Surface reconstruction with Wavelet transformation

Transcription

1 INES th Jubilee IEEE International Conference on Intelligent Engineering Systems June 30-July 2, 2016 Budapest, Hungary Surface reconstruction with Wavelet transformation Petra Balla, Péter Kocsis, György Eigner and Ákos Antal Department of Mechatronics, Optics and Mechanical Engineering Informatics, BUTE, Budapest, Hungary Research and Innovation Center of Obuda University, Physiological Controls Group, Obuda University, Budapest, Hungary Abstract The development in computing power highlights some forgotten algorithms, which were neglected because of their complexity and slowness on early computers. One example is the Wavelet-Transformation Profilometry (WTP) of which successful application is demonstrated in the paper. WTP is a high level signal processing method using orthogonal algorithms for huge datasets. The high performance in quality and running speed makes the described method suitable for medical image processing applications. 1. INTRODUCTION The performance of the computers increased significantly through the development of science that is why many of the mathematical methods had become available to restore the surface, like the Fourier-, Gábor-, S-, Hilbert- or Wavelettransform. The orthogonal trajectories act particularly important part from these, because they can be easily adapted to the environment and give accurate results. Further advantages are that the entire spatial analysis needs only one image and the accuracy can be significantly improved with the function of frequency if there are enough available pixels. The disadvantage is that the higher number of pixels means more calculation that is why the speed slows down, however it can be significantly reduced in the future [1]. Successful use of the Fourier-transform method was first published in 1982 by M. Takeda [2], [3], who used a one-dimensional procedure, but in 1986 a two-dimensional method has been successfully applied by Bone [4]. The paper is structured as follows: first, the regarding Fourier- and Wavelet-transformations are presented. After, the Wavelet-transformation profilometry is demonstrated. In Sec. 5 we present the test results. Finally, the development opportunities and the conclusion are coming. 2. USAGE OF FOURIER-TRANSFORM ANALYSIS The Fourier-Transform Analysis (FTA) has many application fields, including the field of image processing [5]. In order to restore a particular surface a reference sample (g 0 (x, y)) and the deformed surface (g(x, y)) are needed: where r(x, y) and r 0 (x, y)) are the concurrent component of the non-uniform reflections, A n is the weighting factor, f 0 is the carrier frequency, and ϕ(x, y) and ϕ 0 (x, y) are the value of the phase. The Fourier-transform of the resulting complex amount gives the spectrum, and the respective components can be determined by filtration. The phase difference between the modulated signal and reference signal can be calculated by using inverse Fourier-transform: ḡ(x, y) = A 1 r(x, y)e 2πf0x+nϕ(x,y), ḡ 0 (x, y) = A 1 r 0 (x, y)e 2πf0x+nϕ0(x,y). The disadvantage of this process is that it breaks the signals into sinusoidal harmonics (as it can be seen on Fig. 1), that is why only frequency domain analysis may be carried out with it and the time (or dimensional in this case) analysis is not. This means that the Fourier-transform procedure has no temporal localization properties, so the place of the examined frequency component cannot be determined with the function of time. Because of the breakdown of the signal into horizontal waves, if the signal is changed in one place its Fourier-transform changes as well, thus the location of the changes cannot be determined. In addition, the disadvantage that signal analysis errors can occur because the overlapping over and above the counting is difficult because the sine and cosine functions and their long burst values should be calculated, which requires large computational resources. These errors can be fixed with the Wavelet-Transform Analysis (WTA) [6]. Figure 1: a) Square wave approximation, b) Saw tooth wave approximation, c) Triangle wave approximation (2) g(x, y) = r(x, y) A n e 2πf0x+nϕ(x,y), g 0 (x, y) = r 0 (x, y) n= n= A n e 2πf0x+nϕ0(x,y), (1) 3. WAVELET-TRANSFORMATION ANALYSIS The WTA procedure is relatively new, though its foundations have been known at (since) the beginning of the 20th /16/$ IEEE 201

