SPECT Reconstruction

 

Image reconstruction is the process of transforming a set of 2D projections into a 3D image.

 

Some of the more common algorithms for image reconstruction are:

 

1.     Simple Backprojection

2.     Filtered Backprojection

3.     Fourier Transform Reconstruction

4.     Classical Iterative Reconstruction

·         LSIT (Least Squares Iterative Techniques)

·         ART (Algebraic Reconstruction Techniques)

·         SIRT (Simultaneous Iterative Reconstruction Techniques)

·         Gradient and conjugate gradient

5.     New Iterative Reconstruction

·         Maximum Entropy

·         Maximum Likelihood (e.g. Maximum Likelihood-Expectation Maximization - ML-EM)

·         OSEM (Ordered Subset Expectation Maximization)

 

Simple BackProjection Reconstruction

 

The simpliest image reconstruction method.

 

It makes no assumptions about the form of the image before resonstruction.

 

 

 

 

 

 


 


 

 

 


The major problem with simple backprojection reconstruction is that it leaves "extra" counts on the image in the wrong places.

 

·         The "Star" or "Spoke" pattern

 

 

Filtered Backprojection

 

·         Used to remove the "star" or "spoke" artifact from the simple backprojection reconstructed images.

 

·         Can be done before backprojection (pre-filtering) or after backprojection (post-filtering).

 

·         Is usually done in the frequency space domain (to be discussed later).

 

·         However, it can also be done in the spatial domain by a process called convolution.

 

 

 

 


 

 


 


Fourier Transformation Reconstruction

 

The Fourier Transform (FT) is a mathematical operation that changes a projection's data from being a function of counts per pixel to a completely equivalent function of amplitude versus cycles/pixel.

 

This transformed projection is in what is called the "spatial frequency domain".

 

 

 

 

 

 

The reconstruction produces the same image as the filtered backprojection.

 

 

 

 

Iterative Reconstruction Methods

 

 

This method of reconstruction involves solving a set of algebraic equations to reconstruct the image.

 

This method is very computer intensive and until recently the computer power needed to perform this reconstruction hadn't been available in the clinic.

 

 

The general process is as follows:

 

1.     The computer makes and initial "guess" at the form of the reconstructed image and creates an initial "guess" image.

2.     The initial "guess" image matrix is re-projected along the original projections.

3.     The projection of the "guess" image is compared to the real projection data.

4.     The counts/pixel in the image matrix are adjusted until the projection agrees with the real projection data.

5.     Steps 2-4 are repeated for each angle and projection.

6.     The process stops when most of the projections of the image are close to the values of the original projection data (i.e. called convergence).

 

 

 

 

 

Filtered Back-Projection versus Iterative Reconstruction Methods

 

 

Filtered Back-Projection Reconstruction:

·        Resultant Images are reconstructed from direct calculations from collected “projections” of activity.

·        Assumes no “attenuation” of activity

·        Has no mechanism for attenuation corrections.

 

Iterative Reconstruction Methods:

·        Has the ability to model the physical processes of image acquisition.

·        Can correct some of the factors that degrade images in filtered back projection

·        Can correct for attenuation of activity

·        Can correct for depth dependent blurring

·        Can correct for scatter effects

 

 

Some of the most commonly used iterative approaches in SPECT are:

          Early Methods:

·         ART (Algebraic Reconstruction Techniques)

          One of the earliest and simplest methods.

·         Gradient method

Improves the convergence rate by using a more "intelligent" updating of the guess matrix after comparison with the projection data.

·         Conjugate gradient method

                             An improvement on the gradient method

 

          Modern Methods:

·         Maximum Likelihood (e.g. Maximum Likelihood-Expectation Maximization - ML-EM)

Very popular method with good results but computationally very slow and has a slow convergence rate.

·         OSEM (Ordered Subset Expectation Maximization)

                             Newer and Faster than the ML-EM method

 

The main differences between these methods involve:

1.     The choice of the initial "guess" image.

2.     The way the image matrix is changed after comparison to the projection data.

3.     How the approximated image is tested for "convergence" (i.e. how the program determines whether it is close enough to the projection data to stop the iterations.)

 

A Simple Example of the Algebraic Iterative Reconstruction.

 

Lets take this simple 2x2 matrix of data (our "true" image):

 

0

2

1

3

 

Now, let's calculate the simplest projections for this data set:

 

                                                                                                                             1        5       

2
 
0

2
 
2

4
 
1

4
 
3

                                                                                                                             1        5

 

So, if we actually collected this SPECT image, we'd be collecting this projection data and wouldn’t know what the "true" image was.  So, this would be our actual starting point:

 

                                                                                                                             1        5       

2
 
?

2
 
?

4
 
?

4
 
?

                                                                                                                             1        5

The computer will first calculate an initial estimate image.  For this example we will take the projection total of 6 and divide that evenly over the 4 pixels:

1.5

1.5

1.5

1.5

                                                         

Next, the computer calculates this estimated image's projections along one of the axes for this estimate image:

1.5

3
 
1.5

1.5

3
 
1.5

                                                         

Then, the computer compares this projection from the estimated image to the actual projection data for that same projection from the real data:

3
 
2

3
 
4

 

The difference between these two "projections"        is what is used to correct the estimate image:

2 - 3 = -1  => -1/2 = -0.5 per pixel
 

3
 
2

4 - 3 =  1  =>  1/2 = +0.5 per pixel
 

3
 
4

 

So, the second "estimated" image or "corrected estimate image" will be:

1

1

2

2

                                                         

The computer next calculates the projections of columns on this second estimated image:

1

1

2

2

                                                                                                                             3        3

Then the computer compares these projections from the estimated image with the actual collected projection data:

                                                                                                                             3        3

1

5

                                     

The computer calculates the same differences and uses this information to correct the estimated image just like before.

 

So, the 3rd estimated image will be:

0

2

1

3

 

The computer now goes through this entire process again by starting on the projections of the rows and then the columns until there are no (or minimal) differences between the estimated data projections and the true projection data that was collected.

 

 

Example of final checks of the projections:

 

  0

2
 
2

1

4
 
3

                                                         

Then, the computer compares this projection from the estimated image to the actual projection data for that same axis:

2
 
2

4
 
4

 

The difference between these two "projections" is what is used to correct the estimate image:

2 - 2 = 0 

 

2
 
2

4 - 4 =  0

 

4
 
4

 

The same can be done for the projections along the columns.

 

 

This has been a very simplified example.  However, this same process is done for lots and lots of projections at various angles through the sample.

 

 

Note:          Only recently have computers been powerful and quick enough to do this in a clinical setting.