parameters. Accordingly, a first iteration of the method using the Gibb's Sampling Procedures 1 and 2 may be initialized randomly.

[0083] In certain embodiments, the Gibb's Sampling Procedures 1 and 2 may be initialized with a check to verify if the Markov chain has converged to the stationary distribution. In one embodiment, initialization iterations for the Gibb's Sampling Procedures 1 and 2 may be performed for a fixed number of iterations N, for example N = 1000, with an expectation that the Markov chain converges to the stationary distribution. Alternatively, a running mean of each parameter in the Markov chain \P\n,P2n'\\ may be .computed and verified for numerical convergence based upon first order differencing, thus reducing the number of initialization iterations, while enhancing accuracy of the measurements.

[0084] Furthermore, in accordance with aspects of the present disclosure, accuracy of the proposed density estimation method may be a function of a number of voxels n in the identified set of voxels considered for the analysis and the standard deviation of the measurement noise (cr). Additionally, the accuracy of the density estimation may depend upon a difference in the densities of the two individual constituents, (px - p{), and/or a similarity between columns of the ju matrix, which may be measured in terms of a ratio of the smallest to largest Eigen values of the matrix. In certain embodiments, the accuracy of the proposed method may also depend upon an average of volume fractions a,, , andenoted by a and corresponding standard deviations, denoted by <7a. The average of the volume fractions may measure homogeneity of the volume fractions of the individual constituents in the target ROI.

[0085] Subsequently, at step 314, densities corresponding to the individual constituents may be estimated based on the computed posterior distribution, such as described with reference to FIGs. 1-2 and step 312 of FIG. 3. Additionally, confidence intervals corresponding to the density estimates may be determined. In one embodiment, the confidence intervals may be determined using certain selected percentiles of the posterior distribution. Further, a central tendency of the posterior distribution may be computed and used as an estimate of the densities. In certain embodiments, the central tendency of the posterior distribution may correspond to other statistical measurements, such as, median and mode of the density estimates.

[0086] The estimated densities in view of the corresponding confidence intervals and/or the other statistical measurements may then be used to identify the pathology of the target ROI. By way of example, the density estimates may be used to identify infarcted regions in the white matter region in the brain. To that end, in one embodiment, the clinician and/or the operator may input a medically relevant threshold value and/or a threshold range corresponding to the density of fat in the white matter in the brain. Conventionally, fat and blood provide spectrally close responses to X-ray interactions, and thus may complicate density estimation for the individual constituents of the white matter.

[0087] Use of the embodiments of the present method, however, may allow a clinician to discriminate between, for example, fat and blood in the white matter. Typically, density of blood in the white matter remains substantially the same irrespective of pathology of the target ROI. However, density of fat in the white matter in the brain may differ noticeably in normal and infarcted tissues. Embodiments of the present method, thus, may be used for discriminating between fat and blood, while also providing quantitative measurements corresponding to densities of the fat and water for use in diagnosis. To that end, in one embodiment, a probability that the density of fat falls outside (above or below) a medically relevant threshold may be computed. The computed probability may provide medically useful information indicative of presence and/or progression of specific ailments. Accurate estimation of the density of fat using the present method, thus, may provide useful cues for use in diagnosis and/or treatment of the infarcted region.

[0088] FIGs. 4-6 depict an exemplary simulation that may allow for an evaluation of a performance of the present method, as described with reference to FIG. 3.

[0089] FIG. 4 illustrates a diagrammatical representation 400 depicting an exemplary image 402 corresponding to adipose tissue in the neck of a patient. The adipose tissue is a composite material that includes fat and water as corresponding individual constituents. Conventional dual energy imaging techniques may only allow for estimation of the effective density of the adipose tissue, but may not be able to determine respective densities of fat and water.

