We have developed a technique to map the three-dimensional structure of the local interstellar medium using a maximum entropy reconstruction technique. A set of column densities N to stars of known distance can in principle be used to recover a three-dimensional density field n, since the two quantities are related by simple geometry through the equation N = C n, where C is a matrix characterizing the stellar spatial distribution. In practice, however, there is an infinte number of solutions to this equation. We use a maximum entropy reconstruction algorithm to find the density field containing the least information which is consistent with the observations. The solution obtained with this technique is, in some sense, the model containing the minimum structure. We apply the algorithm to several simulated data sets to demonstrate its feasibility and success at recovering ``real density contrasts. This technique can be applied to any set of column densities whose end points are specified. In a subsequent paper we shall describe the application of this method to a set of stellar color excesses to derive a map of the dust distribution, and to soft X-ray absorption columns to hot stars to derive a map of the total density of the interstellar medium.
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