2 P. Balla et al. Surface Reconstruction with Wavelet Transformation century. Although it did not promise much success, thanks to Meyer s and Mallatt s work [5], [7] from late 1980s and early 1990s the research of the wavelets and the further development of the older methods have begun on international level. The advantage is that the WTA is a spectral resolution process, which is not sinusoidal anymore, but divides the signal into the so-called mother wavelets. These wavelet signals have limited length with a mean value of zero. Thanks to them, we can correct the errors of the FTA, because the time analysis - in this case spatial analysis - is also feasible with it. A. Main properties The common property of the WTA and FTA is that both methods measure the similarity between the signal under test and a test function and uses internal multiplying mathematical tools. This can be written as < u, v >= u v cosθ, where if u and v is a unit, then the cosine of the angle between them determines the result, so it is limited into the [ 1, 1] interval. This means that in the analysis u will be the signal under test and v will be the test function. As it was mentioned, the difference between the two procedures is the test function, which represents the recoverable information. In the case of FTA the harmonics of the sine function are regular and smooth, while in case of WTA they are irregular and asymmetric, so the sharp and sudden changes of the signal can be detected easier [8]. The compressions and stretching of the function is called scaling. Our main goal is to measure the similarity between the two functions, which is accomplished with a twovariable equation for the wavelets, using positions and scales. For the three-dimensional profilometry using complex motherwavelets is needed, when the wavelet transform is a complex function of the scale and position. The mentioned motherwavelets generally classified into families which are the basis for the most common grouping. The basis-function of the Wavelet-transformation has to be a compressed and shifted version of a base-wavelet and the following correlations have to be correct, where ψ(t) is the wavelet-function: ψ(t)dt. (3) Here, ψ(ω) = 0 if ω = 0, where ψ(ω) is the Fouriertransform of the wavelet-function. The most important types of them from our application point of view are available in the MATLAB software and we used these in this study: Complex Gaussian Shannon ψ Gaussian (x) = d(c p exp( ix) exp( x 2 )) p dx p, (4) ψ Shannon (x) = f b exp(2πif c x)(sinc(f b x)), (5) Frequency B-spline ψ b spline (x) = ( ( )) fb x f b exp(2πif c x) sinc, m (6) Complex Morlet ( ) 1 x 2 ψ Morlet (x) = (fb 2 exp(2πif cx) exp π)1/4 2fb 2. (7) The value of the compression (a) and the shifting (τ) defines a two variable function, which can be prescribed as F (a, τ) = 1 a f(t)ψ t τ dt, (8) a 1 where ψ(t) is the wavelet-function, ψ t τ the basisfunction of the transformation, and F (a, τ) is the wavelet a a transformation itself. 4. WAVELET-TRANSFORMATION PROFILOMETRY In the field of Wavelet-Transformation Profilometry (WTP) we can distinguish one- and two dimensional methods but the experience shows that the former one is more useful. In this case a line of pixels of a given image are analyzed, and then repeated for each row. After the detection of the phase a demodulation method follows it, which is performed with phase-evaluation technique when we can perform the evaluation with arcus tangent operations. The generated patterns can be prepared by appropriate modification of a sine wave, so the generated phase pattern can be written as the following equation: f(x, y) = a(x, y) + b(x, y)cos(2πf 0 x + ϕ(x, y)). (9) In (9) a(x, y) is the back-light, b(x, y) is the amplitude of the bars, f 0 is the spatial carrier frequency and ϕ(x, y) is the phase modulation of the bars. The one-dimensional continuous wavelet transformation on the test signal creates a complex array, which modulus and phase can be determined with abs(a, τ) = W (a, τ), Im(W (a, τ)) 1 ϕ(a, τ) = tan Re(W (a, τ)), (10) where Im(W (a, τ)) is the imaginary part and Re(W (a, τ)) is the real part of the transformation. After this procedure we used a direct maximum method [9] to evaluate the phase and modulus of every pixel, but this phase map is characterized by tears. That is why necessary to use an unwrapping algorithm [10], which aims to improve or even stop the discontinuity of the signal. The unwrapped phase map is now possible to use (can now be used) for depth information extraction with relative coordinates. In this case we determined the depth values of the pixels (h(x, y)) compared to a defined reference plane. To do that, we needed the phase of the transmitted signal, which can be calculated by φ(x) = φ(x) 2πf 0 x. This can be further formed to be able to evaluate h(x, y) parameter as well [11]: φ(x, y) = 2πf 0h(x, y)d l 0 h(x, y). (11) 202