[0090] Embodiments of the present method, such as, the method described with reference to FIG. 3, however, may allow estimation of the respective densities of fat and water in the adipose tissue of interest using dual energy imaging. To that end, a target ROI 404 in the image 402 corresponding to the adipose tissue may be selected. In one embodiment, an operator may identify the target ROI 404. Subsequently, a set of, for example, 26 voxels in an image slice from the image 402 may be used as an input to the method of FIG. 3. The 26 voxels may correspond to a small neighborhood in the image slice that is selected for density estimation. Further, an embodiment of the present method may be applied at the target ROI 404 for estimating the densities, standard deviation, and volume fractions for the individual constituents of interest, for example, fat and blood in the adipose tissue.

[0091] Particularly, in one embodiment, the respective densities of fat and water may be determined iteratively from corresponding prior distributions. Additionally, standard deviation of noise in an acquisition system configured to acquire projection data used for generating the image 402 may be set to a positive number. Further, volume fractions corresponding to each voxel in the target ROI 404 may also be determined iteratively.

[0092] In certain embodiments, the volume fractions may be determined based on the density of fat and water and the standard deviation of the noise determined in a preceding iteration, a prior distribution of the densities, a prior distribution of the volume fractions and/or the acquired projection data. Further, the densities of fat and water may be determined using a conditional posterior distribution of the densities. In certain embodiments, the conditional posterior distribution of the densities may be determined based on an empirical distribution of the densities, the volume fractions, and the standard deviation of the noise determined in a plurality of iterations. As previously noted, the conditional posterior distribution may also be determined based on the prior distribution of the densities, the prior distribution of the volume fractions, and/or the acquired projection data.

[0093] Additionally, the standard deviation of the noise may be determined using a conditional posterior distribution of the standard deviation of the noise. The conditional posterior distribution of the standard deviation, in turn, may be based on the determined volume fractions, the determined densities, the prior distribution of the densities, the prior distribution of the volume fractions, and/or the acquired projection data.

[0094] In one embodiment, the densities, the volume fractions, and/or the standard deviation of the noise may continue to be iteratively determined until mean values corresponding to each of the determined densities, the volume fractions, and the standard deviation of the noise converge. To ensure accuracy, in certain embodiments, mean values of the densities, the volume fractions, and/or the standard deviation of the noise may be computed for iterations that occur after a determined number of iterations following the first iteration. More specifically, the determined number of iterations may be selected so as to allow stabilization of the determined density, the volume fraction, and the noise values, and in turn, the corresponding mean values. In one example, the determined number of iterations may be selected as ten iterations. Accordingly, the mean values of the densities, the volume fractions, and/or the standard deviation of the noise may be computed starting from the eleventh iteration following the first iteration.

[0095] Moreover, in the present simulation, the mean values may be determined to converge if a difference between mean values of the determined densities in a particular iteration and a preceding iteration is less than a determined threshold. The iterative determination of the densities, the volume fractions, and the standard deviation of the noise may continue until one or more termination criteria are satisfied. To that end, the termination criteria may be determined, for example, using a Gelman-Rubin statistic, running mean convergence, and/or standard deviation convergence. The estimated values corresponding to individual densities of fat and water may then be communicated to a user for use in diagnosing the patient.

[0096] FIG. 5 illustrates a graphical representation 500 depicting exemplary values of individual densities of fat and blood estimated using the present method described with reference to FIGs. 3-4. In the graphical representation 500, reference numeral 502 corresponds to the estimated density of water or perfused blood in the adipose tissues of interest (see FIG. 4). Further, reference numeral 504 corresponds to the estimated density of fat in the adipose tissues of interest.

[0097] As illustrated in the graphical representation 500, density estimates corresponding to fat in the adipose tissues of the neck are constrained to a tighter confidence interval. Further, the adipose tissues are determined to have high fat fraction. Additionally, the density estimates corresponding to fat in the adipose tissues are determined to be substantially homogenous in a small neighborhood of about 26 voxels.

[0098] Reference values corresponding to known densities of fat and blood may be obtained from published medical literature. These reference density values for fat and blood are known to be about 0.9196 grams/cubic centimeter (g/cc) and about 1.06 g/cc, respectively. The reference density values for fat and blood may be compared with density values corresponding to fat and blood measured using an embodiment of the present method to ascertain performance of the present method.

[0099] Accordingly, in one exemplary simulation, the mean density and standard deviation of fat measured using the present method is determined to be about 0.943 g/cc and 0.0047, respectively. Further, the mean and standard deviation of the density of water measured using the present method is determined to be 1.061 g/cc and 0.0317, respectively. Thus, the densities of water and fat determined measured using the present method are determined to be substantially similar to the known densities of water and fat in the adipose tissues of neck. The density estimation using the present method may be further supported by a simultaneous estimation of volume fractions corresponding to the voxels in the target ROI.

[0100] FIG. 6 illustrates a graphical representation 600 depicting exemplary volume fractions corresponding to a small neighborhood of voxels. As previously noted, accuracy of the density values determined using the present method may depend at least in part on an average of volume fractions in the small neighborhood. The average of the volume fractions may measure homogeneity of the volume fractions of the individual constituents in the target ROI. The measured homogeneity, in turn, may be used for a more accurate estimation of the density values corresponding to fat and water.

[0101] Thus, quantitative measurements corresponding to the density and volume fractions associated with the individual constituents of a composite material, such as muscle or adipose tissue, match reference density values reported in medical literature. Substantial similarities between the estimated and known density values, thus, are indicative of accuracy and efficiency of the present method. Particularly, use of density and volume fraction priors and constraining the prior distribution on the volume fractions to be an independent truncated Gaussian distribution improves the accuracy of the density estimation. Additionally, the accuracy of the density estimates validates the assumption of homogeneity in a small neighborhood of voxels in the images.

[0102] Embodiments of the present disclosure, thus, may provide methods and systems for estimating density of individual constituents of a composite material with greater accuracy. Accurate estimation of density and volume fractions corresponding to spectrally close individual constituents, which are typically indistinguishable in conventional density maps, may provide useful indications that may allow for a more informed diagnosis. For example, a quantitative measurement of density of fat in the grey matter may aid in identifying presence and/or severity of infracted grey matter regions with greater accuracy. Additionally, use of embodiments of the present disclosure to provide accurate density estimates directly from processed projection data may circumvent a need for multiple scans and/or surgical intervention to assess a medical condition of a patient.

[0103] It may be noted that the foregoing examples, configurations, and method steps that may be performed by certain components of the present systems may be implemented by suitable code on a processor-based system. These components, for example, may include the control unit 206, the DAS 214, the computing device 216, and/or the image reconstructor 228, and the image processing unit 230 of FIG. 2. Particularly, the steps may be performed using a special-purpose computer, multi-core CPU architecture, distributed cluster systems, general purpose GPU architecture, and/or cloud-based systems. It may also be noted that different implementations of the present disclosure may perform some or all of the steps described herein in different orders or substantially concurrently, that is, in parallel.

[0104] Additionally, the functions may be implemented in a variety of programming languages, including but not limited to Ruby, Hypertext Preprocessor (PHP), Perl, Delphi, Python, Matlab, Freemat, Octave, Interactive Data Language (IDL), FORTRAN, Cuda, openCL, C, C++, and/or Java. Such code may be stored or adapted for storage on one or more tangible, machine-readable media, such as on data repository chips, local or remote hard disks, optical disks (that is, CDs or DVDs), solid-state drives, or other media, which may be accessed by the processor-based system to execute the stored code.

[0105] Although specific features of various embodiments of the present disclosure may be shown in and/or described with respect to some drawings and not in others, this is for convenience only. It is to be understood that the described features, structures, and/or characteristics may be combined and/or used interchangeably in any suitable manner in the various embodiments

[0106] While only certain features of the present disclosure have been illustrated and described herein, many modifications and changes will occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the invention.

