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1 Notations and definitions

1. MLEM Maximum-Likelihood Expectation-Maximization
2. PET Positron Emission Tomography
3. SPECT Single-Photon Emission Computed Tomography
4. $I$ A 2-D image. This represent the original user image at which the HYPR algorithm is applied to.
5. $I_t$ When the original image content changes during the process, we add a subscript to indicate the image $I$ at time instance $t$.
6. $R$ radon transform.
7. $R_{\varphi }$ radon transform used at a projection angle $\varphi$.
8. ${\varphi }_t$ When the projection angle $\varphi$ is not constant but changes with time during the MRI acquisition process, we add a subscript to indicate the angle at time instance $t$.
9. $R_{{\varphi }_t}$ radon transform used at an angle ${\varphi }_t$.
10. $s=R_{\varphi }\left[I\right]$. radon transform applied to an image $I$ at angle $\varphi$. This results in a projection vector $s$.
11. $H$ Forward projection matrix. The Matrix equivalent to the radon transform $R$.
12. $\theta$ Estimate of an image $I$.
13. $H\theta$ Multiply the forward projection matrix $H$ with an image estimate $\theta$.
14. $g=H\theta$ Multiply the forward projection matrix $H$ with an image estimate $\theta$ to obtain a projection vector $g$.
15. $R_{\varphi }^u\left[s\right]$ The inverse radon transform applied in unfiltered mode to a projection $s$ which was taken at angle $\varphi$. This results in a 2D image.
16. $R_{\varphi }^f\left[s\right]$ The inverse radon transform applied in filtered mode to a projection $s$ which was taken at angle $\varphi$. This results in a 2D image.
17. $H^{T}g$ The transpose of the forward projection matrix $H$ multiplied by the projection vector $g$. This is the matrix equivalent of applying the inverse radon transform in an unfiltered mode to a projection $s$ (see item 12 above).
18. $H^{+}g$ The pseudo inverse of the forward projection matrix $H$ being multiplied by the projection vector $g$. This is the matrix equivalent of applying the inverse radon transform in filtered mode to a projection $s$ (see item 13 above).
19. $C$ Composite image generated by summing all the filtered back projections from projections $s_t$ of the original images $I_t$. Hence $C={\displaystyle \sum _{i=1}^N}{R}_{{\varphi }_{t_i}}^f\left[s_{t_i}\right]$
20. $P_t$ The unfiltered backprojection 2D image as a result of applying $R_{{\varphi }_t}^u\left[s_t\right]$ where $s_t$ is projection from user image $I_t$ taken at angle ${\varphi }_t$.
21. $P_{c_t}$ The unfiltered backprojection 2D image as a result of applying $R_{{\varphi }_t}^u\left[s_t\right]$ where $s_t$ is projection from the composite image $C$ taken at angle ${\varphi }_t$.
22. $N_p$ Number of projections used to generate one HYPR frame image. This is the same as the number of projections per one time frame.
23. $N$ The total number of projections used. This is the number of time frames multiplied by $N_p$
24. $J_k$ The $k^{th}$ HYPR frame image. A 2-D image generate at the end of the HYPR algorithm. There will be as many HYPR frame images $J_k$ 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${}^{\circ }$ 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

2 HYPR mathematical formulation

2.1 Original HYPR

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

The projection $s_t$ is obtained by applying radon transform $R$ on the image $I_t$ at some angle ${\varphi }_t$$$s_t=R_{{\varphi }_t}\left[I_t\right]$$ When the original object image does not change with time then we can drop the subscript $t$ from $I_t$ and just write $s_t=R_{{\varphi }_t}\left[I\right]$

The composite image $C$ is found from the filtered back projection applied to all the $s_t$$$C={\displaystyle \sum _{i=1}^N}{R}_{{\varphi }_{t_i}}^f\left[s_{t_i}\right]$$ Notice that the sum above is taken over $N$ and not over $N$. Next a projection $s_c$ is taken from $C$ at angle $\varphi$ as follows$$s_{c_t}=R_{{\varphi }_t}\left[C\right]$$ The the unfiltered back projection 2-D image $P_t$ is generated$$P_t=R_{{\varphi }_t}^u\left[s_t\right]$$ And the unfiltered back projection 2-D image $P_{c_t}$ is found$$P_{c_t}=R_{{\varphi }_t}^u\left[s_{c_t}\right]$$ Then the ratio of $\frac{P_t}{P_{c_t}}$ is summed and averaged over the time frame and multiplied by $C$ to generate a HYPR frame $J$ for the time frame$.$Hence for the $k^{th}$ time frame we obtain$$\begin{array}{rl}{J}_k& =C\left(\frac{1}{N_p}{\displaystyle \sum _{i=1}^{N_p}}\frac{P_{t_i}}{P_{c_{t_i}}}\right)\\ & =\frac{1}{N_p}\left({\displaystyle \sum _{i=1}^N}{R}_{{\varphi }_{t_i}}^f\left[s_{t_i}\right]\right){\displaystyle \sum _{j=1}^{N_p}}\frac{R_{{\varphi }_{t_j}}^u\left[s_{t_j}\right]}{R_{{\varphi }_{t_j}}^u\left[s_{c_{t_j}}\right]}\end{array}$$

