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

Behrouz Mataei; H. Zakeri; F. Moghadas Nejad

doi:10.1007/s11831-017-9230-7

Archives of Computational Methods in Engineering

pp 1–19 | Cite as

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

Authors
Authors and affiliations

Behrouz Mataei
H. ZakeriEmail author
F. Moghadas Nejad

Original Paper

First Online: 20 May 2017

Received: 10 March 2017

Accepted: 06 May 2017

57 Downloads

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.

Keywords

Discrete Wavelet Transform Image Enhancement Continuous Wavelet Transform Multiresolution Analysis Image Denoising

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

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

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Behrouz Mataei
- 1
H. Zakeri
- 2
Email author
F. Moghadas Nejad
- 2

1.Department of EngineeringRazi UniversityKermanshahIran
2.Department of Civil and Environment EngineeringAmirkabir University of TechnologyTehranIran

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

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