An Overview of Multiresolution Analysis for Nondestructive Evaluation of Pavement Surface Drainage

Abstract

Network level drainage assessment of the pavement surface plays a crucial role in controlling and decreasing the accident rate. Hydroplaning is one of the major causes of accidents in wet weather conditions and is the consequence of low drainage quality of pavement surfaces. Since no automated system currently exists for the pavement drainage evaluation, this work was conducted to present a new system to assess drainage process quality. For this aim, the saturation situation was simulated for pavement surface and photo acquisition was carried out on the drainage process of pavement surface after saturation. Finally, image processing method was applied to produce an index related to drainage quality. Using a proper method to enhance and prepare these images for the analysis step and find appreciate feature for the drainage quality is also among the necessities of drainage assessment. After a brief overview of multiresolution analysis, we revise the state-of-the-art of multiresolution analysis methods by discussing assessing parameters for asphalt surface image enhancement in nondestructive evaluation, formulated and fused to allow for a general comparison. In this work, different transform methods are used for asphalt surface image enhancement and a comparison is made between wavelet, curvelet, ridgelet, shearlet, and contourlet transforms by assessing parameters including TIME, PSNR, SNR, MSE, MAE, MSE, UQI, and SSIM. The comparison among the obtained results shows the superiority of shearlet transform over other transforms in providing of processed images with higher quality. Furthermore, it was found that ridgelet transform is more suitable for the jobs which time is the main parameter.

Appendix A: Filter Coefficients

Let $h$ and $g$ be the wavelet decomposition (analysis) filters, where $h~$is a lowpass filter and $g$ is a highpass filter. Let the dual filters ${h'}~$and $g'$ be the wavelet reconstruction (synthesis) filters. The coefficients of the wavelet filters are shown in the following sections:

Note:Wavelets are indexed by the number of vanishing moments; for example, “daubechies 2” has two vanishing moments and four tap filters.

Haar

h0 = 0.7071067812	g0 = −0.7071067812	h′ 0 = 0.7071067812	g′ 0 = 0.7071067812
h1 = 0.7071067812	g1 = 0.7071067812	h′ 1 = 0.7071067812	g′ 1 = −0.7071067812

Daubechies

‘db1’

h0 = 0.7071067812	g0 = −0.7071067812	h′ 0 = 0.7071067812	g′ 0 = 0.7071067812
h1 = 0.7071067812	g1 = 0.7071067812	h′ 1 = 0.7071067812	g′ 1 = −0.7071067812

‘db2’

h0 = −0.1294095226	g0 = −0.4829629131	h′ 0 = 0.4829629131	g′ 0 = −0.1294095226
h1 = 0.2241438680	g1 = 0.8365163037	h′ 1 = 0.8365163037	g′ 1 = −0.2241438680
h2 = 0.8365163037	g2 = −0.2241438680	h′ 2 = 0.2241438680	g′ 2 = 0.8365163037
h3 = 0.4829629131	g3 = −0.1294095226	h′ 3 = −0.1294095226	g′ 3 = −0.4829629131

‘db10’

