Figure 1
A truncated cosine and approximations with Haar and Daubechies 4 scaling functions using different scales. It may be difficult to see the graph of the original truncated cosine.
Figure 2
From 512 × 512 frequency measurements (not shown) 256 × 256 Daubechies 4 scaling functions are computed and used to produce this image. The image is evaluated at scale 10, i.e., in 1024 × 1024 points.
Table 1
Runtime comparisons with the Matlab implementation from [5] for reconstruction with the Daubechies 4 scaling functions. “Size” refers to the change of basis matrix; “init” is the computation of the change of basis matrix; “sol” is the computation of the solution; “Iter.” is the number of iterations by the iterative solver; t/n is the time per iteration in the solver. Time is measured in seconds and “mem” is the memory allocation in megabytes.
| init | sol | Iter | ||||||
|---|---|---|---|---|---|---|---|---|
| Problem | Size | Program | time | mem | time | mem | n | t/n |
| Uniform 1D | 8192 × 4096 | Matlab | 15.0 | 901 | 0.13 | 37 | 9 | 0.01 |
| Julia | 0.34 | 6.8 | 0.04 | 0.7 | 12 | 0.003 | ||
| Jitter 1D | 5463 × 2048 | Matlab | 10.0 | 627 | 0.28 | 59 | 20 | 0.13 |
| Julia | 0.23 | 4.0 | 0.08 | 0.4 | 20 | 0.004 | ||
| Uniform 2D | 5122 × 2562 | Matlab | 0.96 | 59 | 5.2 | 1507 | 9 | 0.58 |
| Julia | 0.15 | 146 | 17.6 | 17 | 16 | 1.10 | ||
| Jitter 2D | 26 244 × 322 | Matlab | 104.8 | 6377 | 8.3 | 1270 | 50 | 0.17 |
| Julia | 2.4 | 165 | 2.4 | 1.3 | 18 | 0.13 | ||
| Spiral | 27 681 × 322 | Matlab | 107.1 | 6670 | 3.7 | 479 | 17 | 0.21 |
| Julia | 2.8 | 261 | 2.2 | 1.4 | 16 | 0.14 | ||