Noise reduction using wavelet cycle spinning: analysis of useful periodicities in the z-transform domain

Cycle spinning (CS) and a'trous algorithms are different implementations of the undecimated wavelet transform (UWT). Both algorithms can be used for UWT and even though the resulting wavelet coefficients are different, they keep a correspondence. This paper describes an analysis of the CS algorithm performed in the z-transform domain, showing the similarities and differences with the a'trous implementation. CS generates more wavelet coefficients than a'trous, but the number of significative and different coefficients is the same in both cases because of the occurrence of a periodic repetition in CS coefficients. Mathematical expressions for the relationship between CS and a'trous coefficients and for CS coefficient periodicities are provided in the z-transform domain. In some wavelet denoising applications, periodicities (present in the coefficients of the CS procedure) can also be found in the performance measure of the processed signals. In particular, in ultrasonic CS denoising applications, periodicities have been appreciated in the signal-to-noise ratio (SNR) of the ultrasonic denoised signals. These periodicities can be used to optimize the number of CS coefficients for an efficient implementation. Two examples showing the periodicities in the SNR are included. A selection of several reduced sets of CS wavelet coefficients has been utilized in the examples, and the SNRs resulting after denoising are analyzed.This work was partially supported by Spanish MCI Project DPI2011-22438 and MEC Project TIN2013-47272-C2-1-R. The translation of this paper was funded by the Universitat Politecnica de Valencia, Spain.Rodríguez-Hernández, MA.; San Emeterio, JL. (2016). Noise reduction using wavelet cycle spinning: analysis of useful periodicities in the z-transform domain. Signal, Image and Video Processing. 10(3):519-526. https://doi.org/10.1007/s11760-015-0762-8S519526103Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, New York (1999)Kovacevic, J., Goyal, V.K., Vetterli, M.: Signal Processing Fourier and Wavelet Representations. http://www.fourierandwavelets.org/SPFWR_a3.1_2012.pdf (2012)Burrus, C.S., Gopinath, R.A., Guo, H.: Introduction to Wavelets and Wavelet Transforms. Prentice-Hall, New Jersey (1998)Kamilov, U., Bostan, E., Unser, M.: Wavelet shrinkage with consistent cycle spinning generalizes total variation denoising. IEEE Signal Process. Lett. 19(4), 187–190 (2012)Kumar, B.K.S.: Image denoising based on non-local means filter and its method noise thresholding. Signal Image Video Process. 7, 1211–1227 (2013)Rezazadeh, S., Coulombe, S.: A novel discrete wavelet transform framework for full reference image quality assessment. Signal Image Video Process. 7, 559–573 (2013)Atto, A.M., Pastor, D., Mercier, G.: Wavelet shrinkage: unification of basic thresholding functions and thresholds. Signal Image Video Process. 5, 11–28 (2011)Yektaii, M., Ahmad, M.O., Bhattacharya, P.: A method for preserving the classifiability of digital images after performing a wavelet-based compression. Signal Image Video Process. 8, 169–180 (2014)Kanumuri, T., Dewal, M.L., Anand, R.S.: Progressive medical image coding using binary wavelet transforms. Signal Image Video Process. 8, 883–899 (2014)Kubinyi, M., Kreibich, O., Neuzil, J., Smid, R.: EMAT noise suppression using information fusion in stationary wavelet packets. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58, 1027–1036 (2011)Abbate, A., Koay, J., Frankel, J., Schroeder, S.C., Das, P.: Signal detection and noise suppression using a wavelet transform signal processor: application to ultrasonic flaw detection. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44, 14–26 (1997)Pardo, E., San Emeterio, J.L., Rodriguez, M.A., Ramos, A.: Noise reduction in ultrasonic NDT using undecimated wavelet transforms. Ultrasonics 44, e1063–e1067 (2006)Pardo, E., Emeterio, J.L., Rodriguez, M.A., Ramos, A.: Shift invariant wavelet denoising of ultrasonic traces. Acta Acust. United Acust. 94, 685–693 (2008)Shensa, M.J.: The discrete wavelet transform: wedding the a trous and Mallat algorithms. IEEE Trans. Signal Process. 40, 2464–2482 (1992)Coifman, R., Donoho, D.: Translation invariant de-noising. In: Antoniadis, A., Oppenheim, G. (eds.) Wavelets and Statistics, Lecture Notes in Statistics, pp. 125–150. Springer, Berlin (1995)Mallat, S.G.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 2, 674–693 (1989)Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms. Commun. Pure Appl. Math. 44, 141–183 (1991)Beylkin, G.: On the representation of operators in bases of compactly supported wavelets. SIAM J. Numer. Anal. 6(6), 1716–1740 (1992)Vaidyanathan, P.P.: Multirate Systems and Filter Banks. Prentice Hall, Englewood Cliffs (1992)Donoho, D.L., Johnstone, I.M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455 (1994)Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90, 1200–1224 (1995)Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D.: Wavelet shrinkage: Asymptotia? J. R. Stat. Soc. Ser. B 57, 301–369 (1995)Karpur, P., Shankar, P.M., Rose, J.L., Newhouse, V.L.: Split spectrum processing: optimizing the processing parameters using minimization. Ultrasonics 25, 204–208 (1997)Lazaro, J.C., San Emeterio, J.L., Ramos, A., Fernandez, J.L.: Influence of thresholding procedures in ultrasonic grain noise reduction using wavelets. Ultrasonics 40, 263–267 (2002)Donoho, D.L.: De-noising by soft thresholding. IEEE Trans. Inf. Theory 41, 613–627 (1995)Johnstone, I.M., Silverman, B.W.: Wavelet threshold estimators for data with correlated noise. J. R. Stat. Soc. 59, 319–351 (1997

