IMAGE COMPRESSION APPLICATION
Image compression minimizes volume of data in an image without loss of image information. Algorithms for image compression remove redundancies that appear in the data [65]. It is possible to further reduce redundancies in multiresolution wavelet compressions with the hierarchical row compression algorithm in this dissertation.
7.1 Image Compression Fundamentals
Data compression is the process of reducing the amount of data to represent an image. The goal is to remove redundant data and still retain quality information to represent the image. There three types of data redundancies: coding redundancy, spatial and temporal redundancy, and irrelevant information. The first redundancy deals with the binary coded bits that represent the intensities in the image. The second redundancy examines the array spatially for replicated correlated pixels. The third and simplest compression is removal of extraneous information that is not used. Irrelevant information compression is the direction of the research [64].
Human visual perception of the image information does not involve a quantitative analysis of the pixels. The pixel values can be modified, within given parameters, without any subjective degradation to the resulting image. For removal of irrelevant information, the digital image has its intensity values examined. Where there are clusters in the intensity values, then an averaging is done to have a single value represent that intensity. There is irreversible loss of information, but the loss is not perceived in the quality of the reconstructed image. This loss of quantitative information is referred to as quantization. There are three techniques for removing an image’s irrelevant information: block transform coding predictive coding, and wavelet coding.
Figure 7.1. Diagram of a block transform encoding process.
In block transform coding, an image is divided into equal size subblocks that do not overlap. Each subblock is processed independently using a 2-D transform, such as the FFT. Then the transform coefficients are quantized. A diagram for the block transform coding process is found in Figure 7.1 [64]. In predictive coding, the pixels of the image are inspected and eliminating any redundancies in closely spaced pixels. The value of a closely spaced pixel is predicted, and the difference between the prediction and the actual value is stored. The differences are quantized and encoded. A diagram for the predictive coding process is found in Figure 7.2 [64].
The wavelet transform maps the spatial domain of an image to a frequency domain, then excessive redundancies in the image are exploited and removed [73, 74]. Wavelet transformations make it easier to compress, transmit and analyze images. The transform coding process is done in four major steps: apply the wavelet transform, detect the
Figure 7.3. Diagram of the typical wavelet transform encoding and decoding process.
threshold, entropy code the quantized transform coefficients, and apply an inverse transform as shown in Figure 7.3 [75]. The difference between the wavelet transform coding and block transform coding methods is that the wavelet transform process does not include the subblock preprocessing stage in block transform coding.
7.2 Wavelets and Row Compression
Mathematical wavelet transforms are used extensively in image compression. Reducing the huge volume of data in a direct image spatial domain is important for transmission or storage. This method of lossy compression of the image is acceptable since the reconstruction of the image need not be exact [74]. The research here focuses on the discrete wavelet transform (DWT) which computes the series expansion coefficients for a function that is comprised of a wavelet function, ψ(x), and a scaling function, ϕ(x) [65]. In the one-dimensional case, the DWT of an image computes the approximation (low frequency) coefficients and the detail (high frequency) coefficients.
A single-level, separable, 2-D orthogonal Daubechies wavelet decomposition of an image forms a 2×2 block partitioned matrixW where each of the four subblocks contain detail or approximation coefficients. In the 2-D DWT, four separable functions are required:
ψ(x, y) = ψ(x)ψ(y), scaling;
ϕH(x, y) = ϕ(x)ψ(y), horizontal edges;
ϕV(x, y) =ψ(x)ϕ(y), vertical edges;
ϕD(x, y) = ϕ(x)ϕ(y), diagonal edges.
Figure 7.4 depicts a three-stage 2-D DWT decomposition of an image with A, H, V, and D
as the low, horizontal, vertical and diagonal bands respectively [64]. Once an image is decomposed by the multiresolution wavelet transform, a hierarchical structure is evident in the resulting decomposed matrix W, where the lower left off-diagonal blocks represent the vertical bands and the upper right off-diagonal blocks represent the horizontal bands of the image. This hierarchical structure can be seen in Figure 7.4. This research exploits common rank structure between the lower off-diagonal blocks to introduce additional zeros. Similarly, this exploitation also applies to the common horizontal band structure in the upper off-diagonal blocks of the wavelet.
Figure 7.4. Wavelet mulitresolution image decomposition where Li is a low band, Vi is a
vertical band,Hiis a horizontal band, andDiis a diagonal band generated from a three-stage
wavelet transformation.
The vertical and horizontal edge off-diagonal blocks in the wavelet decomposition have similarities to the hierarchical matrices. The connection between the wavelet structure and the hierarchical representation is apparent. The row compression algorithm, developed in Chapter 3, applies a sequence of unitary transformations to the the low rank factorization to
introduce zeros into the off-diagonal blocks of the matrix. The row compression algorithm can be amended to operate on the Daubechies DWT common edge blocks, exploiting both the sparsity and rank structure of the wavelet transform. A permutation and subtraction of the vertical and horizontal edges aligns the common edge space for compression. Orthogonal transformations are applied to the common edge space and zeros are introduced. Thus the information about the edges captured by the wavelet transform is compressed.