3 INES th Jubilee IEEE International Conference on Intelligent Engineering Systems June 30-July 2, 2016 Budapest, Hungary If we accept the assumption that the relative distance between the camera and projector is negligibly small compared to the distance from the surface (l0 >> d) the equation is simplified and modifying it we can obtain the equation for the depth information [12]: h(x, y) = φ(x, y)l0. 2πf0 d EF-S For the tested object we choose a loupe, measured its radius with a three-ball spherometer, then we painted it to white. It was necessary because the WTP works only in non-reflecting surfaces. The test object can be seen on Fig. 3. (12) For example in case of a given application, when ϕ(x, y) is constrained to its principal value, either the interval ( π, π] or [0, 2π), it is called wrapped phase. Otherwise it is called unwrapped phase. These can be seen on Fig. 2. Figure 3: The captured picture B. Transformation method Figure 2: a and c: Wrapped phase; b and d: Unwrapped phase [11] 5. T EST FRAMEWORK A. Capturing the image For this method three important parts are needed: Projector, Camera, The object. Two types of projectors were tried, both had advantage and disadvantage as well. The first was a digital one, controlled by a computer, therefore sinus wave with any frequency could to be projected. It had very good color depth, although it could not project real black. That is why the second was an oldtime, analog one. With the help of this, we could project sinus waves in two frequencies only these are in a slide image. It had real good black parts, however, the color of the bulb was yellowish. Due to this effect the amplitude of the wave of the tested picture becomes lower. That is why we used the first one, because the picture made with it contains more information about the surface. The camera was a Canon 350D DS No , the lens was Canon Zoom Lens In order to realize and test the necessary algorithms, we have chosen the Matlab software, because it has one of the highest speeds in mathematical calculations and built-in wavelet families. Fig. 4 shows the initial image, which was captured from the surface of the object (it can be compared with the central region of the sample object on Fig. 3). After loading the image we have to ensure that the picture can be transformed into arithmetical progression. After that the method can be started: choosing the mother wavelet and the level of the WT, we transform each row of the picture, use a maximum ridge search and unwrap the phase. To reconstruct the surface we need to extract a reference from the generated one. For this process another picture about a flat surface was used, but in our case we decided to create an artificial one. To do that, two important information were needed: 1) The frequency of the projected sinus wave, which is known 2) To be sure that the center of the loupe is in the center of the cut picture After using the l0 /(2πf0 d) multiplier on the generated one we have to set X and Y direction multiplier for the artificial one. In the result the parallel edges of the reconstructed surface has to be in the same value. For the evaluation we cut the middle line and searched its radius. 203

4 P. Balla et al. Surface Reconstruction with Wavelet Transformation Table II: X and Y direction radius with Complex fb-spline f b m f c FBSP Rx Ry % was the smallest error (at 385.7mm), which can be reached with the application of Frequency B-spline method with the f b = 2, m = 1, f c = 1 parameter set. However, the algorithm produces various results with high dispersion in the different directions, when the tuning parameters were changed. Table III: X and Y direction radius with Complex Shannon C. Results Figure 4: Ready for processing image The radius of the test object (a usual loupe, which was painted to white) was measured with a three-ball spherometer, and got it as 390,418 mm. Since, it is a symmetrical object, we assumed that radius is the same in every directions (the manufacturing errors were neglected). The goal was from this point to find that methods (from the above listed ones), which is the most appropriate to provide the same result. We tried each applications (Complex Gaussian, Complex Shannon, Frequency B-spline and Complex Morlet), with different tuning parameters. The process was the following: first, the surface reconstructions were produced. After the surface reconstruction, we were able to measure the radius in vertical (Ry) and horizontal (Rx) directions. The results of the tests are shown in Table I-IV: Table I: X and Y direction radius with Complex Morlet f b f c CMOR Rx Ry The best result with the Complex Morley algorithm (386.6mm, error: %) was occurred, when the tuning parameters f b and f c were equal to 1 1, respectively. However, the algorithm only produces close to the original radius values in the vertical (Rx) directions. f b f c SHAN Rx Ry The Complex Shannon method produced the highest error in the radius in both directions. The only assessable result was with f b = 1 and f c = 1 parameters, however, the error was around 3.14% (at 375.4mm). We found that the produced results in the Ry direction were useless - each of them was significantly different then the original radius. Table IV: X and Y direction radius with Complex Gaussian p CGAU Rx Ry It can be seen, that the Complex Gauss method provided the best results in the Rx direction. The radius values are more precise, than in other cases with several parameter sets. We got the best match with p = 3 (a third ordered Gauss method), which means 0.28% error. The radius values in the Y direction were the closest in this case, however, the errors were intolerable high. We found that, the waviness and disturbance effects of the captured picture (as it can be seen on Fig. 5) make the precise operation of the algorithms inaccurate. The solution can be an other filtering (smoothing) method - however, we did not apply such kind of application, yet. From our goals point of view, namely, to try which is the best algorithm which can provide the best approximated radius data the experiments were successful. We was able to recover (approximately) the radius value with small error with every algorithm. Due to the fact, that the test object was symmetrical, a one direction equivalence is enough to estimate the 204

