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. "
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 \(\phi _{t}\)\[ s_{t}=R_{\phi _{t}}\left [ I_{t}\right ] \] When the original object image does not change with time then we can drop the subscript \(t\) from \(I_{t}\) and just write \(s_{t}=R_{\phi _{t}}\left [ I\right ] \)
The composite image \(C\) is found from the ﬁltered back projection applied to all the \(s_{t}\)\[ C={\displaystyle \sum \limits _{i=1}^{N}} R_{\phi _{t_{i}}}^{f}\left [ s_{t_{i}}\right ] \] Notice that the sum above is taken over \(N\) and not over \(N\). Next a projection \(s_{c}\) is taken from \(C\) at angle \(\phi \) as follows\[ s_{c_{t}}=R_{\phi _{t}}\left [ C\right ] \] The the unﬁltered back projection 2-D image \(P_{t}\) is generated\[ P_{t}=R_{\phi _{t}}^{u}\left [ s_{t}\right ] \] And the unﬁltered back projection 2-D image \(P_{c_{t}}\) is found\[ P_{c_{t}}=R_{\phi _{t}}^{u}\left [ s_{c_{t}}\right ] \] 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 {align*} J_{k} & =C\ \left ( \frac {1}{N_{p}}{\displaystyle \sum \limits _{i=1}^{N_{p}}} \frac {P_{t_{i}}}{P_{c_{t_{i}}}}\right ) \\ & =\frac {1}{N_{p}}\left ( {\displaystyle \sum \limits _{i=1}^{N}} R_{\phi _{t_{i}}}^{f}\left [ s_{t_{i}}\right ] \right ) \ {\displaystyle \sum \limits _{j=1}^{N_{p}}} \frac {R_{\phi _{t_{j}}}^{u}\left [ s_{t_{j}}\right ] }{R_{\phi _{t_{j}}}^{u}\left [ s_{c_{t_{j}}}\right ] } \end {align*}
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 \(\phi _{t}\)\[ s_{t}=R_{\phi _{t}}\left [ I_{t}\right ] \] When the original object image does not change with time then we can drop the subscript \(t\) from \(I_{t}\) and just write \(s_{t}=R_{\phi _{t}}\left [ I\right ] \)
The composite image \(C\) is found from the ﬁltered back projection applied to all the \(s_{t}\)\[ C={\displaystyle \sum \limits _{i=1}^{N}} R_{\phi _{t_{i}}}^{f}\left [ s_{t_{i}}\right ] \] Notice that the sum above is taken over \(N\) and not over \(N\). Next a projection \(s_{c}\) is taken from \(C\) at angle \(\phi \) as follows\[ s_{c_{t}}=R_{\phi _{t}}\left [ C\right ] \] The the unﬁltered back projection 2-D image \(P_{t}\) is generated\[ P_{t}=R_{\phi _{t}}^{u}\left [ s_{t}\right ] \] And the unﬁltered back projection 2-D image \(P_{c_{t}}\) is found\[ P_{c_{t}}=R_{\phi _{t}}^{u}\left [ s_{c_{t}}\right ] \] 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 {align*} J_{k} & =C\ \frac {{\displaystyle \sum \limits _{i=1}^{N_{p}}} P_{t_{i}}}{{\displaystyle \sum \limits _{i=1}^{N_{p}}} P_{c_{t_{i}}}}\\ & =C\ \frac {{\displaystyle \sum \limits _{i=1}^{N_{pr}}} R_{\phi _{t}}^{u}\left [ s_{t}\right ] }{{\displaystyle \sum \limits _{i=1}^{N_{pr}}} R_{\phi _{t}}^{u}\left [ s_{c_{t}}\right ] } \end {align*}
We start with the same starting equation used to derive the HYPR formulation as in the above section.\[ s_{t}=H_{\phi _{t}}\left [ I_{t}\right ] +\mathbf {n}\] Where \(\mathbf {n}\) is noise vector from Gaussian distribution with zero mean. \(H_{\phi _{t}}\) is forward projection operator at an angle \(\phi \) 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\[ H^{T}\left [ s_{t}\right ] =H^{T}\left [ H_{\phi _{t}}\left [ I_{t}\right ] +\mathbf {n}\right ] \] Since \(H^{T}\) is linear, the above becomes\[ H^{T}\left [ s_{t}\right ] =H^{T}\left [ H_{\phi _{t}}\left [ I_{t}\right ] \right ] +H^{T}\left [ \mathbf {n}\right ] \] Pre multiply the above with \(I_{t}\)\[ I_{t}\ H^{T}\left [ s_{t}\right ] =I_{t}\ H^{T}\left [ H_{\phi _{t}}\left [ I_{t}\right ] \right ] +I_{t}\ H^{T}\left [ \mathbf {n}\right ] \] Divide both side by \(H^{T}\left [ H_{\phi _{t}}\left [ I_{t}\right ] \right ] \)\[ \frac {I_{t}H^{T}\left [ s_{t}\right ] }{H^{T}\left [ H_{\phi _{t}}\left [ I_{t}\right ] \right ] }=\frac {I_{t}H^{T}\left [ H_{\phi _{t}}\left [ I_{t}\right ] \right ] }{H^{T}\left [ H_{\phi _{t}}\left [ I_{t}\right ] \right ] }+\frac {I_{t}H^{T}\left [ \mathbf {n}\right ] }{H^{T}\left [ H_{\phi _{t}}\left [ I_{t}\right ] \right ] }\] Under the condition that noise vector can be ignored the above becomes (after canceling out the \(H^{T}\left [ H_{\phi _{t}}\left [ I_{t}\right ] \right ] \) terms)\[ \frac {I_{t}H^{T}\left [ s_{t}\right ] }{H^{T}\left [ H_{\phi _{t}}\left [ I_{t}\right ] \right ] }=I_{t}\] Or\[ I_{t}=I_{t}\left ( \frac {H^{T}\left [ s_{t}\right ] }{H^{T}\left [ H_{\phi _{t}}\left [ I_{t}\right ] \right ] }\right ) \] 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 {equation} I_{t}=C\left ( \frac {H^{T}\left [ s_{t}\right ] }{H^{T}\left [ H_{\phi _{t}}\left [ C\right ] \right ] }\right ) \tag {1} \end {equation} 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 \(H^{T}\left [ H_{\phi _{t}}\left [ C\right ] \right ] \) is the unﬁltered backprojection of the projection \(H_{\phi _{t}}\left [ C\right ] \), which is the term \(P_{c_{t}}\) in the last section. Hence we see that (1) is the same equation as \begin {equation} I_{t}=C\frac {P_{t}}{P_{c_{t}}}\tag {2} \end {equation} 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.