19.3 Spectral/Spatial Compression

Since band-to-band correlation is usually very high in hyperspectral imagery, removing such redundant information can achieve a significant compression ratio. Two major approaches are generally used for hyperspectral compression, which are dimensionality reduction by transforms (DRT) and DRBS in Chapter 6. The DR is often accomplished by DRT that compacts information into a small number of components, while the DRBS selects a small number of bands in some sense of optimality to represent data. However, a key issue is how to find an optimal component transform to perform DR for best possible hyperspectral compression or how to effectively select significant bands that can preserve desired information for hyperspectral compression to optimize performance of a designated exploitation application. In other words, the success of DRT and DRBS in hyperspectral compression is determined by how much information is extracted and preserved for the follow-up exploitation data processing after DRT and DRBS. Therefore, DRT and DRBS must be performed by custom-designed criteria for information extraction. This issue can be addressed by hyperspectral compression via dimensionality prioritization in Chapter 20, and by hyperspectral compression via band prioritization in Chapter 21. While the techniques developed in Chapters 20 and 21 can be directly applied to hyperspectral information compression, various versions of DRT and DRBS developed in Chapter 6 are not immediately ready for compression since they are generally developed for DR and not particularly designed for information compression. Therefore, in what follows, we reinvent the wheel by redeveloping the techniques in Chapter 6 for the purpose of hyperspectral information compression.

There are key differences between the spectral/spatial compression presented in this section and 3D-cube compression. One is that our proposed spectral/spatial compression de-couples spectral compression from spatial compression to perform spectral dimensionality reduction prior to 3D-cube compression. Another is that after spectral dimensionality reduction our spectral/spatial compression still performs 3D-cube compression on the spectral dimensionality reduced 3D-cube data compared to only 2D spatial compression being performed on those spectral compressed data by 1D spectral compression. A third difference is that there are indeed two types of spectral compression carried out in our proposed spectral/spatial compression: one is spectral dimensionality reduction and the other is spectral redundancy in 3D-cube compression. As a result, two compression criteria, spectral compression criterion and exploitation-based compression criterion, are needed and must be designed from an exploitation point of view to best fit applications. Finally, a fourth difference is that that according to our experience spectral information is better preserved using dimensionality reduction than using 1D wavelet compression since it offers better de-correlation. Accordingly, on many occasions even spectral dimensionality reduction implemented in conjunction with only 2D spatial compression may outperform 3D-cube compression techniques. One such example is the linear spectral mixture analysis (LSMA)-based hyperspectral image compression (Du and Chang, 2004a) where LSMA is first used to perform spectral compression by transforming an original hyperspectral image cube to a small number of abundance fractional images that are further processed by a follow-up spatial compression. It is interesting to note that many transform coding methods developed in the literature for hyperpspectral image compression generally perform 1D-spectral/2D-spatial compression where a 2D spatial compression technique is applied to individual spectral decorrelated components. However, it has been shown in Ramakrishna (2004), Ramakrishna et al. (2005a, 2005b) that 1D-spectral/3D-cube compression performed slightly better than 1D-spectral/2D-spatial compression. This is because the former performs two types of spectral compression, spectral dimensionality reduction by 1D spectral compression followed by removing spectral redundancy via 1D discrete wavelet in 3D-cube compression as opposed to the latter that is only benefited from 1D spectral dimensionality reduction. As a result, 2D spatial compression is generally not as effective as 3D-cube compression.

