Transform coding is widely used in signal and image compression today. The main idea behind transform coding is to find a new basis for an image, and store the coefficients of these basis vectors, instead of the image itself. The basis vectors should be chosen to maximize energy compactification of the coefficients. The high energy coefficients are quantized finely, while the low energy coefficients are quantized coarsely. A simple Parseval's theorem argument tells us that if the basis is orthogonal, then paying less attention to low energy coefficients won't lead to a great deal of distortion. The net result is a great reduction in storage bits for a given distortion, than simply quantizing the raw image.
Typically, unitary transformations are considered for reasons of simplicity and energy conservation. An image is usually transformed block by block, where each block is typically eight by eight pixels, for example. Each row is transformed, to give a new block, whose columns are then transformed. This is called a separable two dimensional transformation. There are several advantages to transforming the image block by block, as opposed to transforming the entire image at once. Computational complexity is one advantage, progressive transmission is another benefit (transmitting the lowest frequency coefficient of each block first, etc.), and finally spatial locality. Spatial locality is a very desirable trait in image processing, which makes wavelet transformations desirable in many cases.
One drawback of block coding of images, using a unitary transformation on each block, is an undesirable visual effect know as "blocking". Blocking occurs at low bit rates, where the boundary of each block can be seen because of a large discontinuity at the boarder between adjacent blocks. Lapped orthogonal transforms can be used to alleviate the blocking effect, while maintaining a low bit rate, by synthesizing blocks which overlap adjacent blocks by a set amount, and decay at the edge of the synthesis block.
Variations of three transforms will be discussed and compared in the body of this report. The Lapped Orthogonal Transform (LOT) is the basic lapped transform based off of the ubiquitous Discrete Cosine Transform (DCT), which will also be discussed. Finally, a modified LOT, named the Lapped Biorthogonal Transform (LBT) is no longer unitary, but has synthesis functions which decay to zero at the block boundaries. All three of these transforms were simulated in Matlab and compared in terms of distortion rate curves, and visual examination.