Apparatus and method for correcting bias in low-count computed tomography projection data

1. An apparatus, comprising:

processing circuitry configured to

obtain projection data representing an intensity of X-ray radiation detected at a plurality of detector elements after traversing an object, the projection data being corrected for a baseline offset,

correct the projection data by performing a positivity mapping to generate corrected projection data,

perform a logarithm operation on the corrected projection data to generate post-log projection data,

correct the post-log projection data for a bias of the post-log projection data, using the projection data, to generate bias-corrected projection data, and

reconstruct an image of the object using the bias-corrected projection data, wherein

the projection data is linearly related to an exponential of an attenuation coefficient of the X-ray radiation and the post-log projection data is linearly related to the attenuation coefficient of the X-ray radiation.

2. The apparatus according to claim 1 , wherein the processing circuitry is further configured to

estimate the bias of the post-log projection data using a probability density function of noise associated with the intensity of X-ray radiation detected at one or more of the plurality of detector elements, and

generate a look-up table relating a mean value of the projection data to a bias-correction value for the logarithm operation and the positivity mapping, the bias-correction value corresponding to the estimated bias of the post-log projection data.

3. The apparatus according to claim 2 , wherein the processing circuitry is further configured to generate the look-up table by

obtaining calibration data representing an intensity of the X-ray radiation detected at the plurality of detector elements during a calibration scan, the calibration data being corrected for the baseline offset and representing the X-ray radiation generated with a plurality of settings of an X-ray source generating the X-ray radiation,

determining respective mean count values of the calibration data, for each detector element of the plurality of detector elements and for each setting of the plurality of settings of the X-ray source, and generate post-log mean count values by performing the logarithm operation on the respective mean count values,

determining post-log calibration data by performing the positivity mapping and the logarithm operation on the calibration data,

determining bias-correction values representing respective differences between respective expectation values of the post-log calibration data and the corresponding post-log mean count values, the bias-correction values corresponding to the estimated bias of the post-log projection that is estimated using the probability density function derived from the calibration data, and

associating the respective bias-correction values with the corresponding mean count values to generate the look-up table.

4. The apparatus according to claim 2 , wherein the processing circuitry is further configured to generate the look-up table by

obtaining the probability density function of noise using a statistical model of the X-ray radiation detected at the plurality of detector elements,

determining bias-correction values corresponding to the estimated bias of the post-log projection data and representing respective differences between expectation values of post-log data and logarithms of true values, wherein each of the expectation values of post-log data include an expectation value integral over the probability density function with an integrand that includes the logarithm operation of the positivity mapping of a value representing the detected X-ray radiation, and

associating the respective bias-correction values with the corresponding true values to generate the look-up table.

5. The apparatus according to claim 4 , wherein the processing circuitry is further configured to generate the look-up table by

obtaining the probability density function representing the statistical model that is one of a linear combination of a compound Poison model and Gaussian model, a shifted Poisson model, a linear combination of a Poison model and Gaussian model, a mixture model, a Poisson model, and a Gaussian model.

6. The apparatus according to claim 1 , wherein the processing circuitry is further configured to perform the positivity mapping, which is one of a threshold method, an absolute-value method, an exponential shift curve method, a log-tweak method, and a Maximum-likelihood based method.

7. The apparatus according to claim 1 , wherein the processing is further configured to correct the post-log projection data for the bias of the post-log projection data by

estimating the bias of the post-log projection data using a probability density function of noise associated with the intensity of X-ray radiation detected at one or more of the plurality of detector elements,

using a look-up table to look up respective bias values associated with corresponding values of the projection data, wherein, when the look-up table receives an input of a value of the pre-log projection data, the look-up table outputs an associated bias value, and

subtract the respective bias values from the corresponding post-log projection data to generate the bias-corrected projection data.

8. The apparatus according to claim 7 , wherein the processing circuitry is further configured to

denoise the projection data to generate denoised projection data,

estimate a mean value for each of the plurality of detector elements using the denoised projection data, and

input the mean value into the look-up table to generate outputs of the corresponding bias values.

9. The apparatus according to claim 8 , wherein the processing circuitry is further configured to denoise the projection data by performing one of a low-pass filtering method, a linear-smoothing filtering method, an anisotropic diffusion method, a non-local means method, a sinogram restoration method, a local linear minimum mean-squared-error filtering method, a Gaussian estimator method, a count adaptive mean estimator, and a nonlinear filtering method.

10. The apparatus according to claim 1 , wherein the processing circuitry is further configured to bypass the correcting of the projection data by performing a positivity mapping and the correcting for the bias of the post-log projection data, for values of the projection data that exceed a predefined threshold.

11. The apparatus according to claim 1 , wherein the processing circuitry is further configured to correct the projection data by

performing a scatter correction on the projection data to separate the projection data into primary data and scattered data, and

performing a positivity mapping on the primary data to generate the corrected projection data.

