Hyperspectral Image Compression Using Three-Dimensional Wavelet Transformation
6. Experimental Results
Two selected wavelet filter candidates were employed for the 3-D WT. They are the Daubechies_1 (db1) (i.e. the Hard) and Daubechies_3 (db3) (Daubechies and Micchelli, 1994). Because the Daubechies scaling function
f and wavelet
y have the supports [0, 2p-1] and [-p+1, p] (p denotes the vanishing moments), the db1 filters have the supports [0,1] and [0,1] and the db3 filters have the supports [0,5] and [-2,3]. The Lipschitz regularity of the db1 wavelet
y is about 0.3 and the db3 wavelet
y is Lipschitz 1.08. This means that the db3 wavelet
y is continuously differentiable, whereas the db1 wavelet
y is not. The choice of an optimal wavelet is a trade-off between the number of vanishing moments and its support size (Mallat and Hwang, 1992). A high amplitude wavelet coefficient occurs when the wavelet has a support that overlaps a brutal transition like an edge, which should thus be as small as possible to reduce the quantization error. In addition, over smooth regions, the wavelet coefficients are small at fine scales if
y has enough vanishing moments to take advantage of the large Lipschitz regularity. However, the support size of
y increases proportionally to the number of vanishing moments. For large classes of natural images, it has been observed numerically that the distortion rate is optimized with wavelets having three or four vanishing moments. Therefore, to obtain optimal results, the db3 wavelet was adopted. The db1 wavelet was applied for comparison because of its computation simplicity.
A four-level 3-D wavelet decomposition was applied to the test image with the db1 and db3 wavelets respectively. For each decomposition level, we used 256 quantization intervals for the LLL block and tried 0, 4, 8, 16, and 32 intervals for the other blocks, in which 0 interval means that the data block is ignored. With various combinations of using different decomposition levels and quantization intervals, objective measures (SNR) and subjective measures (classification overall accuracy, OA) are presented versus compression ratio (CR).
In order not to make complicated combinations, only data blocks from the highest decomposition level were kept, i.e. data blocks from the lower decomposition levels were always ignored. Therefore, for CR less than 20, a one-level decomposition was applied with the five-quantization intervals on high-resolution blocks to obtain five combinations of compression, which is denoted as the level 1 result. To take a two-level decomposition, high-resolution blocks in the first level were discarded and same five combinations were made for data blocks in the second level to obtain the results from level 2. The results from level 3 and 4 were obtained after the fashion of the previous combinations. Figure 6 shows compression ratios achieved with respect to different combinations, which reveals a clue to determining an appropriate decomposition level. The first decomposition offers a CR less than 20. It cannot produce a CR greater than 10 if one does not discard the high-resolution data. Generally speaking, for most applications, the appropriate CR range should be between 10 and 100. It is not necessary to compress an image by a factor of more than 100. The second decomposition offers a CR in between 20 and 150. Therefore, the level-2 combinations are good choices for hyperspectral image compression. Figure 6 also shows the fact that the db1 wavelet offers a higher CR than the db3 wavelet under the same decomposition and quantization combinations. This fact, however, is not the key factor for judging compression performance.
Fig. 6. Compression ratio versus different combinations of filters, decomposition levels, and quantization intervals.
Figure 7 (a), (b), (c), and (d) show the objective measures of the 1, 2, 3, and 4 level results, respectively. Comparing the db1 and db3 wavelet performance, two concluding remarks can be made. First, the db3 wavelet outperforms the db1 wavelet at a given CR based on the objective evaluation. Second, the SNR drops rapidly in the first decomposition level, but after that, the decrease becomes lesser with respect to the rapid increase in CR. Concerned with practical applications, the first conclusion implies that the db3 wavelet is a better choice than the db1 wavelet. Figure 8 (a), (b), (c), and (d) show the subjective measures of the level 1, 2, 3, and 4 results respectively. Based on a subjective evaluation, the db3 wavelet also obviously outperforms the db1 wavelet. In the level-4 combinations, decompressed images were even not converged in the classification process.
Fig. 7. The SNR measures of the 4 different level combinations versus compression ratio. (a) level 1. (b) level 2. (c) level 3. (d) level 4.
Fig. 8. The classification accuracy measures of the 4 different level combinations versus compression ratio. (a) level 1. (b) level 2. (c) level 3. (d) level 4.
An interesting phenomenon can be observed in Figure 8. OA Values do not decrease in response to the increase in information loss. In each decomposition level, the OA reduces when the number of quantization intervals drops, but it suddenly goes up to a level even higher than the original data when the number of quantization intervals goes to zero (i.e. ignoring all high-resolution data). This phenomenon still has us puzzled. A reasonable explanation is that information loss caused by image compression possesses two different effects: image roughing and smoothing. When the number of quantization intervals drops, the image is roughed. However, if all of the high-resolution data are ignored, the image is smoothed by wavelet decomposition. Hsieh (Hsieh, 1998) demonstrated that spatially smoothing a hyperspectral image improves the classification accuracy. The phenomenon can be explained in response to Hsieh's conclusion. The OA went up to 99% when a four-level decomposition was applied. The reason it reached a higher accuracy than Hsieh's demonstration was due to the 3-D smoothness. However, one should take a conservative view of this conclusion, because the spatial reference data distribution strongly affects classification accuracy. Land-cover classes of ground truth in the test data set are distributed as blocks, and so are Hsieh's data sets. Under this circumstance, the fact that smoothing images increases the classification accuracy (and vice versa) can be expected. However, this will not be true if the class data are not distributed as blocks.
7. Conclusion
An efficient scheme for hyperspectral image compression using 3-D wavelet transformation (WT) was developed. This scheme simultaneously explores useful information in the spatial and spectral dimensions and fully takes advantage of the 3-D nature of hyperspectral data. The 3-D WT yields a good representation of an image cube for compression purposes. The low-frequency WT component contains about 90% of the total energy in most cases. Various blocks and resolution levels can be coded in various ways to improve compression results. Using two-level wavelet decomposition offers compression ratios between 20 and 150 that are appropriate for most applications.
Both objective and subjective evaluations confirm that the db3 wavelet outperforms the db1 wavelet at a given CR. This implies that the db3 wavelet is a better choice than the db1 wavelet for the proposed scheme.
The subjective measure does not decrease in response to an increase in information loss. Information loss caused by image compression possesses two different effects: image roughing and smoothing. Smoothing the image promotes classification accuracy, while roughing the image reduce classification accuracy. Generally speaking, 3-D image compression is a sort of 3-D hyperspectral data feature extraction. However, how 3-D features contribute to classification requires further investigation.
Acknowledgement
We thank the National Science Council, ROC, for the support of this research project: NSC89-2211-E006-084.
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