- HOME

California State University, Fullerton. Summer 2008 page compiled on July 1, 2015 at 10:08pm

1 Notations and deﬁnitions

2 HYPR mathematical formulation

2.1 Original HYPR

2.2 Wright HYPR

3 Derivation of Wright HYPR from normal equation

4 References

2 HYPR mathematical formulation

2.1 Original HYPR

2.2 Wright HYPR

3 Derivation of Wright HYPR from normal equation

4 References

- 1.
- MLEM Maximum-Likelihood Expectation-Maximization
- 2.
- PET Positron Emission Tomography
- 3.
- SPECT Single-Photon Emission Computed Tomography
- 4.
- A 2-D image. This represent the original user image at which the HYPR algorithm is applied to.
- 5.
- When the original image content changes during the process, we add a subscript to indicate the image at time instance .
- 6.
- radon transform.
- 7.
- radon transform used at a projection angle .
- 8.
- When the projection angle is not constant but changes with time during the MRI acquisition process, we add a subscript to indicate the angle at time instance .
- 9.
- radon transform used at an angle .
- 10.
- . radon transform applied to an image at angle . This results in a projection vector .
- 11.
- Forward projection matrix. The Matrix equivalent to the radon transform .
- 12.
- Estimate of an image .
- 13.
- Multiply the forward projection matrix with an image estimate .
- 14.
- Multiply the forward projection matrix with an image estimate to obtain a projection vector .
- 15.
- The inverse radon transform applied in unﬁltered mode to a projection which was taken at angle . This results in a 2D image.
- 16.
- The inverse radon transform applied in ﬁltered mode to a projection which was taken at angle . This results in a 2D image.
- 17.
- The transpose of the forward projection matrix multiplied by the projection vector . This is the matrix equivalent of applying the inverse radon transform in an unﬁltered mode to a projection (see item 12 above).
- 18.
- The pseudo inverse of the forward projection matrix being multiplied by the projection vector . This is the matrix equivalent of applying the inverse radon transform in ﬁltered mode to a projection (see item 13 above).
- 19.
- Composite image generated by summing all the ﬁltered back projections from projections of the original images . Hence
- 20.
- The unﬁltered backprojection 2D image as a result of applying where is projection from user image taken at angle .
- 21.
- The unﬁltered backprojection 2D image as a result of applying where is projection from the composite image taken at angle .
- 22.
- Number of projections used to generate one HYPR frame image. This is the same as the number of projections per one time frame.
- 23.
- The total number of projections used. This is the number of time frames multiplied by
- 24.
- The HYPR frame image. A 2-D image generate at the end of the HYPR algorithm. There will be as many HYPR frame images as there are time frames.
- 25.
- Image ﬁdelity: " (inferred by the ability to discriminate between two images)" reference: The
relationship between image ﬁdelity and image quality by Silverstein, D.A.; Farrell, J.E

Sci-Tech Encyclopedia: Fidelity"The degree to which the output of a system accurately reproduces the essential characteristics of its input signal. Thus, high ﬁdelity in a sound system means that the reproduced sound is virtually indistinguishable from that picked up by the microphones in the recording or broadcasting studio. Similarly, a television system has a high ﬁdelity when the picture seen on the screen of a receiver corresponds in essential respects to that picked up by the television camera. Fidelity is achieved by designing each part of a system to have minimum distortion, so that the waveform of the signal is unchanged as it travels through the system. "

- 26.
- "image quality (inferred by the preference for one image over another)". Same reference as above
- 27.
- TE (Echo Time) "represents the time in milliseconds between the application of the 90 pulse and the peak of the echo signal in Spin Echo and Inversion Recovery pulse sequences." reference: http://www.fonar.com/glossary.htm
- 28.
- TR (Repetition Time) "the amount of time that exists between successive pulse sequences applied to the same slice." reference: http://www.fonar.com/glossary.htm

This mathematics of this algorithm will be presented by using the radon transform notation and not the matrix projection matrix notation.

The projection is obtained by applying radon transform on the image at some angle

When the original object image does not change with time then we can drop the subscript from and just write

The composite image is found from the ﬁltered back projection applied to all the

Notice that the sum above is taken over and not over . Next a projection is taken from at angle as follows

The the unﬁltered back projection 2-D image is generated

And the unﬁltered back projection 2-D image is found

Then the ratio of is summed and averaged over the time frame and multiplied by to generate a HYPR frame for the time frameHence for the time frame we obtain

This mathematics of this algorithm will be presented by using the radon transform notation and not the matrix projection matrix notation. The conversion between the notation can be easily made by referring to the notation page at the end of this report.

The projection is obtained by applying radon transform on the image at some angle

When the original object image does not change with time then we can drop the subscript from and just write

The composite image is found from the ﬁltered back projection applied to all the

Notice that the sum above is taken over and not over . Next a projection is taken from at angle as follows

The the unﬁltered back projection 2-D image is generated

And the unﬁltered back projection 2-D image is found

Now the set of and over one time frame are summed the their ratio multiplied by to obtain the HYPR frame

We start with the same starting equation used to derive the HYPR formulation as in the above section.

Where is noise vector from Gaussian distribution with zero mean. is forward projection operator at an angle at time , and is the original image at time , and is the one dimensional projection vector that results from the above operation.

Now apply the operator to the above equation, we obtain

Since is linear, the above becomes

Pre multiply the above with

Divide both side by

Under the condition that noise vector can be ignored the above becomes (after canceling out the terms)

or

If we select the composite as representing the initial estimate of the true image , the above becomes, after replacing in the R.H.S. of the above equation by

| (1) |

But is the unﬁltered backprojection of the projection , hence this term represents the term shown in the last section, which is the unﬁltered backprojection 2D image, and is the unﬁltered backprojection of the projection , which is the term in the last section. Hence we see that (1) is the same equation as

| (2) |

Once is computed from (1), we can repeat (1) again, using this computed as the new estimate of the true image in the RHS of (1), and repeat the process again.

- 1.
- Dr Pineda, CSUF Mathematics dept. California, USA.
- 2.
- Highly Constrained Back projection for Time-Resolved MRI by C. A. Mistretta, O. Wieben, J. Velikina, W. Block, J. Perry, Y. Wu, K. Johnson, and Y. Wu
- 3.
- Iterative projection reconstruction of time-resolved images using HYPR by O'Halloran et.all
- 4.
- Time-Resolved MR Angiography With Limited Projections by Yuexi Huang1,and Graham A. Wright
- 5.
- GE medical PPT dated 6/6/2008
- 6.
- Book principles of computerized Tomographic imaging by Kak and Staney