12. The apparatus according to claim 1 , wherein the processing circuitry is further configured to

perform a beam-hardening correction on the bias-corrected projection data to generate beam-hardening corrected data, and

reconstruct the image using the beam-hardening corrected data and using one of an iterated reconstruction method, an iterated reconstruction method using a regularization term, and a filtered back-projection method.

13. The apparatus according to claim 1 , wherein the processing circuitry is further configured to refine the bias-corrected projection data by

performing an exponential operation on the bias-corrected projection data to generate another projection data,

correcting the another projection data by performing the positivity mapping to generate another corrected projection data,

performing the logarithm operation on the another corrected projection data to generate another post-log projection data, and

correcting for a bias of the another post-log projection data, using the projection data, to generate another bias-corrected projection data, wherein

the image of the object is reconstructed using the another bias-corrected projection data.

14. An apparatus, comprising:

an X-ray source to radiate X-rays;

a plurality of detector elements, each configured to

detect the X-ray radiation emanating from the X-ray source, after the X-ray radiation traverses an object, and

generate projection data representing an intensity of the X-rays detected at the plurality of detector elements, wherein the projection data is corrected for a baseline offset; and

processing circuitry configured to

correct the projection data by performing a positivity mapping to generate corrected projection data,

perform a logarithm operation on the corrected projection data to generate post-log projection data,

correct the post-log projection data for a bias of the post-log projection data, using the projection data, to generate bias-corrected projection data, and

reconstruct an image of the object from the bias-corrected projection data, wherein

the projection data is linearly related to an exponential of an attenuation coefficient of the X-ray radiation and the post-log projection data is linearly related to the attenuation coefficient of the X-ray radiation.

15. The apparatus according to claim 14 , wherein the bias of the post-log projection data is estimated using a probability density function for noise associated with the intensity of X-ray radiation detected at one or more of the plurality of detector elements.

16. A method, comprising:

obtaining projection data representing an intensity of X-ray radiation detected at a plurality of detector elements after traversing an object, the projection data being corrected for a baseline offset,

correcting the projection data by performing a positivity mapping to generate corrected projection data,

performing a logarithm operation on the corrected projection data to generate post-log projection data,

correcting the post-log projection data for a bias of the post-log projection data, using the projection data, to generate bias-corrected projection data, and

reconstructing an image of the object from the bias-corrected projection data, wherein

the projection data is linearly related to an exponential of an attenuation coefficient of the X-ray radiation and the post-log projection data is linearly related to the attenuation coefficient of the X-ray radiation.

17. The method according to claim 16 , further comprising:

estimating the bias of the post-log projection data using a probability density function of noise associated with the intensity of X-ray radiation detected at one or more of the plurality of detector elements, and

generating a look-up table relating respective values of the projection data to bias-correction values, the bias-correction values corresponding to the estimated bias of the post-log projection data.

18. The method according to claim 17 , wherein the generating of the look-up table further includes

obtaining calibration data representing an intensity of the X-ray radiation detected at the plurality of detector elements during a calibration scan, the calibration data being corrected for the baseline offset and representing the X-ray radiation generated with a plurality of settings of an X-ray source generating the X-ray radiation,

determining respective mean count values of the calibration data, for each detector element of the plurality of detector elements and for each setting of the plurality of settings of the X-ray source, and generate post-log mean count values by performing the logarithm operation on the respective mean count values,

determining post-log calibration data by performing the positivity mapping and the logarithm operation on the calibration data,

determining bias-correction values representing respective differences between respective expectation values of the post-log calibration data and the corresponding post-log mean count values, the bias-correction values corresponding to the bias of the post-log projection estimated using the probability density function derived from the calibration data, and

associating the respective bias-correction values with the corresponding mean count values to generate the look-up table.

19. The method according to claim 17 , wherein the generating of the look-up table further includes

obtaining the probability density function for noise using a statistical model of the X-ray radiation detected at the plurality of detector elements,

determining bias-correction values corresponding to the estimated bias of the post-log projection data and representing respective differences between expectation values of post-log data and logarithms of true values, wherein each of the expectation values of post-log data include an expectation value integral over the probability density function with an integrand that includes the logarithm operation of the positivity mapping of a value representing the detected X-ray radiation, and

associating the respective bias-correction values with the corresponding true values to generate the look-up table.

20. The method according to claim 16 , wherein the correcting the post-log projection data includes

estimating the bias of the post-log projection data using a probability density function of noise associated with the intensity of X-ray radiation detected at one or more of the plurality of detector elements,

correcting for the bias of the post-log projection data using a look-up table to look up respective bias values associated with corresponding values of the projection data, the bias values corresponding to the estimated bias of the post-log projection data, wherein, when the look-up table receives an input of a pre-log projection data value, the look-up table outputs an associated bias value, and

subtracting the respective bias values from the corresponding post-log projection data to generate the bias-corrected projection data.

21. A non-transitory computer readable storage medium including executable instruction, wherein the instructions, when executed by circuitry, cause the circuitry to perform the method according to claim 16 .