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

1 Terminologies

2 Introduction

3 Review of HYPR

3.1 High level schematic of original HYPR with many projections per time frame

4 HYPR mathematical formulation

4.1 Original HYPR

4.1.1 Schematic diagram of original HYPR algorithm

4.2 Wright HYPR

4.2.1 Schematic diagram of Wright HYPR algorithm

4.2.2 On the diﬀerence between original HYPR and Wright-HYPR

4.3 I-HYPR

4.3.1 Schematic diagram of I-HYPR algorithm

5 WI-HYPR (Iiterative HYPR based on Wright-HYPR ﬂow)

5.1 Schematic diagram of WI-HYPR

2 Introduction

3 Review of HYPR

3.1 High level schematic of original HYPR with many projections per time frame

4 HYPR mathematical formulation

4.1 Original HYPR

4.1.1 Schematic diagram of original HYPR algorithm

4.2 Wright HYPR

4.2.1 Schematic diagram of Wright HYPR algorithm

4.2.2 On the diﬀerence between original HYPR and Wright-HYPR

4.3 I-HYPR

4.3.1 Schematic diagram of I-HYPR algorithm

5 WI-HYPR (Iiterative HYPR based on Wright-HYPR ﬂow)

5.1 Schematic diagram of WI-HYPR

Let be a 2D image. Let be the forward projection matrix (implemented as Radon transformation in practice) which when applied to the image at some projection angle at time will result in a 1D projection image .

Since the image itself will change with time (blood ﬂows, etc...) therefore we need to associated a time index as well with the 2D image itself. Hence we write from now on, to indicate the 2D image at time . Notice that the image as a whole does not move (relative to ﬁxed initerial frame of reference) but the image content can change as described above.

To avoid confusion in what follows, I will use the notation of to indicate a multiplication between 2 matrices element by element and will use to indicate the normal matrix by matrix or matrix by vector multiplication. And will use to indicate division between vectors or matrices being done element by element.

We will analyze the following HYPR algorithms: Original HYPR, Wright-Hyper, I-HYPR and a new variation of I-HYPR based on on the Wright-HYPR. This new variation is diﬀerent from I-HYPR since in I-HYPR the iteration is made on a composite image based on the Original HYPR construction while in this variation, the composite image is constructed is with the Write-HYPR algorithm. We will call this variation IW-HYPR.

For each algorithm we show the schematic ﬂow chart and the mathematical description based on matrix formulation and an algorithm for the implementation. At the end, we will run a simulation of each of the above 4 algorithm from the same set of images and attempt to describe the ﬁnding and compare the results.

In the mathematical formula we derive an expression of the HYPR image as a function of the set of original images and the set of the forward projection matrices .

Before we discuss the Mathematical formulation, we need to better undertstand how to generate a set of projections from a well deﬁned image which we can express mathematically. For this purpose the following diagram shows the 3 possible cases in which projections can occure at.

We need to generate an analytical expression as a function of time of a simple object shape for each of the following 3 cases.

Given a set of projections over a number of time frames say and assuming there are projections generated per a time frame, the data set will consists of projections in total. Within one time frame, a number of projections can be collected. These projections within each time frame will generate one HYPR frame using the HYPR algorithm. A time frame can have only one projection, but normally a time frame will have much larger number of projections.

For example, assuming there are time frames, and projections generated by time frame, then the total number of projections is In this case, we will obtain HYPR frame images at the end.

HYPR starts by constructing a composite image from all the projections in the data set ( in the above example). This is done by computing the ﬁltered backprojection of all the projections into one image called .

Next, we process the projections from each time frame. For each time frame we generate a set of projections from by perform a radon tranform (forward projection) on at the same angle corresponding to the projection being processed. This will generate set of projections called the projections. Hence there will be such projections per each time time.

Next the ratio of the each projection over the corresponding projection is found. This ratio is done pixel by pixel. Then each such ratio is multiplied by the composite image generating a a new set of size of weighted composite images. Now to generate a HYPR frame image, the average of these weighted compsite images is taken. The average is carried over each time frame at a time. Hence the ﬁrst weighted compsite images will generate one HYPR frame, and the next weighted compsite images will generate the next HYPR frame and so on, resulting in HYPR frames.