h0 = −0.0000132642	g0 = −0.0266700579	h′ 0 = 0.0266700579	g′ 0 = −0.0000132642
h1 = 0.0000935887	g1 = 0.1881768001	h′ 1 = 0.1881768001	g′ 1 = −0.0000935887
h2 = −0.0001164669	g2 = −0.5272011889	h′ 2 = 0.5272011889	g′ 2 = −0.0001164669
h3 = −0.0006858567	g3 = 0.6884590395	h′ 3 = 0.6884590395	g′ 3 = 0.0006858567
h4 = 0.0019924053	g4 = −0.2811723437	h′ 4 = 0.2811723437	g′ 4 = 0.0019924053
h5 = 0.0013953517	g5 = −0.2498464243	h′ 5 = −0.2498464243	g′ 5 = −0.0013953517
h6 = −0.0107331755	g6 = 0.1959462744	h′ 6 = −0.1959462744	g′ 6 = −0.0107331755
h7 = 0.0036065536	g7 = 0.1273693403	h′ 7 = 0.1273693403	g′ 7 = −0.0036065536
h8 = 0.0332126741	g8 = −0.0930573646	h′ 8 = 0.0930573646	g′ 8 = 0.0332126741
h9 = −0.0294575368	g9 = −0.0713941472	h′ 9 = −0.0713941472	g′ 9 = 0.0294575368
h10 = −0.0713941472	g10 = 0.0294575368	h′ 10 = −0.0294575368	g′ 10 = −0.0713941472
h11 = 0.0930573646	g11 = 0.0332126741	h′ 11 = 0.0332126741	g′ 11 = −0.0930573646
h12 = 0.1273693403	g12 = −0.0036065536	h′ 12 = 0.0036065536	g′ 12 = 0.1273693403
h13 = −0.1959462744	g13 = −0.0107331755	h′ 13 = −0.0107331755	g′ 13 = 0.1959462744
h14 = −0.2498464243	g14 = −0.0013953517	h′ 14 = 0.0013953517	g′ 14 = −0.2498464243
h15 = 0.2811723437	g15 = 0.0019924053	h′ 15 = 0.0019924053	g′ 15 = −0.2811723437
h16 = 0.6884590395	g16 = 0.0006858567	h′ 16 = −0.0006858567	g′ 16 = 0.6884590395
h17 = 0.5272011889	g17 = −0.0001164669	h′ 17 = −0.0001164669	g′ 17 = −0.5272011889
h18 = 0.1881768001	g18 = −0.0000935887	h′ 18 = 0.0000935887	g′ 18 = 0.1881768001
h19 = 0.0266700579	g19 = −0.0000132642	h′ 19 = −0.0000132642	g′ 19 = −0.0266700579

‘db15’

h0 = 0.0000000613	g0 = −0.0045385374	h′ 0 = 0.0045385374	g′ 0 = 0.0000000613
h1 = −0.0000006317	g1 = 0.0467433949	h′ 1 = 0.0467433949	g′ 1 = 0.0000006317
h2 = 0.0000018113	g2 = −0.2060238640	h′ 2 = 0.2060238640	g′ 2 = 0.0000018113
h3 = 0.0000033630	g3 = 0.4926317717	h′ 3 = 0.4926317717	g′ 3 = −0.0000033630
h4 = −0.0000281333	g4 = −0.6458131404	h′ 4 = 0.6458131404	g′ 4 = −0.0000281333
h5 = 0.0000257927	g5 = 0.3390025355	h′ 5 = 0.3390025355	g′ 5 = −0.0000257927
h6 = 0.0001558965	g6 = 0.1932041396	h′ 6 = −0.1932041396	g′ 6 = 0.0001558965
h7 = −0.0003595652	g7 = −0.2888825966	h′ 7 = −0.2888825966	g′ 7 = 0.0003595652
h8 = −0.0003734824	g8 = −0.0652829528	h′ 8 = 0.0652829528	g′ 8 = −0.0003734824
h9 = 0.0019433240	g9 = 0.1901467140	h′ 9 = 0.1901467140	g′ 9 = −0.0019433240
h10 = −0.0002417565	g10 = 0.0396661766	h′ 10 = −0.0396661766	g′ 10 = −0.0002417565
h11 = −0.0064877346	g11 = −0.1111209360	h′ 11 = −0.1111209360	g′ 11 = 0.0064877346
h12 = 0.0051010004	g12 = −0.0338771439	h′ 12 = 0.0338771439	g′ 12 = 0.0051010004
h13 = 0.0150839180	g13 = 0.0547805506	h′ 13 = 0.0547805506	g′ 13 = −0.0150839180
h14 = −0.0208100502	g14 = 0.0257670073	h′ 14 = −0.0257670073	g′ 14 = −0.0208100502
h15 = −0.0257670073	g15 = −0.0208100502	h′ 15 = −0.0208100502	g′ 15 = 0.0257670073
h16 = 0.0547805506	g16 = −0.0150839180	h′ 16 = 0.0150839180	g′ 16 = 0.0547805506
h17 = 0.0338771439	g17 = 0.0051010004	h′ 17 = 0.0051010004	g′ 17 = −0.0338771439
h18 = −0.1111209360	g18 = 0.0064877346	h′ 18 = −0.0064877346	g′ 18 = −0.1111209360
h19 = −0.0396661766	g19 = −0.0002417565	h′ 19 = −0.0002417565	g′ 19 = 0.0396661766
h20 = 0.1901467140	g20 = −0.0019433240	h′ 20 = 0.0019433240	g′ 20 = 0.1901467140
h21 = 0.0652829528	g21 = −0.0003734824	h′ 21 = −0.0003734824	g′ 21 = −0.0652829528
h22 = −0.2888825966	g22 = 0.0003595652	h′ 22 = −0.0003595652	g′ 22 = −0.2888825966
h23 = −0.1932041396	g23 = 0.0001558965	h′ 23 = 0.0001558965	g′ 23 = 0.1932041396
h24 = 0.3390025355	g24 = −0.0000257927	h′ 24 = 0.0000257927	g′ 24 = 0.3390025355
h25 = 0.6458131404	g25 = −0.0000281333	h′ 25 = −0.0000281333	g′ 25 = −0.6458131404
h26 = 0.4926317717	g26 = −0.0000033630	h′ 26 = 0.0000033630	g′ 26 = 0.4926317717
h27 = 0.2060238640	g27 = 0.0000018113	h′ 27 = 0.0000018113	g′ 27 = −0.2060238640
h28 = 0.0467433949	g28 = 0.0000006317	h′ 28 = −0.0000006317	g′ 28 = 0.0467433949
h29 = 0.0045385374	g29 = 0.0000000613	h′ 29 = 0.0000000613	g′ 29 = −0.0045385374

