CTsynth

The problem

A forward model predicts measurements from a material distribution. The inverse problem asks which material distributions are consistent with the measurements. These questions are related, but a differentiable simulator does not automatically make the inverse problem well posed.

The mathematical idea

A schematic spectral transmission model integrates energy-dependent attenuation along a ray:

Here $S(E)$ represents the incident spectrum and detector weighting, and $\mu(x,E)$ is the attenuation field. This expression illustrates the modeling question; it is not a specification of every implementation detail.

Implementation and validation

CTsynth explores differentiable spectral CT physics, forward projection, numerical adjoints, and material decomposition. An important numerical check compares the forward operator and its adjoint through the inner-product identity $\langle Ax,y\rangle=\langle x,A^*y\rangle$. Gradient checks and decomposition benchmarks address different parts of the problem.

Identifiability and limitations

A small measurement residual need not imply correct material recovery. Spectral information, measurement geometry, noise, and the chosen material basis all affect what can be recovered. Benchmark values and experiment figures are not included here because validated artifacts have not yet been supplied.

Next questions

Which ambiguities arise from the physics, and which come from discretization or optimization? Which additional measurements would distinguish otherwise similar material configurations?