Image Noise Reduction and Solution of Unconstrained Minimization Problems via New Conjugate Gradient Methods
Abstract
:1. Introduction
2. Deriving the New Parameter Based on the Quadratic Model
Algorithm 1. The new conjugate gradient algorithm for minimizing. |
. |
then stop. |
by (9) and (10). |
by (19). |
. |
and go to stage 1. |
3. Convergence Analysis of the Uniformly Convex Function
4. Numerical Results
- If was not satisfied.
- If iterations exceed 2000.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Image | Noise Level r (%) | FR-Method | BT1-Method | BT2-Method | BT3-Method | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
NI | NF | PSNR (dB) | NI | NF | PSNR (dB) | NI | NF | PSNR (dB) | NI | NF | PSNR (dB) | ||
Le | 50 | 82 | 153 | 30.5529 | 42.0 | 90.0 | 30.5077 | 55.0 | 109.0 | 30.4726 | 30.0 | 60.0 | 30.779 |
70 | 81 | 155 | 27.4824 | 45.0 | 97.0 | 27.3425 | 56.0 | 111.0 | 27.5176 | 53.0 | 107.0 | 27.2491 | |
90 | 108 | 211 | 22.8583 | 53.0 | 113.0 | 22.9824 | 58.0 | 115.0 | 23.0099 | 54.0 | 109.0 | 22.8871 | |
Ho | 50 | 52 | 53 | 30.6845 | 30.0 | 63.0 | 35.2072 | 35.0 | 70.0 | 34.9453 | 36.0 | 72.0 | 35.1792 |
70 | 63 | 116 | 31.2564 | 32.0 | 66.0 | 30.9014 | 39.0 | 78.0 | 30.7493 | 29.0 | 58.0 | 30.9249 | |
90 | 111 | 214 | 25.287 | 36.0 | 74.0 | 25.1023 | 52.0 | 103.0 | 25.267 | 48.0 | 96.0 | 25.1356 | |
El | 50 | 35 | 36 | 33.9129 | 24.0 | 48.0 | 33.8687 | 30.0 | 58.0 | 33.862 | 26.0 | 51.0 | 33.9353 |
70 | 38 | 39 | 31.864 | 17.0 | 32.0 | 31.9634 | 30.0 | 58.0 | 31.7931 | 34.0 | 68.0 | 31.7348 | |
90 | 65 | 114 | 28.2019 | 39.0 | 80.0 | 28.2067 | 44.0 | 86.0 | 28.0416 | 44.0 | 88.0 | 28.1316 | |
c512 | 50 | 59 | 87 | 35.5359 | 28.0 | 60.0 | 35.296 | 34.0 | 69.0 | 35.862 | 26.0 | 51.0 | 35.3528 |
70 | 78 | 142 | 30.6259 | 34.0 | 72.0 | 30.6113 | 39.0 | 79.0 | 30.6145 | 34.0 | 68.0 | 30.6749 | |
90 | 121 | 236 | 24.3962 | 47.0 | 98.0 | 24.9266 | 50.0 | 101.0 | 24.8411 | 44.0 | 88.0 | 24.8521 |
BT1 | BT2 | BT3 | FR | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
No. | Function | DIM | Initial | NOI | NOF | CPUT | NOI | NOF | CPUT | NOI | NOF | CPUT | NOI | NOF | CPUT |
1 | QUARTICM | 100 | (11,…,11) | 4 | 101 | 0.002562 | *** | *** | *** | *** | *** | *** | 17 | 108 | 0.00045 |
2 | QUARTICM | 1000 | (11,…,11) | *** | *** | *** | 4 | ## | 0.00287 | 6 | 202 | 0.00248 | 18 | 122 | 0.000572 |
3 | BIGGSB1 | 2 | (3,3) | 1 | 3 | 0.002401 | 1 | 3 | 0.00294 | 1 | 3 | 0.00303 | 1 | 3 | 0.002224 |
4 | BIGGSB1 | 2 | (11,11) | 1 | 3 | 0.002209 | 1 | 3 | 0.00192 | 1 | 3 | 0.00164 | 1 | 3 | 0.002617 |
5 | QUADRATIC QF | 2 | (0.01,0.01) | 2 | 4 | 0.002074 | 2 | 4 | 0.0038 | 2 | 4 | 0.00349 | 2 | 4 | 0.00311 |
6 | QUARTC | 100 | (11,…,11) | 4 | 101 | 0.010275 | *** | *** | *** | *** | *** | *** | 17 | 108 | 0.011307 |
