Back Projection Reconstruction for CT, MRI and Nuclear Medicine

1. Back Projection Reconstructionfor CT, MRI and Nuclear Medicine F33AB5

2. CT collects Projections

3. Introduction • Coordinate systems • Crude BPR • Iterative reconstruction • Fourier Transforms • Central Section Theorem • Direct Fourier Reconstruction • Filtered Reconstruction

4. To produce an image the projections are back projected

5. Crude back projection • Add up the effect of spreading each projection back across the image space. • This assumes equal probability that the object contributing to a point on the projection lay at any point along the ray producing that point. • This results in a blurred image.

7. Crude v filtered BPR 90 360 Crude BPR Filtered BPR

8. Sinograms r r q Stack up projections

9. Solutions • Two competitive techniques • Iterative reconstruction • better where signal to noise ratio is poor • Filtered BPR • faster • Explained by Brooks and di Chiro in Phys. Med. Biol. 21(5) 689-732 1976.

10. Coordinate system • Data collected as series of • parallel rays, at position r, • across projection at angle . • This is repeated for various angles of .

11. Attenuation of ray along a projection • Attenuation occurs exponentially in tissue. • (x) is the attenuation coefficient at position x along the ray path.

12. Definition of a projection • Attenuation of a ray at position r, on the projection at angle , is given by a line integral. • s is distance along the ray, at position r across the projection at angle .

13. Coordinate systems • (x,y) and (r,s) describe the distribution of attenuation coefficients in 2 coordinate systems related by . • where i =1..M for M different projection orientations • angular increment is  = /M.

14. Crude back projection • Simply sum effects of back-projected rays from each projection, at each point in the image.

15. Crude back projection • After crude back projection, the resulting image, *(x,y), is convolution of the object ((x,y)) with a 1/r function.

16. Convolution • Mathematical description of smearing. • Imagine moving a camera during an exposure. Every point on the object would now be represented by a series of points on the film: the image has been convolved with a function related to the motion of the camera 

17. Iterative Technique • Guess at a simulated object on a PxQ grid (j, where j=1PxQ), • Use this to produce simulated projections • Compare simulated projections to measured projections • Systematically vary simulated object until new simulated projections look like the measured ones.

18. For your scanner calculate jj(r,i), the path length through the jth voxel for the ray at (r,i) • j need only be estimated once at the start of the reconstruction, • j is zero for most pixels for a given ray in a projection 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 j=0 2=0.1 7=1.2

19. The simulated projections are given by: • j is mean simulated attenuation coefficient in the jth voxel.

20. 1 2 3 8 9 4 7 6 5 10 6 9 19 15 15 10 13 14 6 2 2 2 7 7 21 7 18 6 6 6 16 17 2 15 15 15 Object and projections First ‘guess’ From Physics of Medical Imaging by Webb

21. To solve • Analytically, construct P x Q simultaneous equations putting (r,i) equal to the measured projections, p(r,i): • this produces a huge number of equations • image noise means that the solution is not exact and the problem is 'ill posed’ • Instead iterate: modify j until (r,i) looks like the real projection p(r,i).

22. Iterating • Initially estimate j by projecting data in projection at  = 0 into rows, or even simply by making whole image grey. • Calculate (r,i) for each i in turn. • For each value of r and , calculate the difference between (r,) and p(r,). • Modify i by sharing difference equally between all pixels contributing to ray.

23. 1 1 2 2 3 3 8 8 9 9 4 4 7 7 6 6 5 5 10 6 10 19 15 15 15 10 13 122/3 14 6 2 2 22/3 1 2 21/3 7 7 6 72/3 21 7 71/3 18 6 6 62/3 5 6 61/3 16 17 2 15 15 15 16 17 12 Object First ‘guess’ Next iteration

24. Fourier Transforms • Imagine a note played by a flute. • It contains a mixture of many frequency sound waves (different pitched sounds) • Record the sound (to get a signal that varies in time) • Fourier Transforming this signal will give the frequencies contained in the sound (spectrum) Time Frequency

25. y ky FT x kx Fourier transforms of images • A diffraction pattern is the Fourier transform of the slit giving rise to it

26. y ky FT x kx Central Section theorem • The 1D Fourier transform of a projection through an object is the same as a particular line through 2DFT of the object. • This particular line lies along the conjugate of the r axis of the relevant projection. Projection

27. Direct Fourier Reconstruction • Fourier Transform of each projection can be used to fill Fourier space description of object.

28. Direct Fourier Reconstruction • BUT this fills in Fourier space with more data near the centre. • Must interpolate data in Fourier space back to rectangular grid before inverse Fourier transform, which is slow.

29. Relationship between object and crude BPR results • Crude back projection from above: • Defining inverse transform of projection as: • then

30. The right hand side has been multiplied and divided by k so that it has the form of a 2DFT in polar coordinates • k conjugate to r • k conjugate to r • the integrating factor is kdrd dxdy

31. Crude back projected image is same as the true image, except Fourier amplitudes have been multiplied by (magnitude of spatial frequency)-1. • Physically because of spherical sampling. • Mathematically because of changes in coordimates.

32. Filtered BPR • Multiplying 2 functions together is equivalent to convolving the Fourier Transforms of the functions. • Fourier transform of (1/k) is (1/r) • Multiplying FT of image with 1/k is same as convolving real image with 1/r • ie BPR has effect we supposed.

33. Filtered BPR • Therefore there are two possible approaches to deblurring the crude BPR images: • Deconvolve multiplying by f (1/f x f = 1) in Fourier domain. • Convolve with Radon filter in the image domain, to overcome effect of being filtered with 1/r by crude BPR.