coiflets 1

h0 = −0.0156557281	g0 = 0.0727326195	h′ 0 = −0.0727326195	g′ 0 = −0.0156557281
h1 = −0.0727326195	g1 = 0.3378976625	h′ 1 = 0.3378976625	g′ 1 = 0.0727326195
h2 = 0.3848648469	g2 = −0.8525720202	h′ 2 = 0.8525720202	g′ 2 = 0.3848648469
h3 = 0.8525720202	g3 = 0.3848648469	h′ 3 = 0.3848648469	g′ 3 = −0.8525720202
h4 = 0.3378976625	g4 = 0.0727326195	h′ 4 = −0.0727326195	g′ 4 = 0.3378976625
h5 = −0.0727326195	g5 = −0.0156557281	h′ 5 = −0.0156557281	g′ 5 = 0.0727326195

Biorthogonal 1.1

h0 = 0.7071067812	g0 =  = −0.7071067812	h′ 0 = 0.7071067812	g′ 0 = 0.7071067812
h1 = 0.7071067812	g1 = 0.7071067812	h′ 1 = 0.7071067812	g′ 1 = −0.7071067812

Reverse biorthogonal 1.1

h0 = 0.7071067812	g0 = −0.7071067812	h′ 0 = 0.7071067812	g′ 0 = 0.7071067812
h1 = 0.7071067812	g1 = 0.7071067812	h′ 1 = 0.7071067812	g′ 1 = −0.7071067812

Symlets 2

h0 = −0.1294095226	g0 = −0.4829629131	h′ 0 = 0.4829629131	g′ 0 = −0.1294095226
h1 = 0.2241438680	g1 = 0.8365163037	h′ 1 = 0.8365163037	g′ 1 = −0.2241438680
h2 = 0.8365163037	g2 = −0.2241438680	h′ 2 = 0.2241438680	g′ 2 = 0.8365163037
h3 = 0.4829629131	g3 = −0.1294095226	h′ 3 = −0.1294095226	g′ 3 = −0.4829629131

