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Highly Constrained Image Reconstruction (HYPR)

Nasser M. Abbasi

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

Contents

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

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.
It  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 ϕ  radon transform used at a projection angle ϕ  .
8.
ϕt  When the projection angle ϕ  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
  ϕt   radon transform used at an angle ϕ
 t  .
10.
s = R ϕ[I]  . radon transform applied to an image I  at angle ϕ  . This results in a projection vector s  .
11.
H  Forward projection matrix. The Matrix equivalent to the radon transform R  .
12.
𝜃  Estimate of an image I  .
13.
H 𝜃  Multiply the forward projection matrix H  with an image estimate 𝜃  .
14.
g = H 𝜃  Multiply the forward projection matrix H  with an image estimate 𝜃  to obtain a projection vector g  .
15.
Ru [s]
  ϕ  The inverse radon transform applied in unfiltered mode to a projection s  which was taken at angle ϕ  . This results in a 2D image.
16.
  f
R ϕ[s]  The inverse radon transform applied in filtered mode to a projection s  which was taken at angle ϕ  . This results in a 2D image.
17.
HT 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 st  of the original images It  . Hence       N
     ∑    f
C =     R ϕti [sti]
     i=1
20.
Pt  The unfiltered backprojection 2D image as a result of applying Ruϕt [st]  where st  is projection from user image It  taken at angle ϕt  .
21.
Pct   The unfiltered backprojection 2D image as a result of applying Ruϕt [st]  where st  is projection from the composite image C  taken at angle ϕt  .
22.
Np  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 Np
24.
Jk  The kth  HYPR frame image. A 2-D image generate at the end of the HYPR algorithm. There will be as many HYPR frame images Jk  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

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 ϕt

st = R ϕt [It]

When the original object image does not change with time then we can drop the subscript t  from  I
  t  and just write st = R ϕt [I]

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

      N
     ∑    f
C  =    R ϕti [sti]
     i=1

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

sct = Rϕt [C ]

The the unfiltered back projection 2-D image Pt  is generated

Pt = Ruϕt [st]

And the unfiltered back projection 2-D image Pc
   t   is found

P  = Ru  [s ]
 ct    ϕt  ct

Then the ratio of Pt-
Pct   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 kth  time frame we obtain

pict

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 st  is obtained by applying radon transform R  on the image It  at some angle ϕt

st = R ϕt [It]

When the original object image does not change with time then we can drop the subscript t  from  It  and just write st = R ϕt [I]

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

      N
     ∑    f
C  =    R ϕti [sti]
     i=1

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

sct = Rϕt [C ]

The the unfiltered back projection 2-D image Pt  is generated

Pt = Ruϕt [st]

And the unfiltered back projection 2-D image Pct   is found

       u
Pct = R ϕt [sct]

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

pict

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  = H   [I] + n
 t     ϕt  t

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

Now apply the HT  operator to the above equation, we obtain

HT  [s] = HT  [H   [I ] + n]
      t         ϕt  t

Since HT  is linear, the above becomes

  T         T             T
H   [st] = H  [H ϕt [It]] + H [n ]

Pre multiply the above with It

It HT [st] = It HT [H ϕt [It]] + It HT [n ]

Divide both side by HT [H ϕt [It]]

--ItHT-[st]-    ItHT--[Hϕt-[It]]-  --ItHT--[n-]-
HT  [H   [I ]] = HT  [H   [I ]] + HT  [H   [I ]]
       ϕt t            ϕt t           ϕt  t

Under the condition that noise vector can be ignored the above becomes (after canceling out the HT  [Hϕt [It]]  terms)

     T
--ItH---[st]-- = It
HT  [H ϕt [It]]

or

       (             )
         --HT--[st]---
It = It  HT [H   [I]]
               ϕt  t

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

       (     T       )
It = C   --H---[st]---
         HT [H ϕt [C]]
(1)

But HT  [st]  is the unfiltered backprojection of the projection st  , hence this term represents the term      Pt  shown in the last section, which is the unfiltered backprojection 2D image, and HT  [H ϕt [C ]]  is the unfiltered backprojection of the projection H ϕ [C ]
   t  , which is the term Pc
  t   in the last section. Hence we see that (1) is the same equation as

It = C Pt-
       Pct
(2)

Once It  is computed from (1), we can repeat (1) again, using this computed It  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