Wavelets are used extensively in image processing due to the localized frequency information that can be conveyed by the wavelet transform. This and other characteristics of wavelet transforms can be exploited very effectively for the compression of images. We apply the wavelet transform to digital holograms of three-dimensional objects. Our digital holograms are complex-valued signals captured using phase-shift interferometry. Speckle gives them a white noise-like appearance with little correlation between neighboring pixels. In our analyses we concentrate on the discrete wavelet transform and Haar dyadic bases. We achieve compression through quantization of the wavelet transform coefficients. We quantize the discrete wavelet coefficients in each of the subbands depending on the dynamic range of the coefficients in that subband. Finally, we losslessly encode these subbands to quantify the high compression ratios achieved. We outline the three issues that need to be dealt with in order to improve the compression ratio of wavelet based techniques for particular applications as (i) determining a good criterion for ascertaining the coefficients that have to be retained, (ii) determining a quantization strategy and quantization error appropriate to one's particular application, and (iii) compression of the bookkeeping data.