Back Projection Reconstruction for CT, MRI and Nuclear Medicine F33AB5

CT collects Projections

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

To produce an image the projections are back projected

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.

Crude v filtered BPR 90 360 Crude BPR Filtered BPR

Sinograms r r q Stack up projections

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.

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

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

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 .

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.

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

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

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 

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.

j need only be estimated once at the start of the reconstruction, 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

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

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

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).

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.

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

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

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

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. kx ky y x FT Projection

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

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.

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

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

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.

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.

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.