Discrete Meyer

h0 = 0	g0 = 0.0000000000	h′ 0 = −0.0000000000	g′ 0 = 0
h1 = −0.0000000000	g1 = 0.0000000085	h′ 1 = 0.0000000085	g′ 1 = 0.0000000000
h2 = 0.0000000085	g2 = 0.0000000111	h′ 2 = −0.0000000111	g′ 2 = 0.0000000085
h3 = −0.0000000111	g3 = −0.0000000108	h′ 3 = −0.0000000108	g′ 3 = 0.0000000111
h4 = −0.0000000108	g4 = −0.0000000607	h′ 4 = 0.0000000607	g′ 4 = −0.0000000108
h5 = 0.0000000607	g5 = −0.0000001087	h′ 5 = −0.0000001087	g′ 5 = −0.0000000607
h6 = −0.0000001087	g6 = −0.0000000820	h′ 6 = 0.0000000820	g′ 6 = −0.0000001087
h7 = 0.0000000820	g7 = 0.0000001178	h′ 7 = 0.0000001178	g′ 7 = −0.0000000820
h8 = 0.0000001178	g8 = 0.0000005506	h′ 8 = −0.0000005506	g′ 8 = 0.0000001178
h9 = −0.0000005506	g9 = 0.0000011308	h′ 9 = 0.0000011308	g′ 9 = 0.0000005506
h10 = 0.0000011308	g10 = 0.0000014895	h′ 10 = −0.0000014895	g′ 10 = 0.0000011308
h11 = −0.0000014895	g11 = 0.0000007368	h′ 11 = 0.0000007368	g′ 11 = 0.0000014895
h12 = 0.0000007368	g12 =−0.0000032054	h′ 12 = 0.0000032054	g′ 12 = 0.0000007368
h13 = 0.0000032054	g13 = −0.0000163127	h′ 13 = −0.0000163127	g′ 13 = −0.0000032054
h14 = −0.0000163127	g14 = −0.0000655431	h′ 14 = 0.0000655431	g′ 14 = −0.0000163127
h15 = 0.0000655431	g15 = −0.0006011502	h′ 15 = −0.0006011502	g′ 15 = −0.0000655431
h16 = −0.0006011502	g16 = 0.0027046721	h′ 16 = −0.0027046721	g′ 16 = −0.0006011502
h17 = −0.0027046721	g17 = 0.0022025341	h′ 17 = 0.0022025341	g′ 17 = 0.0027046721
h18 = 0.0022025341	g18 = −0.0060458141	h′ 18 = 0.0060458141	g′ 18 = 0.0022025341
h19 = 0.0060458141	g19 = −0.0063877183	h′ 19 = −0.0063877183	g′ 19 = −0.0060458141
h20 = −0.0063877183	g20 = 0.0110614964	h′ 20 = −0.0110614964	g′ 20 = −0.0063877183
h21 = −0.0110614964	g21 = 0.0152700151	h′ 21 = 0.0152700151	g′ 21 = 0.0110614964
h22 = 0.0152700151	g22 = −0.0174234341	h′ 22 = 0.0174234341	g′ 22 = 0.0152700151
h23 = 0.0174234341	g23 = −0.0321307940	h′ 23 = −0.0321307940	g′ 23 = −0.0174234341
h24 = −0.0321307940	g24 = 0.0243487459	h′ 24 = −0.0243487459	g′ 24 = −0.0321307940
h25 = −0.0243487459	g25 = 0.0637390243	h′ 25 = 0.0637390243	g′ 25 = 0.0243487459
h26 = 0.0637390243	g26 = −0.0306550920	h′ 26 = 0.0306550920	g′ 26 = 0.0637390243
h27 = 0.0306550920	g27 = −0.1328452004	h′ 27 = −0.1328452004	g′ 27 = −0.0306550920
h28 = −0.1328452004	g28 = 0.0350875557	h′ 28 = −0.0350875557	g′ 28 = −0.1328452004
h29 = −0.0350875557	g29 = 0.4445930028	h′ 29 = 0.4445930028	g′ 29 = 0.0350875557
h30 = 0.4445930028	g30 = −0.7445855923	h′ 30 = 0.7445855923	g′ 30 = 0.4445930028