19.3.1 Dimensionality Reduction by Transform-Based Spectral Compression

Despite the fact that a hyperspectral image can be viewed as a 3D image cube, there are several major unique features that a hyperspectral image distinguishes itself from being viewed as a 3D image cube. The first and foremost is spectral features provided by hundreds of contiguous spectral channels. Unlike pure voxels in a 3D image, a hyperspectral image pixel vector is specified by a wide range of wavelengths in a third dimension that characterizes the spectral properties of a single pixel vector. Using the spectral profile captured in the spectral domain a single pixel vector in a 3D hyperspectral image cube can be solely analyzed by its spectral characterization. Another important unique feature provided by hyperspectral imagery is that many material substances of interest can be only explored by their spectral properties, not spatial properties such as small wastes in environmental pollution, chemical/biological agent detection in bioterrorism, camouflaged combat vehicles, and decoys in surveillance applications. In addition, certain targets such as chemical plumes, biological agents, which are considered to be relatively small with no rigid shapes but yet provide significant information, generally cannot be processed by rigid object-based image processing or identified by prior knowledge. Instead, these targets can be only captured and characterized by their spectral properties. Therefore, when a compression ratio is high, whether or not a hyperspectral image compression technique is effective may not be necessarily determined by its spatial compression as do most compression techniques in image processing. This is because small and subtle targets such as subpixel and mixed pixel targets may be very likely sacrificed by low-bit rate compression due to their limited spatial presence. Under such a circumstance, we need rely on spectral compression to retain these targets. Accordingly, separating spectral compression from 3D compression may be more desirable and effective than 3D-cube compression performing spectral and spatial information all together simultaneously in the sense that both JPEG2000 Part II and 3D-SPIHT codec perform spectral and spatial compression using separable transformations (i.e., 1D linear transform or 1D wavelet packet transform in the spectral dimension and 2D discrete wavelet transform (DWT) in the spatial dimensions) as a one-shot operation. In what follows, several transform-based spectral compression-based approaches to dimensionality reduction are developed for this purpose.

19.3.1.1 Determination of Number of PCs/ICs to be Retained

One of primary obstacles to implement PCA/ICA is to determine how many principal components (PCs) or independent components (ICs) are significant for information preservation. In the past, the number of PCs/ICs is determined by the amount of signal energy calculated from data variances that correspond to eigenvalues. Unfortunately, it was shown (Chang, 2003a; Chang and Du, 2004) that using accumulated sums of eigenvalues as a criterion to determine the number of PCs/ICs was not reliable and also not accurate in most cases in hyperspectral imagery. This is because subtle objects such as small targets, anomalies generally contribute little energies to eigenvalues that may not be retained in the first few PCs/ICs. In order to mitigate this dilemma, the concept of virtual dimensionality (VD) developed by Chang (2003b) and Chang and Du (2004) and also detailed in Chapter 5 can serve as a purpose to meet this need. If we assume that each spectrally distinct signature is accommodated by a single PC/IC, then the total number of PCs/ICs required to accommodate all the spectrally distinct signatures will be VD.

19.3.1.2 PCA (ICA)/2D Compression

PCA/2D compression is probably the most popular and commonly used in hyperspectral compression. It first uses PCA to spectrally de-correlate information of second-order statistics among all spectral bands and then followed up by a 2D compression technique to perform spatial compression on each of spectrally de-correlated bands so as to achieve hyperspectral data compression. The number of PCs, denoted by q to be retained can be determined by VD. These q PCs are then compressed by a 2D compression technique as code streams for data transmission. Then the corresponding 2D decompression technique is further applied to de-compress the received transmitted code streams for final exploitation applications. A similar approach using ICA/2D compression can be also implemented in exactly the same fashion that PCA/2D compression does. Figure 19.5 shows a block diagram that describes a process including a new component introduced by VD as a preprocessing prior to PCA/ICA for the purpose of estimating the number of PCs/ICs, q to be retained after spectral dimensionality reduction. The system in Figure 19.5 is referred to as PCA(ICA)/2D-SPHIT and PCA(ICA)/2D-JPEG2000, respectively.

Figure 19.5 PCA(ICA)/2D compression system.

An algorithm to implement the system in Figure 19.5 is described a follows:

PCA(ICA)/2D Compression Algorithm

It should be noted that if there is no signal source transmission required, steps 2 and 3 can be skipped and step 4 described in step 5 should be replaced by step 2. Since JPEG2000 and SPHIT have been shown to be most promising and effective compression techniques in the literature and produce nearly the same results, either of these two compression techniques specified in Figure 19.5 can be used for compression.

19.3.1.3 PCA (ICA)/3D Compression

When 2D compression is implemented in conjunction with PCA/ICA in Figure 19.5, a naive assumption is made on the fact that all spectral information can be nearly de-correlated by PCA/ICA so that the loss of spectral information caused by such a transform can be ignored without significant impact on compression performance. Unfortunately, this is generally not true. First of all, PCA/ICA is usually used to perform spectral dimensionality reduction for de-correlation not necessarily for spectral redundancy removal. As a consequence, using 2D compression techniques can only compress 2D spatial information not spectral information. To address this issue, a 3D compression is needed. Figure 19.6 modifies the block diagram in Figure 19.5 by replacing 2D JPEG and 2D-SPHIT with 3D-Multicomponent JPEG 2000 and 3D-SPHIT, which are referred to as PCA(ICA)/3D-SPIHT, PCA(ICA)/3D-multicomponent JPEG 2000, respectively.