We claim:

1. A method for imaging, comprising: acquiring projection data corresponding to a target using X-rays generated at a plurality of energy levels; identifying at least two individual constituents corresponding to the target; identifying a set of voxels comprising the at least two individual constituents in at least a pair of basis material decomposition images reconstructed using the acquired projection data; determining a prior distribution of densities corresponding to the at least two individual constituents in the identified set of voxels; ascertaining at least an upper bound on variation of volume fractions corresponding to the at least two individual constituents in the identified set of voxels; computing a posterior distribution of the densities corresponding to the at least two individual constituents based on the determined prior distribution of densities, the ascertained upper bound on variation of volume fractions, or a combination thereof; and estimating densities corresponding to the at least two individual constituents based on the computed posterior distribution.

2. The method of claim 1, wherein the target comprises a binary composite material.

3. The method of claim 1, further comprising assessing a medical condition of a subject based on the estimated densities, wherein the target comprises a biological tissue.

4. The method of claim 1, further comprising discriminating between the at least two individual constituents based on the estimated densities, wherein the at least two individual constituents comprise spectrally similar materials.

5. The method of claim 1, wherein identifying the at least two individual constituents comprises receiving user input indicative of the two individual constituents, using determined information correlating the target and corresponding individual constituents, or a combination thereof.

6. The method of claim 1, wherein identifying the set of voxels comprises filtering the projection data based on determined properties corresponding to the at least two individual constituents corresponding to the target.

7. The method of claim 1, wherein identifying the set of voxels comprises: receiving user input indicating a region of interest in at least the pair of basis material decomposition images; and selecting the set of voxels based on the indicated region of interest.

8. The method of claim 1, wherein identifying the set of voxels comprises: reconstructing at least a pair of monochromatic images from the projection data acquired using X-rays generated at the plurality of energy levels; receiving user input indicating a region of interest in at least the pair of basis material decomposition images; and selecting the set of voxels in at least the pair of monochromatic images based on the indicated region of interest.

9. The method of claim 1, wherein determining the prior distribution of densities comprises receiving user input indicative of one or more values corresponding to the prior distribution, automatically setting the prior distribution of densities using known prior distribution information corresponding to the at least two individual constituents, or a combination thereof.

10. The method of claim 1, wherein determining the prior distribution of the densities comprises determining a Gaussian distribution of the densities with corresponding means and standard deviations.

11. The method of claim 1, wherein ascertaining at least the upper bound on variation of volume fractions comprises receiving user input indicative of the upper bound, using a previously determined value of the upper bound corresponding to the at least two individual constituents in the identified set of voxels, or a combination thereof.

12. The method of claim 1, wherein computing the posterior distribution of the densities comprises using a Markov Chain Monte-Carlo procedure.

13. The method of claim 12, further comprising simulating the Markov chain such that a stationary distribution of the Markov chain provides an estimate of the posterior distribution of the densities corresponding to the at least two individual constituents.

14. The method of claim 13, further comprising generating the Markov chain using a Metropolis Hasting method.

15. The method of claim 14, further comprising generating the Markov chain using a Gibb's Sampler.

16. The method of claim 15, further comprising performing a first iteration for computing the posterior distribution, wherein the first iteration comprises: initializing the densities of the individual constituents based on corresponding prior distributions; and initializing standard deviation of noise in an acquisition system to be a positive number, wherein the acquisition system is configured to acquire the projection data.

17. The method of claim 16, further comprising:

(a) determining the volume fractions corresponding to each of the voxels in the target based on the density of the individual constituents determined in a preceding iteration, the standard deviation of the noise determined in the preceding iteration, the prior distribution of the densities, the prior distribution of the volume fractions, and the projection data;

(b) determining the densities of the individual constituents using a conditional distribution of the densities based on the determined volume fractions, the standard deviation of the noise, the prior distribution of the densities, the prior distribution of the volume fractions, and the projection data;

(c) determining the standard deviation of the noise using a conditional distribution of the standard deviation of the noise based on the determined volume fractions, the determined densities, the prior distribution of the densities, the prior distribution of the volume fractions, and the projection data; and

(d) repeating steps (a)-(c) to compute the posterior distribution of the densities, the volume fractions, noise of the acquisition system corresponding to the identified set of voxels, or combinations thereof.