h31 = 0.7445855923	g31 = 0.4445930028	h′ 31 = 0.4445930028	g′ 31 = −0.7445855923
h32 = 0.4445930028	g32 = 0.0350875557	h′ 32 = −0.0350875557	g′ 32 = 0.4445930028
h33 = −0.0350875557	g33 = −0.1328452004	h′ 33 = −0.1328452004	g′ 33 = 0.0350875557
h34 = −0.1328452004	g34 = −0.0306550920	h′ 34 = 0.0306550920	g′ 34 = −0.1328452004
h35 = 0.0306550920	g35 = 0.0637390243	h′ 35 = 0.0637390243	g′ 35 = −0.0306550920
h36 = 0.0637390243	g36 = 0.0243487459	h′ 36 = −0.0243487459	g′ 36 = 0.0637390243
h37 = −0.0243487459	g37 = −0.0321307940	h′ 37 = −0.0321307940	g′ 37 = 0.0243487459
h38 = −0.0321307940	g38 = −0.0174234341	h′ 38 = 0.0174234341	g′ 38 = −0.0321307940
h39 = 0.0174234341	g39 = 0.0152700151	h′ 39 = 0.0152700151	g′ 39 = −0.0174234341
h40 = 0.0152700151	g40 = 0.0110614964	h′ 40 = −0.0110614964	g′ 40 = 0.0152700151
h41 = −0.0110614964	g41 = −0.0063877183	h′ 41 = −0.0063877183	g′ 41 = 0.0110614964
h42 = −0.0063877183	g42 = −0.0060458141	h′ 42 = 0.0060458141	g′ 42 = −0.0063877183
h43 = 0.0060458141	g43 = 0.0022025341	h′ 43 = 0.0022025341	g′ 43 = −0.0060458141
h44 = 0.0022025341	g44 = 0.0027046721	h′ 44 = −0.0027046721	g′ 44 = 0.0022025341
h45 = −0.0027046721	g45 = −0.0006011502	h′ 45 = −0.0006011502	g′ 45 = 0.0027046721
h46 = −0.0006011502	g46 = −0.0000655431	h′ 46 = −0.0000655431	g′ 46 = −0.0006011502
h47 = 0.0000655431	g47 = −0.0000163127	h′ 47 = −0.0000163127	g′ 47 = −0.0000655431
h48 = −0.0000163127	g48 = −0.0000032054	h′ 48 = 0.0000032054	g′ 48 = −0.0000163127
h49 = 0.0000032054	g49 = 0.0000007368	h′ 49 = 0.0000007368	g′ 49 = −0.0000032054
h50 = 0.0000007368	g50 = 0.0000014895	h′ 50 = −0.0000014895	g′ 50 = 0.0000007368
h51 = −0.0000014895	g51 = 0.0000011308	h′ 51 = 0.0000011308	g′ 51 = 0.0000014895
h52 = 0.0000011308	g52 = 0.0000005506	h′ 52 = −0.0000005506	g′ 52 = 0.0000011308
h53 = −0.0000005506	g53 = 0.0000001178	h′ 53 = 0.0000001178	g′ 53 = −0.0000005506
h54 = 0.0000001178	g54 = −0.0000000820	h′ 54 = 0.0000000820	g′ 54 = 0.0000001178
h55 = 0.0000000820	g55 = −0.0000001087	h′ 55 = −0.0000001087	g′ 55 = −0.0000000820
h56 = −0.0000001087	g56 = −0.0000000607	h′ 56 = 0.0000000607	g′ 56 = −0.0000001087
h57 = 0.0000000607	g57 = −0.0000000108	h′ 57 = −0.0000000108	g′ 57 = −0.0000000607
h58 = −0.0000000108	g58 = 0.0000000111	h′ 58 = −0.0000000111	g′ 58 = −0.0000000108
h59 = −0.0000000111	g59 = 0.0000000085	h′ 59 = 0.0000000085	g′ 59 = 0.0000000111
h60 = 0.0000000085	g60 = 0.0000000000	h′ 60 = −0.0000000000	g′ 60 = 0.0000000085
h61 = −0.0000000000	g61 = 0	h′ 61 = 0	g′ 61 = 0.0000000000