Sparse representation of astronomical images is discussed. It is shown that a significant gain in sparsity is achieved when particular mixed dictionaries are used for approximating these types of images with greedy selection strategies. Experiments are conducted to confirm (i) the effectiveness at producing sparse representations and (ii) competitiveness, with respect to the time required to process large images. The latter is a consequence of the suitability of the proposed dictionaries for approximating images in partitions of small blocks. This feature makes it possible to apply the effective greedy selection technique called orthogonal matching pursuit, up to some block size. For blocks exceeding that size, a refinement of the original matching pursuit approach is considered. The resulting method is termed “self-projected matching pursuit,” because it is shown to be effective for implementing, via matching pursuit itself, the optional backprojection intermediate steps in that approach.
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SR and Execution Time, in Seconds, for Approximating a Complete Image with Different Approaches and Different Sized Blocks Partitioning the Imagea
OMP2D
MP2D
Block Size
SR
Secs
SR
Secs
SR
Secs
SR
Secs
21.69
51
21.69
61
20.52
60
19.55
56
27.63
98
27.61
115
26.41
99
24.49
93
29.15
233
29.08
209
28.28
200
25.79
205
29.97
506
29.88
392
29.30
382
26.46
387
30.36
1065
30.24
666
29.80
640
26.90
660
30.78
2041
30.60
1055
30.25
1015
27.20
1032
16.85
79
16.85
85
16.49
89
15.95
79
20.51
163
20.50
185
19.84
154
18.73
147
21.27
413
21.23
362
20.61
354
19.12
328
21.59
916
21.52
694
20.93
666
19.16
653
21.70
1989
21.59
1494
21.02
1145
19.13
1139
21.68
4031
21.53
2477
21.04
1919
19.11
1853
The top half of the table corresponds to the results for the nebula at publication size resolution () pixels. The bottom half contains the results for the galaxy image at the equivalent resolution ( pixels).
Table 2.
Average SR () and Average Processing Time () per Image (in Seconds) for Approximating, up to a PSNR of 45 dB, a Set of 55 Images from the HST Web Sitea
OMP2D
MP2D
Block Size
12.36
11.26
12.46
13.70
12.18
12.2
11.72
12.37
14.35
38.11
14.42
45.23
14.13
34.22
13.21
28.52
14.94
113.27
14.96
111.44
14.74
91.11
13.59
70.66
15.23
326.09
15.22
237.79
15.05
207.47
13.78
134.73
15.36
839.47
15.31
447.20
15.17
397.81
13.83
239.10
Both quantities are displayed against the block size partitioning the images. The average size of the images in the set is pixels. Note: the average times per image are also the averages of five independent runs for each given block size.
Table 3.
Mean () and Variance () of the SR Obtained with Different Mixed Dictionaries, by Partitioning the Images into Blocks of Size 8, 16, 24, and 32 and Applying the OMP2D Approach with RDCT-RDBS, RDCT-RDW, and RDCT-RR Dictionariesa
RDCT-RDBS
RDCT-RDW
RDCT-RR
DCT
DWT
Block Size
12.36
6.2
11.28
5.7
10.99
5.5
7.94
4.5
6.39
4.9
14.35
8.9
12.38
7.5
12.07
7.2
6.13
4.4
6.39
4.9
14.94
9.7
12.52
7.8
12.29
7.5
5.66
4.3
6.39
4.9
15.23
10.0
12.56
7.9
12.42
7.6
5.39
4.2
6.39
4.9
The results from the approximation with the transforms DCT and WT are also shown.
Tables (3)
Table 1.
SR and Execution Time, in Seconds, for Approximating a Complete Image with Different Approaches and Different Sized Blocks Partitioning the Imagea
OMP2D
MP2D
Block Size
SR
Secs
SR
Secs
SR
Secs
SR
Secs
21.69
51
21.69
61
20.52
60
19.55
56
27.63
98
27.61
115
26.41
99
24.49
93
29.15
233
29.08
209
28.28
200
25.79
205
29.97
506
29.88
392
29.30
382
26.46
387
30.36
1065
30.24
666
29.80
640
26.90
660
30.78
2041
30.60
1055
30.25
1015
27.20
1032
16.85
79
16.85
85
16.49
89
15.95
79
20.51
163
20.50
185
19.84
154
18.73
147
21.27
413
21.23
362
20.61
354
19.12
328
21.59
916
21.52
694
20.93
666
19.16
653
21.70
1989
21.59
1494
21.02
1145
19.13
1139
21.68
4031
21.53
2477
21.04
1919
19.11
1853
The top half of the table corresponds to the results for the nebula at publication size resolution () pixels. The bottom half contains the results for the galaxy image at the equivalent resolution ( pixels).
Table 2.
Average SR () and Average Processing Time () per Image (in Seconds) for Approximating, up to a PSNR of 45 dB, a Set of 55 Images from the HST Web Sitea
OMP2D
MP2D
Block Size
12.36
11.26
12.46
13.70
12.18
12.2
11.72
12.37
14.35
38.11
14.42
45.23
14.13
34.22
13.21
28.52
14.94
113.27
14.96
111.44
14.74
91.11
13.59
70.66
15.23
326.09
15.22
237.79
15.05
207.47
13.78
134.73
15.36
839.47
15.31
447.20
15.17
397.81
13.83
239.10
Both quantities are displayed against the block size partitioning the images. The average size of the images in the set is pixels. Note: the average times per image are also the averages of five independent runs for each given block size.
Table 3.
Mean () and Variance () of the SR Obtained with Different Mixed Dictionaries, by Partitioning the Images into Blocks of Size 8, 16, 24, and 32 and Applying the OMP2D Approach with RDCT-RDBS, RDCT-RDW, and RDCT-RR Dictionariesa
RDCT-RDBS
RDCT-RDW
RDCT-RR
DCT
DWT
Block Size
12.36
6.2
11.28
5.7
10.99
5.5
7.94
4.5
6.39
4.9
14.35
8.9
12.38
7.5
12.07
7.2
6.13
4.4
6.39
4.9
14.94
9.7
12.52
7.8
12.29
7.5
5.66
4.3
6.39
4.9
15.23
10.0
12.56
7.9
12.42
7.6
5.39
4.2
6.39
4.9
The results from the approximation with the transforms DCT and WT are also shown.