Bush__The_Essential_Physics_for_Medical_Imaging - Biomedical ...

esents an individual measurement of 11. In addition to the value of 11 for each ray,the reconstruction algorithm also "knows" the acquisition angle and position in thedetector array corresponding to each ray. Simple backprojection starts with anempty image matrix (an image with all pixels set to zero), and the 11 value from eachray in all views is smeared or backprojected onto the image matrix. In other words,the value of 11 is added to each pixel in a line through the image corresponding tothe ray's path.Simple backprojection comes very close to reconstructing the CT image asdesired. However, a characteristic IIr blurring (see latet discussion) is a byproductof simple backprojection. Imagine that a thin wire is imaged by a CT scannerperpendicular to the image plane; this should ideally result in a small pointon the image. Rays not running through the wire will contribute little to theimage (11 = 0). The backprojected rays, which do run through the wire, will convergeat the position of the wire in the image plane, but these projections runfrom one edge of the reconstruction circle to the other. These projections (i.e.,lines) will therefore "radiate" geometrically in all directions away from a pointinput. If the image gray scale is measured as a function of distance away from thecenter of the wire, it will gradually diminish with a 1/r dependency, where r isthe distance away from the point. This phenomenon results in a blurred imageof the actual object (Fig. 13-28) when simple backprojection is used. A filteringstep is therefore added to correct this blurring, in a process known as filteredbackprojection.Filtered Backprojection ReconstructionIn filtered backprojection, the raw view data are mathematically filtered beforebeing backprojected onto the image matrix. The filtering step mathematicallyreverses the image blurring, restoring the image to an accurate representation of theobject that was scanned. The mathematical filtering step involves convolving theprojection data with a convolution kernel. Many convolution kernels exist, and differentkernels are used for varying clinical applications such as soft tissue imagingor bone imaging. A typical convolution kernel is shown in Fig. 13-28. The kernelrefers to the shape of the filter function in the spatial domain, whereas it is commonto perform (and to think of) the filtering step in the frequency domain. Muchof the nomenclature concerning filtered backprojection involves an understandingof the frequency domain (which was discussed in Chapter 10). The Fourier transform(FT) is used to convert a function expressed in the spatial domain (millimeters)into the spatial frequency domain (cycles per millimeter, sometimes expressedas mm- I ); the inverse Fourier transform (FT-I) is used to convert back. Convolutionis an integral calculus operation and is represented by the symbol ®. Let p(x)represent projection data (in the spatial domain) at a given angle (p(x) is just onehorizontal line from a sinogram; see Fig. 13-20), and let k(x) represent the spatialdomain kernel as shown in Fig. 13-28. The filtered data in the spatial domain,p' (x), is computed as follows:p' (x) = p(x) ® k(x)The difference between filtered backprojection and simple backprojection isthe mathematical filtering operation (convolution) shown in this equation above. Infiltered backprojection, pi (x) is backprojected onto the image matrix, whereas in