How can projections help reconstruct an unseen interior?

An image of an interior can be reconstructed from measurements that never directly label its individual locations. Tomography works with projections: measurements combining contributions along paths through an object. The central mathematical problem is inversion, recovering a hidden spatial distribution from the measured combinations.

The illustration is a nonmedical two-by-two attenuation model with cell values one, two, three and four. Attenuation describes how a material reduces a transmitted beam. In an ideal transmission model, taking a logarithm of the intensity ratio converts multiplicative transmission into an additive integral of attenuation along the beam path. Thus a projection is a path-weighted sum, not generally the raw brightness reaching a detector.

Horizontal paths through the unit cells give row sums three and seven. Vertical paths give column sums four and six. Those measurements do not determine this grid uniquely: four nonnegative integer grids fit them. Writing the upper-left entry as a leaves the other entries as three minus a, four minus a, and three plus a, with a between zero and three.

An additional main-diagonal path crosses two cells, with length square root of two inside each cell. Dividing that measured projection by square root of two gives the diagonal sum five. Since that sum is three plus twice a, a must be one, resolving the illustrated grid.

Practical reconstruction uses many paths and accounts for noise, sampling, geometry and modeling assumptions. More measurements help only when they supply useful independent information. This small exact system exposes the relationship between projections and an inferred image; it is not a medical scan or guidance about imaging care.