3.1 Transformation: Wavelet-Based Transform

In compression, the transformation step aims at a decorrelation of the data, compacting in this way the energy of the signal into few significant transform coefficients. The wavelet transform has been widely adopted in the field of image compression for its good property of spatial-frequency localization, which is also essential for compressing holographic data. The classical 1D discrete wavelet transform (1D-DWT) can be simply summarized as applying a pair of low-pass and high-pass filters to the input signal, followed by a downsampling of each filter output. A multiresolution decomposition can be obtained by further decomposing the low-frequency subband. For an image, a separable DWT is commonly used by applying the 1D-DWT once along the rows and once along the columns. Consequently, four subbands can be obtained by one decomposition: an approximation subband named LL and three detail subbands named HL, LH, and HH corresponding, respectively, to the horizontal, vertical, and diagonal orientations. Moreover, many different filters can be used—for example, the simplest 5/3 filter bank, has the “perfect reconstruction” property while the well-known Daubechies 9/7 filter bank has high compression efficiency (Daubechies, 1992).

3.2 Quantization

The quantization step is a lossy process which reduces the precision of the data samples into a discrete set of levels. The general quantization process is shown in Fig. 3.2.

Generally speaking, two kinds of quantization methods can be distinguished, based on uniform and adaptive quantizers. Moreover, quantization can operate on scalar or vector data samples, referred to as scalar quantization (SQ) and vector quantization (VQ), respectively.

Uniform scalar quantization The simplest example of SQ is uniform scalar quantization (USQ), by which an input sample x_i is mapped to a quantized value ${\widehat{x}}_i$ by

$\begin{array}{l}\hfill {\widehat{x}}_i=⌊\frac{x_i}{Q_{\mathrm{step}}}⌋Q_{\mathrm{step}},\end{array}$

where Q_step is the quantization step size and ⌊⋅⌋ is the rounding operator.

Adaptive scalar quantization Adaptive scalar quantization (ASQ) has been proposed to reduce quantization errors for input values which are not uniformly distributed. In this context, the Lloyd-Max algorithm (Max, 1960; Lloyd, 1982) aims at minimizing the mean squared error (MSE) resulting from quantization.

Vector quantization In the case of VQ, the Linde-Buzo-Gray algorithm (LBG-VQ) (Linde et al., 1980) is also designed to iteratively minimize MSE, but on multidimensional inputs. Two criteria should be satisfied: (1) the nearest neighbor condition, in other words, all vectors that are closer to a specific centroid than any other centroids, should be contained in one region and approximated to this specific centroid as output; and (2) a centroid should be the average of all the vectors in one region.

3.3 Entropy Coding

After the transformation and quantization steps, the quantized transform coefficients need to be encoded, in other words, converted to a binary codestream. Basically, the principle of entropy coding is to assign variable length codewords (VLC) to input symbols to exploit the nonuniform statistical distribution of these symbols, hence resulting in an overall reduced number of bits.

As previously stated, wavelet-based schemes are especially interesting. In this case, the property of quality scalability can effectively be achieved. For this purpose, the coding scheme should encode the coefficients in decreasing order of their importance with reduced accuracies. During the decoding procedure, the decoded image is first approximated by the most significant coefficients, then gradually refined by the less significant ones. The development of wavelet-based embedded codecs has been the subject of numerous research works and readers can find an overview of the corresponding state of the art in Bildkompression and der Technik (2003).

The embedded zerotree wavelet (EZW) coding (Shapiro, 1993) and its extended version named set partitioning in hierarchical trees (SPIHT) (Said and Pearlman, 1996) have been proven to be very effective for image compression. Particularly, SPIHT has been applied for compressing holographic data given its good performance. Both schemes are based on three principles: partial ordering by magnitude with a set partitioning sorting algorithm, ordered bit plan transmission, and exploitation of cross-scale similarities of the wavelet coefficients. However, they differ in the way subsets of coefficients are partitioned. Basically, the zerotree concept based on raster scan in EZW is improved by using sorted lists in SPIHT instead.

The basic concepts used in SPIHT are now briefly introduced.Figure 3.3 shows the definition of the zerotree or spatial-orientation tree. The arrows are oriented from the parent node to the offspring nodes. Each node of the tree corresponds to a pixel and has either zero or four offsprings with same spatial orientation in the next finer level. In practical implementation, four kinds of subsets are used: one set containing all the roots and three other sets, respectively, containing all the offsprings, descendants, and indirect descendants of the entry node. Three ordered lists are used to decide the order in which the subsets are tested for significance, which are called list of insignificant sets (LIS), list of insignificant pixels (LIP), and list of significant pixels (LSP), respectively. The descendant sets of the roots are added in LIS for initialization.

SPIHT aims at improving the zerotree concept by replacing the raster scan with sorted lists. During the sorting pass, the pixels in LIP are tested and moved to LSP if they are significant. Sets in LIS are evaluated following the LIS order. A set is removed from LIS when it is found to be significant. New subsets with more than one element are added back to the LIS and the element of the removed sets are added to the LIP or LSP depending on their significance. The pixels in LSP are refined with an additional bit in the refinement pass. The two passes are iterated until the threshold for testing the significance becomes lower than the minimum wavelet coefficient.

3.4 JPEG 2000

JPEG 2000 is the state-of-the-art standard for still image coding (Taubman and Marcellin, 2001; Schelkens et al., 2009). On top of very high coding efficiency, JPEG 2000 also provides with a number of highly desirable features such as seamless progressive transmission by resolution or quality, lossy to lossless compression, random code stream access, continuous-tone and bi-level compression, and region of interest. It has been widely adopted in several application domains, including digital cinema and broadcasting environments, medical imaging, surveillance, and cultural heritage archives.

Figure 3.4 shows the typical schematic of JPEG 2000, which consists of five main steps: image tiling, wavelet transform, quantization, embedded code-blocks encoding, and rate-distortion optimization. Image tiling is useful to handle very large images. DWT is then applied. Two wavelet transforms are defined in the core coding system: the irreversible floating-point CDF 9/7 transform kernel and the reversible integer 5/3 transform kernel. Each subband is then subdivided into code-blocks, which are independently encoded. In turn, each code-block is quantized and entropy coded in a bitplane-by-bitplane process using a context-based adaptive binary arithmetic encoding. Before assembling the individual bitstreams into a single, final bitstream, each code-block bitstream is truncated by the Lagrangian rate-distortion optimal technique. The truncated bitstreams are then concatenated together to form the final bitstream. This technique is named embedded block coding with optimal truncation (EBCOT) (Taubman, 2000).

The code-block structure facilitates random access in the data, a property which can often be useful. Moreover, unlike most alternative coding schemes, JPEG 2000 compression can be both lossy and lossless, always produces an intrinsically progressive bitstream, and can compress a region of interest with higher quality. Finally, the versatility of the JPEG 2000 encoding architecture makes it an appealing solution when considering the compression of holographic data.