#research/imaging/techniques #radiation/detection-and-imaging See also: [[Simple Backprojection (SBP)]] # Filtered Backprojection Motivation: applications of image reconstruction algorithms for [[Compton Imaging]]. Has applications in medical imaging as well! Filtered backprojection takes [[Simple Backprojection (SBP)]] many different backprojected images and applies a filter of some user's choice to remove artifacts to reconstruct an image. This model, for [[Compton Imaging]] for example, is given mathematically as: $\hat{\bf{f}} = \bf{B}^{-1} \hat{f}_{sbp} = \bf{B}^{-1}\bf{T}^ty = \bf{B}^{-1}\bf{T}^tTf = f $ where $\bf{B} = T^tT$ and represents the $J\times J$ [[Point Spread Function]] matrix of the system. Having a low number of projections (i.e. not enough data) in the sampling data can lead to artifacts. the pasted Figure (13.6) shows how too few angular measurements on the Shepp-Logan phantom for the brain can lead to artifacts as seen with the lower number of projection angles (180 vs. 20) ![[Pasted image 20220714112716.png]] ## Choice of Filter The choice of filter in a system is dependent on the application. In Medical Imaging, a choice of filter might include a high-pass filter or a low-pass filter to isolate specific frequencies of an image. The mechanics of this involve using a [[Convolution]] between the filtering function and the image being filtered. The process of choosing a filter is always changing, so there is merit to repeatedly returning to assess whether the chosen filter is useful or not. While CT often make use of a linear-ramp filter, this cannot be applied to gamma cameras, due to the $4\pi$ measurement space [^1]. [^1]:https://cztlab.engin.umich.edu/wp-content/uploads/sites/187/2015/03/20060907-thesis.pdf ## Some Strengths and Weaknesses of FBP - Strengths: - Fast calculations: Based on the Fast Fourier Transform (FFT) and a single simple back-projection measurement. There are therefore very few parameters to adjust. - Conceptually very simple. Take a [[Simple Backprojection (SBP)]] and apply a filter as you go. - Reconstruction behavior is well-understood - Typically works well for complete and "good" data. - Weaknesses - Large number of projections are required - Full angular range is required - Only a modest amount of noise can be tolerated. - Mandates a fixed scan of geometries. Other geometries require different inversion formulas - does not make use of any prior knowledge of the system, such as non-negativity, anatomical constraints (medical imaging), etc.