Introduction

The principle of digital holography is to illuminate a sample (by transmission or reflection) and to record the interference between the wave emitted by the object (the object wave) and a reference wave. The problem is to reconstruct a complex image (intensity and phase) of the object wave near the sample from the digital hologram.

Main Contribution

We have formalized the acquisition process and decomposed the inverse problem into two partial problems that can be solved independently: the first one, nonlinear, is the determination of the complex wave in the hologram plane; the second one, linear, is the (inverse) propagation of this wave to determine the complex image in the object plane (inverse Fresnel transform).

We have made a mathematical study of the Fresnel transform (uncertainty relations, invariances) and proposed a new class of wavelets that are optimally adapted to this transformation—the Fresnelet basis. The Fresnelet transform allows us to improve the reconstructed image quality, to remove artifacts (zero order), and also to find the optimal focusing distance.