# Tradeoffs between measurement residual and reconstruction error in inverse problems with prior information

## Abstract

In many inverse problems with prior information, the measurement residual and the reconstruction error are two natural metrics for reconstruction quality, where the measurement residual is defined as the weighted sum of the squared differences between the data actually measured and the data predicted by the reconstructed model, and the reconstruction error is defined as the sum of the squared differences between the reconstruction and the truth, averaged over some a priori probability space of possible solutions. A reconstruction method that minimizes only one of these cost functions may produce unacceptable results on the other. This paper develops reconstruction methods that control both residual and error, achieving the minimum residual for any fixed error or vice versa. These jointly optimal estimators can be obtained by minimizing a weighted sum of the residual and the error; the weights are determined by the slope of the tradeoff curve at the desired point and may be determined iteratively. These results generalize to other cost functions, provided that the cost functions are quadratic and have unique minimizers; some results are obtained under the weaker assumption that the cost functions are convex. This paper applies these results to a model problem from biomagnetic source imagingmore »

- Authors:

- Lawrence Berkeley Lab., CA (United States). Life Sciences Div.

- Publication Date:

- Research Org.:
- Lawrence Berkeley Lab., CA (United States)

- Sponsoring Org.:
- USDOE, Washington, DC (United States)

- OSTI Identifier:
- 106621

- Report Number(s):
- LBL-37402; CONF-9507171-1

ON: DE96000120; TRN: AHC29525%%122

- DOE Contract Number:
- AC03-76SF00098

- Resource Type:
- Technical Report

- Resource Relation:
- Conference: Experimental and numerical methods for solving ill-posed problems, San Diego, CA (United States), 9-14 Jul 1995; Other Information: PBD: Jun 1995

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 55 BIOLOGY AND MEDICINE, BASIC STUDIES; BRAIN; IMAGE PROCESSING; DIAGNOSTIC TECHNIQUES; THEORETICAL DATA; MAGNETIC FIELDS; ERRORS; PROBABILISTIC ESTIMATION; WEIGHTING FUNCTIONS

### Citation Formats

```
Hughett, P.
```*Tradeoffs between measurement residual and reconstruction error in inverse problems with prior information*. United States: N. p., 1995.
Web. doi:10.2172/106621.

```
Hughett, P.
```*Tradeoffs between measurement residual and reconstruction error in inverse problems with prior information*. United States. doi:10.2172/106621.

```
Hughett, P. Thu .
"Tradeoffs between measurement residual and reconstruction error in inverse problems with prior information". United States.
doi:10.2172/106621. https://www.osti.gov/servlets/purl/106621.
```

```
@article{osti_106621,
```

title = {Tradeoffs between measurement residual and reconstruction error in inverse problems with prior information},

author = {Hughett, P.},

abstractNote = {In many inverse problems with prior information, the measurement residual and the reconstruction error are two natural metrics for reconstruction quality, where the measurement residual is defined as the weighted sum of the squared differences between the data actually measured and the data predicted by the reconstructed model, and the reconstruction error is defined as the sum of the squared differences between the reconstruction and the truth, averaged over some a priori probability space of possible solutions. A reconstruction method that minimizes only one of these cost functions may produce unacceptable results on the other. This paper develops reconstruction methods that control both residual and error, achieving the minimum residual for any fixed error or vice versa. These jointly optimal estimators can be obtained by minimizing a weighted sum of the residual and the error; the weights are determined by the slope of the tradeoff curve at the desired point and may be determined iteratively. These results generalize to other cost functions, provided that the cost functions are quadratic and have unique minimizers; some results are obtained under the weaker assumption that the cost functions are convex. This paper applies these results to a model problem from biomagnetic source imaging and exhibits the tradeoff curve for this problem.},

doi = {10.2172/106621},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Thu Jun 01 00:00:00 EDT 1995},

month = {Thu Jun 01 00:00:00 EDT 1995}

}