Figure 19.6 PCA(ICA)/3D compression system.

PCA(ICA)/3D Compression Algorithm

It should be noted that if there is no signal source transmission required, steps 2 and 3 can be skipped and step 4 described in step 5 should be replaced by step 2.

19.3.1.4 Inverse PCA (Inverse ICA)/2D Compression

In Figures 19.5 and 19.6, an exploitation application is directly applied to compressed hyperspectral image data, which is a reduced dimensional image cube formed by q PCs/ICs. As an alternative, an exploitation application can be also applied to a reconstructed image data of the original L-dimensional data space by q PCs/ICs via an inverse transformation. In this case, an inverse transform of PCA (IPCA) or an inverse transform of ICA (IICA) is further applied to a q-PCs/ICs image cube to reconstruct a 3D image that has the same number of spectral bands as the original image has, L. Such an approach is referred to as IPCA (IICA)/2D compression depicted in Figure 19.7 where JPEG 2000 and 2D-SPIHT can be used as 2D compression techniques.

Figure 19.7 IPCA/2D compression system.

A detailed implementation of IPCA(IICA)/2D compression is summarized as follows.

IPCA(IICA)/2D Compression Algorithm

It should be noted that if there is no signal source transmission required, steps 4 and 5 can be skipped and step 5 described in step 6 should be replaced by step 2 without encoding.

19.3.1.5 Inverse PCA (Inverse PCA)/3D Compression

In analogy with IPCA (IICA)/2D compression, two IPCA (IICA)/3D compression systems can be also implemented, referred to as IPCA (IICA)/3D-SPIHT and IPCA (IICA)/3D-Multicomponent JPEG2000. In other words, PCA/ICA is first used to de-correlate a hyperspectral image for spectral compression. Then VD determines the number of PCs/ICs, denoted by q, which must be retained for compression. Then a 3D compression technique is applied to an image cube formed by q PCs/ICs for further compression. Finally, an inverse transform of PCA/ICA is applied to de-compressed q PCs/ICs-formed image cube to reconstruct a 3D image with the same number of spectral bands as the original image has, L, for exploitation applications. A block diagram of IPCA (IICA)/3D compression is depicted in Figure 19.8.

Figure 19.8 IPCA/3D compression system.

The details of implementing IPCA (IICA)/3D compression are briefly described as follows.

IPCA (IICA)/3D Compression Algorithm

Furthermore, if there is no signal source transmission required, steps 4 and 5 can be skipped and step 5 described in step 6 should be replaced by step 2 without encoding. In addition, two 3D compression techniques, 3D-SPIHT and 3D-multicomponent JPEG2000, can be used in the above algorithm. However, since 3D SPIHT requires dimensions to be multiples of 2ⁿ⁺¹ with n being the number of levels in wavelet decomposition, IPCA/3D-SPIHT may not be applicable when the number of spectral bands does not meet this constraint.

19.3.1.6 Mixed Component Transforms for Hyperspectral Compression

As noted, the sample covariance matrix used by PCA is of second-order statistics. PCA is considered as a second-order statistics transform that can only preserve information characterized by second-order statistics through transformation. In many applications preserving information of second-order statistics is generally not sufficient in substance characterization such as small objects, rare targets, etc, which cannot be generally captured by second-order statistics, but rather by statistics of order higher than 2. Under such a circumstance, PCA may not be effective. By contrast, ICA is developed to capture information characterized by statistical independence that may help to resolve this dilemma. For ICA to be effective two assumptions must be satisfied. One crucial assumption is that all the signal sources must be random sources. Since a linear sum of a finite number of Gaussian signal sources is still Gaussian, a second critical assumption is that at most one signal source can be Gaussian. However, due to this particular assumption, ICA is able to capture information that is characterized by non-Gaussianity whose statistics goes beyond second-order statistics. Because of that, PCA and ICA actually perform mutual disjoint transformations and complement each other. This leads to a brief that combining and mixing both PCA and ICA to form a single transform may yield better compression and desired performance.