2.2 Wright HYPR

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

The projection $s_t$ is obtained by applying radon transform $R$ on the image $I_t$ at some angle ${\varphi }_t$$$s_t=R_{{\varphi }_t}\left[I_t\right]$$ When the original object image does not change with time then we can drop the subscript $t$ from $I_t$ and just write $s_t=R_{{\varphi }_t}\left[I\right]$

The composite image $C$ is found from the filtered back projection applied to all the $s_t$$$C={\displaystyle \sum _{i=1}^N}{R}_{{\varphi }_{t_i}}^f\left[s_{t_i}\right]$$ Notice that the sum above is taken over $N$ and not over $N$. Next a projection $s_c$ is taken from $C$ at angle $\varphi$ as follows$$s_{c_t}=R_{{\varphi }_t}\left[C\right]$$ The the unfiltered back projection 2-D image $P_t$ is generated$$P_t=R_{{\varphi }_t}^u\left[s_t\right]$$ And the unfiltered back projection 2-D image $P_{c_t}$ is found$$P_{c_t}=R_{{\varphi }_t}^u\left[s_{c_t}\right]$$ Now the set of $P_t$ and $P_{c_t}$ over one time frame are summed the their ratio multiplied by $C$ to obtain the $k^{th}$ HYPR frame $$\begin{array}{rl}{J}_k& =C\frac{{\displaystyle \sum _{i=1}^{N_p}}{P}_{t_i}}{{\displaystyle \sum _{i=1}^{N_p}}{P}_{c_{t_i}}}\\ & =C\frac{{\displaystyle \sum _{i=1}^{N_{pr}}}{R}_{{\varphi }_t}^u\left[s_t\right]}{{\displaystyle \sum _{i=1}^{N_{pr}}}{R}_{{\varphi }_t}^u\left[s_{c_t}\right]}\end{array}$$

3 Derivation of Wright HYPR from normal equation

We start with the same starting equation used to derive the HYPR formulation as in the above section.$$s_t=H_{{\varphi }_t}\left[I_t\right]+\mathbf{n}$$ Where $\mathbf{n}$ is noise vector from Gaussian distribution with zero mean.  $H_{{\varphi }_t}$ is forward projection operator at an angle $\varphi$ at time $t$, and $I_t$ is the original image at time $t$, and $s_t$ is the one dimensional projection vector that results from the above operation.

Now apply the $H^T$ operator to the above equation, we obtain$$H^T\left[s_t\right]=H^T[H_{{\varphi }_t}\left[I_t\right]+\mathbf{n}]$$ Since $H^T$ is linear, the above becomes$$H^T\left[s_t\right]=H^T\left[H_{{\varphi }_t}\left[I_t\right]\right]+H^T\left[\mathbf{n}\right]$$ Pre multiply the above with $I_t$$$I_t{H}^T\left[s_t\right]=I_t{H}^T\left[H_{{\varphi }_t}\left[I_t\right]\right]+I_t{H}^T\left[\mathbf{n}\right]$$ Divide both side by $H^T\left[H_{{\varphi }_t}\left[I_t\right]\right]$$$\frac{I_t{H}^T\left[s_t\right]}{H^T\left[H_{{\varphi }_t}\left[I_t\right]\right]}=\frac{I_t{H}^T\left[H_{{\varphi }_t}\left[I_t\right]\right]}{H^T\left[H_{{\varphi }_t}\left[I_t\right]\right]}+\frac{I_t{H}^T\left[\mathbf{n}\right]}{H^T\left[H_{{\varphi }_t}\left[I_t\right]\right]}$$ Under the condition that noise vector can be ignored the above becomes (after canceling out the $H^T\left[H_{{\varphi }_t}\left[I_t\right]\right]$ terms)$$\frac{I_t{H}^T\left[s_t\right]}{H^T\left[H_{{\varphi }_t}\left[I_t\right]\right]}=I_t$$ Or$$I_t=I_t\left(\frac{H^T\left[s_t\right]}{H^T\left[H_{{\varphi }_t}\left[I_t\right]\right]}\right)$$ If we select the composite $C$ as representing the initial estimate of the true image $I_t$, the above becomes, after replacing $I_t$ in the R.H.S. of the above equation by $C$$$\begin{array}{}\text{(1)}& I_t=C\left(\frac{H^T\left[s_t\right]}{H^T\left[H_{{\varphi }_t}\left[C\right]\right]}\right)\end{array}$$ But $H^T\left[s_t\right]$ is the unfiltered backprojection of the projection $s_t$, hence this term represents the term $P_t$ shown in the last section, which is the unfiltered backprojection 2D image, and $H^T\left[H_{{\varphi }_t}\left[C\right]\right]$ is the unfiltered backprojection of the projection $H_{{\varphi }_t}\left[C\right]$, which is the term $P_{c_t}$ in the last section. Hence we see that (1) is the same equation as $$\begin{array}{}\text{(2)}& I_t=C\frac{P_t}{P_{c_t}}\end{array}$$ Once $I_t$ is computed from (1), we can repeat (1) again, using this computed $I_t$ as the new estimate of the true image in the RHS of (1), and repeat the process again.

4 References

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