7 | EXT PENALTY | 8000 | (1,1,…,1) | *** | *** | *** | *** | *** | *** | 3 | 30 | 0.00287 | *** | *** | *** |
8 | DIAGONAL 6 | 1000 | (0.5,…,0.5) | 5 | 33 | 0.017629 | 5 | 32 | 0.02154 | *** | *** | *** | 11 | 12 | 0.003807 |
9 | DIAGONAL 6 | 10,000 | (0.5,…,0.5) | 5 | 56 | 0.05131 | *** | *** | *** | *** | *** | *** | *** | *** | *** |
10 | DIAGONAL 6 | 50,000 | (0.5,…,0.5) | 3 | 33 | 0.13378 | *** | *** | *** | 4 | 58 | 0.23391 | *** | *** | *** |
11 | EXT DENSCHNB | 10,000 | (1,1,…,1) | 1 | 3 | 0.001311 | 1 | 3 | 0.00078 | 1 | 3 | 0.00074 | 1 | 3 | 0.00144 |
12 | EXT DENSCHNB | 50,000 | (1,1,…,1) | 1 | 3 | 0.00241 | 1 | 3 | 0.00282 | 1 | 3 | 0.00293 | 1 | 3 | 0.002363 |
13 | EXT DENSCHNB | 100,000 | (1,1,…,1) | 1 | 3 | 0.005923 | 1 | 3 | 0.00398 | 1 | 3 | 0.00579 | 1 | 3 | 0.005875 |
14 | MATYAS | 2 | (1,1) | 6 | 17 | 0.001374 | 6 | 17 | 0.00073 | 6 | 17 | 0.00078 | 7 | 36 | 0.000423 |
15 | MATYAS | 2 | (0.5,0.5) | 6 | 17 | 0.00052 | 6 | 17 | 0.00083 | 6 | 17 | 0.00059 | 7 | 36 | 0.000422 |
16 | BRENT | 2 | (11,11) | 1 | 3 | 0.000396 | 1 | 3 | 0.0004 | 1 | 3 | 0.00049 | 1 | 3 | 0.00071 |
17 | BRENT | 2 | (13,13) | 1 | 3 | 0.000602 | 1 | 3 | 0.00068 | 1 | 3 | 0.00045 | 1 | 3 | 0.000803 |
18 | BRENT | 2 | (3,3) | 1 | 3 | 0.000679 | 1 | 3 | 0.00037 | 1 | 3 | 0.00054 | 1 | 3 | 0.000375 |
19 | Rotated Ellipse 2 | 2 | (0.5, -1) | 13 | 21 | 0.001451 | 13 | 21 | 0.00137 | 13 | 21 | 0.00127 | 10 | 17 | 0.000349 |
20 | Rotated Ellipse 2 | 2 | (5,-5) | 1 | 3 | 0.000438 | 1 | 3 | 0.00044 | 1 | 3 | 0.00056 | 1 | 3 | 0.000501 |
21 | DIAGONAL 1 | 2 | (1,1) | 13 | 24 | 0.005212 | 12 | 19 | 0.00367 | 12 | 20 | 0.00372 | 15 | 23 | 0.003264 |
22 | DIAGONAL 2 | 2 | (1,1) | 14 | 19 | 0.003093 | 9 | 12 | 0.00236 | 10 | 13 | 0.00229 | 12 | 13 | 0.00164 |
23 | Aluffi-Pentini | 2 | (1,1) | 5 | 8 | 0.000786 | 5 | 8 | 0.00073 | 5 | 8 | 0.00119 | 6 | 10 | 0.000516 |
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Hassan, B.A.; Moghrabi, I.A.R.; Ameen, T.A.; Sulaiman, R.M.; Sulaiman, I.M. Image Noise Reduction and Solution of Unconstrained Minimization Problems via New Conjugate Gradient Methods. Mathematics 2024, 12, 2754. https://doi.org/10.3390/math12172754
Hassan BA, Moghrabi IAR, Ameen TA, Sulaiman RM, Sulaiman IM. Image Noise Reduction and Solution of Unconstrained Minimization Problems via New Conjugate Gradient Methods. Mathematics. 2024; 12(17):2754. https://doi.org/10.3390/math12172754
Chicago/Turabian StyleHassan, Bassim A., Issam A. R. Moghrabi, Thaair A. Ameen, Ranen M. Sulaiman, and Ibrahim Mohammed Sulaiman. 2024. "Image Noise Reduction and Solution of Unconstrained Minimization Problems via New Conjugate Gradient Methods" Mathematics 12, no. 17: 2754. https://doi.org/10.3390/math12172754