The following diagram illustrates the above where we used the original HYPR algorithm. Later on, we describe in detailes the diﬀerent HYPR algorithms variations. In the following diagram we show 4 projections per time frame and a total of 3 time frames. This results in 3 HYPR frames.

The projection is obtained by applying forward projection on the image , Hence we write

Next, the set of are combined and a ﬁltered backprojection is applied to the result to generate a composite image

Where is operator for the ﬁltered backprojection.

The composite image can be written as

Where in the number of projections. Now applying forward projection to at angle will generate a projection Hence

Let be the ratio where this division is being carried out element by element between the two projections.

Hence

Now apply unﬁltered backprojection on the above projection ratio to obtain an unﬁltered 2D image and then mutiply that with the composit image to obtain a HYPR frame

Hence HYPR image is

We see that the above expression for HYPR image depends only on and .

Therefore, given a set of images and a set of projection angles we can compute the forward projection matrices (analytically we can do this for simple shapes such as a disk rotating around a unit circle for example). And once the set of matrices is computed, equation (1) could then be computed to obtain a HYPR image.

We can then generate the same HYPR images by performing backprojection (ﬁltered an unﬁltered) using the Fourier transform method. The algorithm for the backprojection (ﬁltered) is known and given on page 62 of Kak and Slaney book which I will post a scan of. Or we could simply use the radon/iradon for the implementaion of ,, where corresponds to applying radon on image at angle and corresponds to applying iradon on with ﬁlter 'none' and corresponds to applying iradon on projection with a speciﬁed ﬁlter.

The projection is obtained by applying forward projection on the image at time Hence we write

Where in the above the 2D image needs to be ﬁrst converted to a 1D vector as was done in the ﬁrst assignment by folding the 2D image into a 1D column vector.

Let be an psudoinverse of which performs a ﬁltered backprojection when applied on a projection vector and let be the transpose of the matrix which performs an unﬁltered backprojection on .

The composite image can be written as

where in the number of projections. Now, each unﬁltered backprojection is

Now applying forward projection to at angle will generate a projection, call it

Hence

Now applying an unﬁltered backprojection on will result in hence

Let be the ratio where this division is being carried out element by element between the two images.

Hence

Hence one HYPR frame is

Hence HYPR image is

The diﬀerence between the original HYPR and Wright-HYPR can be seen in the following simpliﬁed diagram. We see than in the original HYPR, the ratio is performed on the 1-D projections, then the unﬁltered backprojection is applied on the resulting 1-D set of images. In the Wright-HYPR algorithm, the unﬁltered backprojection is ﬁrst applied to the set of the 1-D projections and then the ratio is performed on the result 2-D set of images.

This is an iterative method where the composite image itself is updated and a new HYPR image determined with the hope of obtaining a better HYPR image (how to determine how many iterations? need to read more on this) as more iterations are performed. Let the number of iterations by therefore this algorithm will generate composite images.

This is the mathematical formulation for I-HYPR

The projection is obtained by applying forward projection on the image at time In the original HYPR, is what is refered to projection. Hence we write

The composite image at iteration (1) can be written as

Where in the number of projections. Now applying forward projection to at angle will generate a projection

Hence

Let be the ratio where this division is being carried out element by element between the two projections.

Hence

Now apply unﬁltered backprojection on the above projection ratio to obtain an unﬁltered 2D image and then mutiply that with the composit image to obtain a HYPR frame

Hence HYPR image at interation (1) is

Now use as the composite image for the next iteration, we obtain

Hence

and

Now apply unﬁltered backprojection on the above projection ratio to obtain an unﬁltered 2D image and then mutiply that with the composit image to obtain a HYPR frame

Hence HYPR image at interation (2) is

But hence the above becomes

But , hence the above becomes

Repeate this processes by setting and generate . Continue untill where is number of iterations or untill some other convergence criteria is achived.