VISUALIZATION OF ASTRONOMICAL IMAGES
USING WAVELET TRANSFORMS
Shahram Latifi (PI)
Emma Regentova(Co-I)
Donseng Yao(Graduate student)
September 18, 2000
Grant Number : NAG5-3994
Period Covered: March 1, 2000- August 1, 2000
I. TECHNICAL REPORT
This year, we have focused on exploiting wavelet transforms for visualization of astronomical images. We have analyzed current methods and developed new techniques for
The objective of this task is to obtain enlarged version of the original image that can be viewed in subtle details. Wavelet transform of the image is used to double image resolution.
Results:
Presentation:
In Figure 1, an image is presented at the step before the last step to obtain full resolution image, that is, at the first level of multilevel decomposition based on Haar wavelet.
Final step of inverse transform yields the full resolution image as given in Figure 2. The full resolution image itself can be considered as LL band for doubled resolution image. The rows and columns of LH, HH and HL subbands of the first level of decomposition are interlaced by zeroes, which yield artificially created subbands (see Figure 3). Then, inverse wavelet transform is implemented that results in doubled resolution image. In second approach, we perform the interpolation between adjacent pixels in rows and columns subsequently. The average values of adjacent pixels are taken. Figures 4 and 5 show doubled resolution images obtained using these two approaches for Haar and Daubechie?s 2 filters respectively.
We have employed Haar, Daubechies and biorthogonal transforms and studied quality of images for interlacing by zeroes and interpolation. Table 1 displays results, where SNR is evaluated with respect to the image of original size, that is, full resolution image.
2. Denoising
The objective is to obtain higher quality for either full or doubled resolution image.
Results:
thresholding wavelet coefficients of the high-frequency subbands.
terms of SNR.
Presentation:
Filtering of the original image with Gaussian noise added (Figure 6) is shown in Figures 7a and 7b. Daubechie?s long filters (length 4) yield better results, that can be explained by smoothing properties of the long filters.
3.Visualization
The objective of this task is to display an image at different levels of detail. As a result, the shape of galaxies or nebulae can be observed, as well as their concentration and extension. Finally, an image can be represented by the levels of intensity, that is objects of either same peak and mean intensities or different depth are grouped for visualization.
Results:
The technique is developed that allows for viewing objects based on the intensity levels and levels of details, that gives the appreciation of the shape of galaxies and nebulae.
Presentation:
One level transform is applied. LL subband is processed by a sliding window of the selected size. Mean value of LL coefficients is computed within a block. If difference between mean values in adjacent blocks are within predefined threshold, then same mean value is assigned to all coefficients in the analyzed blocks.
For the original image shown in Figure 2, the window size is set to 32x32 pixels and the threshold value is 0.05(see Figure 8, Daubechie?s 2). The coefficients of the detail subbands are set to zero and inverse transform is applied. As a result, the smooth version of original image is obtained (Figure 9). Then, wavelet transform of the smoothed image is performed. The histogram of the coefficients of LL subband is computed, and standard variation (sigma) is determined, and coefficients within - sigma, + sigma are binned into k groups and substituted by upper border value in the group (see Figure 10). Then inverse transform is implemented. The result of such a processing gives presentation of the extended but fuzzy objects at different levels of "depth"(see Figures 10-15).
We also work on the density analysis to segment objects with respect to both intensity and concentration of the edge points to provide more subtle structural and morphological analyses. For example, the swings of the spiral galaxies can be recognized. To segment an image, the coefficients of the detail subbands are filtered and for the remainder the value above local maxima are taken as the edge point location. Then number of edge points within sliding window of the predefined size is computed. Histogram of the different numbers are binned according to the given number of different regions.
II. STAFFING ACTIVITY
No change.
III. HARDWARE, SOFTWARE, AND WWW
We acquired 2 Dell WORKSTATIONS :
IV. REFERENCES
V. PUBLICATIONS
VI. FIGURES FOR PRESENTATION
Figure 1 Wavelet Decomposition : first level
Figure 2 Full Resolution Image
Figure 3 Artificially created level: Interlacing by zeroes
Figure 4 Resolution doubling : interlacing by zeroes (a. Haar, b. Daubecie?s 2)
a)
b)
Table 1
|
Wavelet transform |
SNR |
|
|
Interlacing by zeroes |
Interpolation |
|
|
Haar |
40.5415 |
36.7647 |
|
Db2 |
24.9086 |
24.9197 |
|
Db3 |
24.9221 |
24.9298 |
|
Db4 |
24.9666 |
24.9744 |
|
Bior1.3 |
24.9100 |
24.9252 |
|
Bior1.5 |
24.9263 |
24.9408 |
|
Bior2.2 |
24.9711 |
24.9744 |
|
Bior2.4 |
25.0539 |
25.0559 |
Figure 5 Resolution doubling : interpolation (a. Haar ; b. Daubechie?s 2)
a)
b)
Figure 6 Original Image with the Gaussian noise (10 sigma)
Figure 7 Denoising (a. Haar (SNR = 36,63), b. Daubechie?s 4(SNR= 37.25))
a)
b)
Figure 8 Smoothed image of LL subband(low resolution image)
Figure 9 Inverse transform of smoothed image
Figure 10 Thresholded coefficients of detail subbands (a. HL and b. LH subbands)
a)
Figure 11 Object at the level 1 ( threshold = t0)
Figure 12 Object at the level 2 (threshold = t0 + delta)
Figure 13 Object at the level 3 ( threshold = t0 + 2*delta)
Figure 14 Object at the level 3 (threshold = t0 + 3*delta)
Figure 15 Object at the level 5 (threshold = t0 + 4*delta)
