The reconstruction problem is ill-posed because it does not satisfy one or more of the three conditions for a well-posed problem: existence, uniqueness, and stability of the solution. For example, in CT imaging, if the number of projections is insufficient or the angular range is incomplete, then there may be no solution or multiple solutions that fit the measurements. Moreover, even if there is a unique solution, it may be highly sensitive to noise or errors in the measurements, leading to artifacts or distortions in the reconstructed image.

To resolve this problem in the computer, regularization and approximation methods are used. These methods introduce some prior information or constraints on the solution space to reduce its dimensionality and complexity1. For example, one can assume that the solution has some smoothness properties or sparsity patterns that can be enforced by adding a regularization term to the cost function2. Alternatively, one can approximate the inverse operator by using a truncated singular value decomposition (SVD) or a generalized inverse matrix that reduces its condition number1.

The resolution limits of image reconstruction depend on several factors, such as measurement noise level, sampling rate, detector size, and geometry. One way to derive these limits is to use Fourier analysis and examine how different spatial frequencies are affected by these factors. For example, in CT imaging, one can use the Radon transform and Fourier slice theorem to relate projections and images in the frequency domain. Then one can estimate how much information is lost due to noise filtering or limited angular range by looking at how different frequency components are attenuated or missing. Similarly, in MRI imaging, one can use k-space analysis and Nyquist theorem to relate Fourier coefficients and images in frequency domain. Then one can estimate how much information is lost due to undersampling or aliasing by looking at how different frequency components are distorted or overlapped.
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