Non-linear Fresnelet approximation for interference term suppression in digital holography

We present a zero-order and twin image elimination algorithm for digital Fresnel holograms that were acquired in an off-axis geometry. These interference terms arise when the digital hologram is reconstructed and corrupt the result. O ur algorithm is based on the Fresnelet transform, a wavelet-like transform that uses basis functions tailor-made for digi tal holography. We show that in the Fresnelet domain, the coeffic ients associated to the interference terms are separated bo th spatially and with respect to the frequency bands. We propose a method to suppress them by selectively thresholding the Fresnelet coefficients. Unlike other methods that operate i n the Fourier domain and affect the whole spacial domain, our method operates locally in both space and frequency, allowing for a more targeted processing. When an object is illuminated with a coherent light source, the transmitted or reflected wave carries information on the sample's properties. In the close vicinity of the object, th e light intensity is related to its reflectance or attenuatio n while the phase is related to its thickness. Light sensors, such as CCDs, measure the intensity of the incoming light but are unable to capture its phase. This crucial information is therefore lost. From a mathematical point of view, the measurement of the wave is equivalent to evaluating the squared modulus of the complex scalar field in the acquisition plane, an operatio n which clearly discards the phase. Holography overcomes this limitation and makes it possible to record the whole information of the wavefront (ampli- tude and phase) for later restitution. The hologram measures the intensity of the object wave's interference with a refe rence wave. In the so-called off-axis geometry, the reference wave and object wave travel in slightly different directions gi ving rise to interference fringes. To reproduce the object wave, the chemically processed hologram is illuminated with a recon- struction beam which is diffracted. Three diffraction orde rs may be distinguished: the +1 order which is an exact replica of the object wave, the undiffracted zero-order, and the ¡1 order. In digital holography,1-3 the photographic plate is replaced by a CCD camera. The hologram is stored in the computer as a digital image and the reconstruction process is carried out by simulating the physical diffraction phenomenon. Since wave propagation can be modeled with good accuracy in the Fresnel régime by the Fresnel transform, it can be easily implemented. Digital holography's advantages are that it i s fast (digital holograms may be acquired at video rate) and that it does not involve any chemical processing of the holographic plate or tedious alignment of the reconstruction beam. But most important, quantitative measurements may be performed since the object wave's amplitude and phase are recon- structed digitally. However, since digital recording medi a have a lower resolution than those used in classical holography, the fringes spacing must be larger in order to be resolved. This means that the reference beam's angle cannot be as high. As a consequence, the three diffracted waves do, at least partially, overlap during reconstruction. So far, only algorithms have been proposed that either filter the relevant information in the Frequency domain,4-6 or, that take advantage of the spatial separation of the diff erent orders after propagation. However, neither approach is completely satisfactory, since either the reconstructed w ave's bandwidth or its field of view are drastically limited. Here, we derive a non-linear signal approximation algorithm that takes advantage of the interference terms' separation in bo th ¤ E-mail: michael.liebling AT epfl.ch, Tel.: +41 21 693 51 43, Fax.: +41 21 693 37 01.