The standard separable two-dimensional (2-D) wavelet transform (WT) has recently achieved a great success
in image processing because it provides a sparse representation of smooth images. However, it fails to capture
efficiently one-dimensional (1-D) discontinuities, like edges or contours. These features, being elongated and
characterized by geometrical regularity along different directions, intersect and generate many large magnitude
wavelet coefficients. Since contours are very important elements in visual perception of images, to provide a
good visual quality of compressed images, it is fundamental to preserve good reconstruction of these directional
features. We propose a construction of critically sampled perfect reconstruction transforms with directional
vanishing moments (DVMs) imposed in the corresponding basis functions along different directions, called directionlets.
We also demonstrate the outperforming non-linear approximation (NLA) results achieved by our transforms and we show how to design and implement a novel efficient space-frequency quantization (SFQ) compression algorithm using directionlets. Our new compression method beats the standard SFQ both in terms of mean-square-error (MSE) and visual quality, especially in the low-rate compression regime. We also show that our compression method, does not increase the order of computational complexity as compared to the standard SFQ algorithm.