Image Boundary Extension With Mean Value for Cosine–Sine Modulated Lapped/Block Transforms

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We present a novel image boundary extension and mean value extension (MVE) for directional lapped transforms, particularly cosine–sine modulated lapped transforms (CSMLTs). Lapped transforms are usually used with an extension technique, such as periodic extension (PE) or symmetric extension (SE), for nonexpansive convolution at signal boundaries. When directional textures (oblique lines and curves) appear at the 2D signal (image) boundaries, both PE and SE produce directional discontinuities, which degrade the sparsity of the transformed coefficients, especially in the case of directional lapped transforms. MVE reduces the discontinuities for directional textures better than PE or SE do; it thus improves the efficiency of the sparse representation based on directional lapped transforms. Moreover, to reduce computational costs compared with those of directional lapped transforms, we introduce new directional block transforms called cosine–sine modulated block transforms. These new transforms are derived from a minimum tile processing (tiling) of $M$ -band CSMLTs with $2M$ filter lengths and nonexpansive convolutions. The resulting directional block transforms, particularly in the case of the MVE, have richer directional selectivity and have better performance compared with a discrete Fourier transform as shown in experiments.