In order to investigate such a mixed (m,n)-PCA/ICA transform that combines the first m PCs with n ICs, three issues need to be addressed (Chai et al. 2007). One is how many components are required for such a mixed (m,n)-PCA/ICA transform. The second issue is what n ICs should be selected for the (m,n)-PCA/ICA transform. Unlike PCA, which ranks PCs according to eigenvalues with decreasing order, ICA does not prioritize the ICs it generates. Selecting appropriate n ICs is crucial for the (m,n)-PCA/ICA transform. The third issue is how to combine two different sets of projection vectors, PCA-generated eigenvectors and ICA-generated projection vectors, that are not necessarily same vectors. Each of these three issues will be resolved as follows.

The first issue can be addressed by VD that has been used in PCA and ICA discussed in previous sections. With the help of VD, we can assume that VD-estimated n_VD = q is the total number of components needed in the (m,n)-PCA/ICA transform with q = m + n.

The second issue is to rank ICs by different criteria in a similar manner that PCA does for its PCs using variance as a criterion. This issue is addressed in Chapter 618. Of particular interest is the automatic target generation process (ATGP) that will be used in the proposed (m,n)-PCA/ICA transform.

To address the third issue, it first determines how many PCs resulting from PCA will be selected, denoted by m, and let be the eigenvectors that generate the first m PCs. Then all data samples are then projected to a space orthogonal to the space spanned by the m eigenvectors . Assume that this resulting orthogonal subspace is denoted by . Then the ATGP-based ICA is applied to the space to find the first n ICs with their corresponding projection vectors, denoted by . Combining the m eigenvectors with the n projection vectors yields a new set of basis vectors for our desired (m,n)-PCA/ICA transform where for , for and q = m + n.

Once all the three issues are resolved, a mixed (m,n)-PCA/ICA transform can be developed for spectral/spatial compression as follows.

Mixed (m,n)-PCA/ICA Compression Algorithm

1. Use n_VD to estimate the number of components needed to be retained for spectral compression, denoted by q.

2. Perform PCA to find eigenvalues and their corresponding eigenvectors and retain m PCs with the m largest eigenvalues.

3. Define an eigenvalue diagonal matrix by and an eigenvector matrix by . A whitening matrix is then used to sphere the mean-removed data matrix X. Let the resulting matrix be denoted by .

4. Apply FastICA using the n = p − m ATGP-generated pixel vectors as initial projection vectors to the sphered data to generate n projection vectors denoted by as well as p − 1 ICs. Let Z denote the projection matrix formed by with dimensionality of and define .

5. Form a new image cube Y by the m PCs and the n ICs and let .

6. Apply IPCA, and add the mean back to reconstruct X. Since q is generally much smaller than L, , in which case the inverse of W is always taken as its pseudo inverse.

Figure 19.9(a) depicts a block diagram of mixed (m,n)-PCA/ICA transform for data compression where the I(PCA/ICA) in Figure 19.9(b) denotes the inverse of mixed (m,n)-PCA/ICA transform via the projection vector . It should be noted that if there is no signal source transmission required, the process of encoding-decoding and reconstruction image described in the lower part of the diagram can be skipped.

Figure 19.9 Structure of mixed (m,n)-PCA/ICA transform.

19.3.2 Dimensionality Reduction by Band Selection-Based Spectral Compression

Another approach to spectral dimensionality reduction is band selection, called DRBS. Since the same treatment carried out for DRT can be applied to DRBS, only the implementations of algorithms are described as follows. However, it should be noted that since there are no images that can be reconstructed in the original data space from BS-compressed q-band images, no similar algorithms corresponding to Figures 19.7, 19.8 and 19.9(b) can be derived for BS/2D or 3D compression.

BS/2D Compression Algorithm

BS/3D-cube Compression Algorithm

Figure 19.10 depicts a block diagram of the BS/2D or 3D spectral/spatial compression. It should be noted that if there is no signal source transmission required, the process of encoding-decoding described in the diagram can be skipped.

Figure 19.10 DRBS/3D compression process.