In this Section the CR resulting of the self contained folded images, If constructed by the
“pixel” scheme is established.
Although the “pixel” scheme stores the vectors of indices with a different procedure to the “ad hoc” method, the increase in the range of indices resulting with larger dictionaries
or blocks is still expected to increase the number of bits Nbf2 required to store the indices.
The Experiment below calculates the size of the CR of the self contained folded image If when the “pixel” scheme is used to fold the indices from approximations of both the astronomical and natural images, made with the 8 dictionaries Dc,2i , i = 1, . . . , 8 in com-bination with the 3 block sizes N = 8, 16 and 24. This CR is then compared with that produces when the “ad hoc” method was used over the same set images, dictionaries and block sizes N in Experiment 4.6.
The “pixel” scheme directly codes the indices instead of first transforming and then quantizing them. Therefore it is expected that the number of bits Nbf2 required to store the image If2 ∈ RNr3×Nc3 produced by the “pixel” scheme will be less than the number of bits Nbf2,△ required to store If2,△ ∈ RNr2×Nc2 produced by the “ad hoc” method. This will result in a larger overall CR for the self contained folded image built by the “pixel”
scheme than the self contained folded image made with the “ad hod” method, which is investigated in the Experiment below.
4.8.1 Experimental Overview
The Experiment in this Section applies the same procedure as the Experiment in Section 4.6 except that instead of applying the “ad hoc” method for folding the indices in the image If1 the indices are stored by the “pixel” scheme, to produce the image If2. Therefore the CR for the self contained folded image constructed by the “pixel” scheme to store the indices is calculated as CR = uNp
Nbf1+Nbf2.
The pseudorandom number genereator required by this method was initialized by the seed s3 = 5.8453 × 107.
Results
The average CR (¯xCR) over the 55 astronomical and natural images, when processed to produce the self contained folded image with the “pixel” method to store the indices, is shown by the solid lines respectively in Figures 4.4 and 4.5. These results for the CR can be summarized as follows:
1) The ¯xCR shown in Figures 4.4 and 4.5 for the self contained folded images within the “pixel” scheme is significantly higher than the ¯xCR for the self contained folded images with the “ad hoc” method for all combinations of dictionary and block size N .
2) Figure 4.4 shows that the ¯xCR, over the astronomical images, for all N increase with
the dictionary size. Therefore a one tailed paired sample t-test was performed to see if the ¯xCR produced by folding with the largest dictionary Dc,28 was significantly higher than the smaller dictionaries Dc,2i , i = 1, . . . , 7. The results of this test are shown in Appendix E.2.3, where it is shown that the average CR produced by the largest dictionary Dc,28 is significantly higher than the average CR produced by all of the the smaller dictionaries with N = 16 and significantly higher than the average CR produced by all of the dictionaries except Dc,26 when N = 8 and 24, for a 95%
confidence level.
3) Figure 4.5 shows that the ¯xCR, over the natural images again increases with the dictionary size. Therefore a one tailed paired sample t-test was performed to see if the ¯xCR produced by folding with the largest dictionary Dc,28 was significantly higher than the smaller dictionaries Dc,2i , i = 1, . . . , 7. The results of this test are shown in Appendix E.2.3 where it is shown that the average CR produced by the largest dictionary Dc,28 is significantly higher than the two smallest dictionaries Dc,2i , i = 1, 2 when N = 8, and significantly higher than the three smallest dictionaries Dc,2i , i = 1, . . . , 3 when N = 16 and 24, for a 95% confidence level.
The general results above concerning the “pixel” method is used are identical to the results of the “ad hoc” method is used, with the difference being that the “pixel” method produced a higher ¯xCR over both image sets for all dictionaries Dc,2i , i = 1, . . . , 8 and block sizes N = 8, 16 and 24.
Discussion
Self contained folded images were constructed by both the “ad hoc” and “pixel” methods, resulting in a CR > 1 for all the images tested. The highest ¯xCR occured for images which applied the “pixel” method for storing the indices.
To get an idea of whether through the process of folding, the storage requirements of the image coefficients and indices are increased, over simply coding them, the results for N = 24 from this Experiment were compared with those from the Experiment in Section 2.8. It is important to note that the results compared in the discussion below were produced by two different dictionaries, however as the comparisson is just to get and indication, the difference in the ¯xSR can be ignored.
Both experiments were performed over the 45 astronomical and natural images which were initially approximated to a PSNRa = 45 + 4.5 × 10−3. The ¯xCR (and ¯xSR) results for the astronomical and natural images, constructed with the “ad hoc” method were
respectively 8.15 (¯xSR = 15.41) and 4.10 (¯xSR = 6.32). The corresponding ¯xCR (and
¯
xSR) for the “pixel” method, were respectively, 5.15 (¯xSR = 13.58) and 2.31 (¯xSR = 6.10). Ignoring the reduction in the ¯xSR in the initial approximations, resulting from approximating with the RDC-RDBS2 instead of the TS3 dictionary, it is clear that the folding procedure introduces some storage overhead.