5 INES th Jubilee IEEE International Conference on Intelligent Engineering Systems June 30-July 2, 2016 Budapest, Hungary approximate radius value. Fig. 6 represents the best matched reconstructed radius (belongs to the Complex Gaussian algorithm, with p = 3). As it can see, the waviness - at least in the Rx direction - is almost invisible and we have got a clear result. Figure 5: Reconstructed surface Figure 6: Tested radius 6. DEVELOPMENT OPPORTUNITIES First of all we are working on terminating the waving. There are already some methods for this, but all of them reduced the other direction accuracy. After we reach a suitable preciseness we would like to use it in biomechatronics tasks, like measuring the spinal curvature, which is now a quickly developing research of our University [13], [14]. are going to try to use this method by real examinations as well. Medical application could be our target area, especially the field of spinal deformations, where our group has also encouraging outcomes using other optical measurement methods. ACKNOWLEDGMENT Gy. Eigner acknowledge the support of the Robotics Special College of Obuda University and the Doctoral School of Applied Informatics and Applied Mathematics of Obuda University. The research was also supported by the Research and Innovation Center of Obuda University. REFERENCES [1] R.M. Aciu and H. Ciocarlie, Runtime Translation of the Java Bytecode to OpenCL and GPU Execution of the Resulted Code, ACTA Pol Hung, vol. 13, no. 3, pp , [2] M. Takeda, H. Ina, and S. Kobayashi, Fourier-Transform Method of Fringe-Pattern Analysis for Computer-Based Topography and Interferometry, J Opt Soc Am, vol. 72, no. 1, pp , [3] M. Takeda, K. Mutoh, and S. Kobayashi, Fourier-Transform Profilometry for the Automatic-Measurement of 3-D Object Shapes, Appl Optics, vol. 22, no. 24, pp , [4] D.J. Bone, H.A. Bachor, and R.J. Sandeman, Fringe-Pattern Analysis Using a 2-D Fourier-Transform, Appl Optics, vol. 25, no. 10, pp , [5] S.G. Mallat, A theory for multiresolution signal decomposition: The wavelet representation, University of Pennsylvania, Tech. Rep., [6] A.Z. Abid, M.A. Gdeisat, D.R. Burton, and M.J. Lalor, Ridge extraction algorithms for one-dimensional continuous wavelet transform: a comparison, Journal of Physics - Conference Series, vol. 76, no. 1, pp. 1 7, [7] Y. Meyer, Wavelets, Algorithms & Applications, 1st ed. Philadelphiam, USA: SIAM, [8] P. Hariharan, Basics of Interferometry, 2nd ed. San Diego, USA: Academic Press, [9] A.Z. Abid, M.A. Gdeisat, D.R. Burton, and M.J. Lalor. (2016) A comparison between wavelet fringe analysis algorithms. [Online]. Available: photon06archive.iopconfs.org/fasig%203%20wed% doc [10] A. Rene, W.L.H. Carmona, and T. Brun, Characterization of Signals by the Ridges of Their Wavelet Transforms, IEEE T Signal Proces, vol. 45, no. 10, pp , [11] R. Talebi, J. Johnson, and A. Abdel-Daye, Binary code pattern unwrapping technique on fringe projection method, in Proceedings of the 17th International Conference on Image Processing, Computer Vision, & Pattern Recognition (IPCV 13), p. P7. [12] Z. Zhang and J. Zhong, Applicability analysis of wavelet- transform profilometry, Opt Express, vol. 21, pp , [13] P. Balla, G. Manhertz, and A. Ákos, Diagnostic moiré image evaluation in spinal deformities - accepted for publication, Opt Appl, vol. 46, no. 4. [14] P. Balla, K. Prommer, and A. Ákos, Investigation of digital Moire pictures for follow-up of patients with spinal deformities (in Hungarian), Biomechanica Hungarica, vol. 7, no. 1, pp , CONCLUSION In this paper we successfully applied the WTP in surface identification. According to our results in this example the Complex Gaussian wavelet proved to be the best solution, however for the complete statement further examinations are needed. If our further results are as appropriate as these, we 205

6 P. Balla et al. Surface Reconstruction with Wavelet Transformation 206