18. The method of claim 17, wherein the determining the posterior distribution of the densities comprises determining an empirical distribution of the densities, the volume fractions, the standard deviation of the noise determined in a plurality of iterations.

19. The method of claim 17, further comprising iteratively determining the densities, the volume fractions, the standard deviation of the noise until one or more termination criteria are satisfied.

20. The method of claim 17, further comprising: computing a probability of the densities corresponding to the at least two individual constituents being outside a determined threshold using the posterior distribution; and assessing a condition of a subject comprising the target based on the computed probability.

21. The method of claim 17, further comprising iteratively determining the densities, the volume fractions, and the standard deviation of the noise until corresponding means of the determined densities, the volume fractions, and the standard deviation of the noise converge.

22. The method of claim 21, further comprising computing the corresponding means of the determined densities, the volume fractions, the standard deviation of the noise for one or more iterations following a determined number of iterations from the first iteration.

23. The method of claim 1, further comprising estimating confidence intervals corresponding to the densities based on the computed posterior distribution.

24. The method of claim 23, wherein estimating the confidence intervals comprises computing the confidence intervals based on the posterior distribution.

25. The method of claim 23, further comprising computing a central tendency of the posterior distribution as an estimate of the densities.

26. The method of claim 1, further comprising estimating one or more statistical measures corresponding to the densities based on the computed posterior distribution.

27. The method of claim 1, further comprising using a modified Gibb's Sampling Procedure for simulating a Markov chain such that a prior distribution of volume fractions corresponding to the at least two individual constituents in the identified set of voxels is a truncated Gaussian distribution.

28. An imaging system, comprising: a data acquisition system configured to acquire projection data corresponding to a target using X-rays generated at a plurality of energy levels; an image processing unit operatively coupled to the data acquisition system and configured to: identify at least two individual constituents corresponding to the target; identify a set of voxels comprising the at least two individual constituents in at least a pair of basis material decomposition images reconstructed using the acquired projection data; determine a prior distribution of densities corresponding to the at least two individual constituents in the identified set of voxels; ascertain at least an upper bound on variation of volume fractions corresponding to the at least two individual constituents in the identified set of voxels; compute a posterior distribution of the densities corresponding to the at least two individual constituents based on the determined prior distribution of densities, the ascertained upper bound on variation of volume fractions, or a combination thereof; and estimate densities corresponding to the at least two individual constituents based on the computed posterior distribution.

29. The imaging system of claim 28, wherein the imaging system comprises a computed tomography system, a single source imaging system, a multi-source imaging system, a multi-detector imaging system, a photon counting and energy discriminating detector imaging system, an X-Ray system, a positron emission tomography scanner, a single photon emission computed tomography scanner, or combinations thereof.

30. A computed tomography (CT) system, comprising: at least one radiation source configured to generate X-rays at a plurality of energy levels; a detector assembly operatively coupled to the at least one radiation source and configured to detect the X-rays generated from the radiation source; an image processing unit operatively coupled to the detector assemblyand configured to: identify at least two individual constituents corresponding to the target; identify a set of voxels comprising the at least two individual constituents in at least a pair of basis material decomposition images reconstructed using the acquired projection data; determine a prior distribution of densities corresponding to the at least two individual constituents in the identified set of voxels; ascertain at least an upper bound on variation of volume fractions
j corresponding to the at least two individual constituents in the identified set of voxels;
compute a posterior distribution of the densities corresponding to the at least two individual constituents based on the determined prior distribution of densities, the ascertained upper bound on variation of volume fractions, or a combination thereof; and estimate densities corresponding to the at least two individual constituents based on the computed posterior distribution.