True 3D reconstruction in digital holography

We examine the nature of the reconstructed 3D image as obtained by replay (or back-propagation) of the object wave from the hologram recording plane to the original object volume. While recording of a hologram involves transferring information from a 3D volume to a 2D detector, the replay of the hologram involves creating information in a set of 3D voxels from a much smaller number of 2D detector pixels, which on a first look appears to be surprising. We point out that the hologram replay process is a Hermitian transpose (and not inverse) of the hologram formation process and therefore only provides an approximation to the original 3D object function. With the knowledge of this Hermitian transpose property, we show how one may realize true 3D image reconstruction via a regularized optimization algorithm. The numerical illustrations of this optimization approach as presented here show excellent slice-by-slice tomographic 3D reconstruction of the original object under the weak scattering approximation. In particular, the reconstructed 3D image field has near-zero numerical values at voxels where the original object did not exist. We note that 3D image reconstruction of this kind cannot be achieved by the traditional physical replay process. In this sense, the proposed methodology for digital holographic image reconstruction goes beyond numerically mimicking the physical process involved in traditional film based holographic replay. The reconstruction approach may find potential applications in a number of digital holographic imaging systems.