Fig 1.
(a) Heaviside function. (b) two approximated Heaviside functions with ξ = 1 (blue solid line) and ξ = 0.01 (red dash line).
Fig 2.
The flow chart of the proposed work.
Table 1.
Comparison with AHFM algorithm and MAHFM algorithm(factor = 2).
Table 2.
Quantitative results by different super-resolution algorithms (factor = 2).
Table 3.
Quantitative results by different super-resolution algorithms (factor = 3).
Fig 3.
Results of “baboon” with the upscaling factor s = 3.
(a) LR image, (b) bicubic, (c) 11’IPOL [2], (d) 14’TIP [10], (e) 10’TIP [12], (f) AHFM [7], (g) MAHFM.
Fig 4.
Results of “peppers” with the upscaling factor s = 2.
(a) LR image, (b) ground truth, (c) bicubic, (d) 11’IPOL [2], (e) 14’TIP [10], (f)10’TIP [12], (g) AHFM [7], (h) MAHFM.
Fig 5.
Comparisons between MAHFM and AHFM with factor s = 2.
(a) MAHFM (RMSE = 8.7501, PSNR = 29.2905 dB). (b) AHFM [7] (RMSE = 9.0810, PSNR = 28.9681 dB). (c) residual (For better visualization, we add 0.5 to the intensities of the residual).
Fig 6.
Comparisons between two iterative refinement methods.
(a) Reerror. (b) Reerror1.
Fig 7.
(a) The RMSE trend of the MAHFM algorithm as the different parameter varies when other parameters are roughly estimated (To simplify the experiment, e1, e2, e3 represents ξ1, ξ2, ξ3, respectively in this figure). (b) The RMSE trend of the MAHFM algorithm as the parameter ξ2 varies, where ξ1 = 0.8, ξ3 = 10−4.
Table 4.
Parameters of MAHFM algorithm.
Fig 8.
(a) Computation time of Algorithm 2 (l = 1, k = 2) and Algorithm 3 vs. upscaling factor for the low-resolution image “butterfly” with LR size 60 × 60. (b) Computation time of Algorithm 2 (l = 1, k = 2) and Algorithm 3 vs. the size of low-resolution image “butterfly”, the size of low-resolution image is increased from 30 × 30 to 110 × 110 and the upscaling factor is always set to be 4. Note that, to reduce the instability of Matlab, the computation time is the average of 10 runs.
Fig 9.
(a) Computation time of MAHFM (i.e., Algorithm 5) vs. upscaling factors for the low-resolution image “butterfly” with LR size 60 × 60. (b) Computation time of MAHFM vs. the size of low-resolution image “butterfly”, the size of low-resolution image is increased from 30 × 30 to 110 × 110 and the upscaling factor is always set to be 4. Note that, at the common point with Fig 8 (i.e., upscaling factor 4 and 60 × 60 LR size), it has slightly different computation due to the instability of Matlab, thus we present the computation time by the average of 10 runs to reduce the gap.
Fig 10.
Results of “peppers” with the upscaling factor s = 2 via the representation of Haar wavelets.
Table 5.
Comparison with super resolution based on the basis of wavelets and Heaviside function.