Hyperspectral Image Compression Using Three-Dimensional Wavelet Transformation
Yi-Hsing Tseng, Hung-Kuan Shih, Pai-Hui Hsu
Department of Surveying Engineering
National Cheng Kung University, Tainan,
Taiwan, R.O.C.
E-mail: tseng@mail.ncku.edu.tw
Keywords: Hyperspectral Image, Image Compression, 3-D Wavelet Transformation, Classification
Abstract
A three-dimensional (3-D) wavelet compression method for hyperspectral images is proposed. This method applies a separable 3-D wavelet transformation (WT) to hyperspectral datacubes resulting in wavelet coefficients for 3-D multi-resolution image representation. The coefficients are then compressed using the optimal scalar quantization. Finally, Huffman coding is applied to save storage space. Experimenting on an AVIRIS image is performed with various combinations of wavelet banks, transformation levels, and quantization intervals. The decompressed images are evaluated objectively and subjectively, based on signal-to-noise ratio (SNR) and classification accuracy measures, respectively. The results show that a two-level 3-D WT offers a compressed ratio (CR) in between 20 and 150, which would be suitable for most applications. A comparison of two wavelets shows that the db3 wavelet outperforms the db1 wavelet based on both objective and subjective evaluations. The results also show an interesting phenomenon in which classification accuracy does not drop in response to an increase in information loss. Information loss caused by image compression possesses two different effects: image roughing due to quantization and smoothing due to decomposition. In this data set, image smoothing promotes classification accuracy, while image roughing reduces classification accuracy.
1. Introduction
Hyperspectral images provide much richer and finer spectral information than traditional multispectral images, however the volume of generated data is dramatically increased. Image compression will be essential for economical of distribution when spaceborne hyperspectral data are regularly available. Image compression reduces storage requirements, network traffic, and therefore improves efficiency. However, the satisfactory image compression requires a large compression ratio and a small amount of information loss. A lossless compression is the best choice as long as the compression ratio is acceptable, but it usually cannot offer a satisfactory compression ratio due to the limitation of image entropy. Typical hyperspectral data, such as images obtained with the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS), are stored as 16-bit words, and have an average entropy of about 9 bits/pixel (Ryan and Arnold, 1997). Therefore, in most cases, lossless hyperspectral image compression hardly achieves a compression ratio greater than 2. To obtain significant image compression, lossy compression is preferable to a lossless compression. Under these circumstances, preserving the most useful information when compressing an image to an acceptable size becomes the central issue of hyperspectral image compression (Qian et al., 2000).
Hyperspectral data sets are referred to as datacubes because of their 3-dimensional (3-D) nature (two spatial dimensions and one spectral dimension). Conventional image compression techniques explore useful information within the two spatial dimensions. Among these techniques, wavelet transformation (WT) has been proven to be very efficient for 2-D image coding (Antonini et al., 1992). Compared to the popular method, discrete cosine transform (DCT), WT extracts not only frequency information, but also spatial information. For multispectral image compression, band images are usually compressed separately with a 2-D image coding technique, so that spectral redundancy will not be explored. A hybrid scheme was proposed to solve this problem using a two-step transformation (Markas and Reif, 1993). First, a spectral transformation, such as principal component transformation (PCT) or DCT, is applied to decorrelate spectral information. Second, 2-D image compression is applied to each transformed component. Removing spectral redundancy is achieved by compressing the subordinate components more than the dominant components. Although the hybrid scheme, in a sense, may be one of the most suitable methods for hyperspectral image compression, it does not fully take the advantage of the 3-D nature of hyperspectral data. A 3-D WT is therefore proposed to simultaneously explore the most representative information in the spatial and spectral dimensions. Figure 1 shows a block diagram of this 3-D wavelet compression method.
Fig 1. 3-D wavelet compression process.
2. 3-D Wavelet Transformation
A wavelet transform (WT) decomposes a signal into a series of smooth signals and the associated detailed signals at different resolution levels (Mallat, 1989). At each level, the smooth signal and associated detailed signal have all of the information necessary to reconstruct the smooth signal at the next higher resolution level. The transformed signal has both time and frequency information from the original signal and provides a good representation for coding. The WT application to image compression has shown promising results (Antonini et al., 1992).
A multiresolution WT involves two functions: a mother wavelet
y and a scaling function
f . Let the normalized scale factor be 2
j and the translation be k×2
j (where j, k
Î Z), then the dilated and translated wavelets will be:
Also, the dilated and translated scale functions will be:
For fixed j, the
fj,k are orthonormal. The spaces V
j spanned by the
fj,k describe successive approximation spaces,
¼V
2Ì V
1Ì V
0Ì V
-1Ì V
-2 ¼, each with resolution 2
j. For each j, the
yj,k span a detail space W
j which is exactly the orthogonal complement in V
j-1 of V
j:
It has been proven that any scaling function is specified by a discrete filter called a conjugate mirror filter. The causality V
j+1 Ì V
j is verified using a conjugate mirror filter:
Similarity, the causality W
j+1 Ì V
j is verified by:
The scaling coefficients
áf,
fj,kñ, denoted by a
jk, characterize the projection of f onto V
j. The detailed coefficients
áf,
yj,kñ, denoted by d
jk, describe the information lost when going from an approximation of f with resolution 2
j-1 to the coarser resolution 2
j. If f(x)
ÎL
2(R), it can be decomposed into a series of smooth signals and the associated detailed signals at different resolution levels:
Based on (4) and (5), one obtains
Substituting (8) and (9) into (6) and (7) respectively:
