Reconstructability

Reconstructability is the degree to which the original state, structure, or data of a system can be inferred from observed data, measurements, or partial information. It depends on identifiability (whether a unique reconstruction exists), the completeness and noise level of the observations, and the validity of the assumptions about the system. High reconstructability means a reconstruction mapping from observations to the original state is possible in a stable way; low reconstructability means multiple plausible reconstructions exist or small observation errors cause large changes in the inferred state.

In fields such as signal processing, imaging, statistics, and forensics, reconstructability governs what can be learned or recovered. For example, electromagnetic or acoustic signals can be reconstructed from samples or projections; medical CT and MRI rely on reconstruction algorithms; data can be reconstructed from compressed representations when sparsity or structure is present; in forensics, event sequences may be reconstructed from partial traces.

Assessing reconstructability uses identifiability analysis, information-theoretic limits, and algorithmic reconstruction theory. Tools include rank conditions, convexity and uniqueness proofs, Fisher information, Cramér–Rao bounds, and empirical validation. In compressed sensing, sparsity enables reliable reconstruction from far fewer measurements than traditional sampling. In tomography, inversion formulas recover images from projection data.

Limitations include non-uniqueness, sensitivity to noise, model misspecification, data corruption, and computational complexity. Reconstructability is often a trade-off between data availability, model fidelity, and computational resources. It is a consideration in data privacy as well, since higher reconstructability can increase the risk of exposing original information from processed data.

Overall, reconstructability quantifies the inferential potential of observed data and guides the design of data collection, experiments, and algorithms.