Improved Denoising Method for Ultrasonic Echo with Mother Wavelet Optimization and Best-Basis Selection

Weak features of ultrasonicnondestructive test signals are usually immersed in noisy signals. So, in this paper, we proposed an improved scheme for noise reduction and feature extraction based on discrete wavelet transform. The basis of the mother wavelet was selected to be matched to a given signal. Three different constraints were presented to minimize the error between the denoised and the given signal. It should be mentioned that such an optimum wavelet can represent the signal more compactly with a few large coefficients which can be considered as the signal features. Standard signals and simulated ultrasonic echo were used to evaluate the performance of the presented algorithms. Signal to error ratio was used to compare the designed wavelet performance with that of standard wavelets. Simulation results revealed that the proposed method outperformed the other presented methods and even standard wavelets. The results also has shown that the signal-based noise reduction algorithms make the feature extraction more reliable. Finally, the performance of the proposed algorithm was compared with other methods from different literatures

Reduced Cycle Spinning Method for the Undecimated Wavelet Transform

[EN] The Undecimated Wavelet Transform is commonly used for signal processing due to its advantages over other wavelet techniques, but it is limited for some applications because of its computational cost. One of the methods utilized for the implementation of the Undecimated Wavelet Transform is the one known as Cycle Spinning. This paper introduces an alternative Cycle Spinning implementation method that divides the computational cost by a factor close to 2. This work develops the mathematical background of the proposed method, shows the block diagrams for its implementation and validates the method by applying it to the denoising of ultrasonic signals. The evaluation of the denoising results shows that the new method produces similar denoising qualities than other Cycle Spinning implementations, with a reduced computational cost.This research was funded by grants number PGC2018-09415-B-I00 (MCIU/AEI/FEDER, UE) and TEC2015-71932-REDT.Rodríguez-Hernández, MA. (2019). Reduced Cycle Spinning Method for the Undecimated Wavelet Transform. Sensors. 19(12):1-16. https://doi.org/10.3390/s19122777S1161912Signal Processing Fourier and Wavelet Representationshttp://www.fourierandwavelets.org/SPFWR_a3.1_2012.pdfZhao, H., Zuo, S., Hou, M., Liu, W., Yu, L., Yang, X., & Deng, W. (2018). A Novel Adaptive Signal Processing Method Based on Enhanced Empirical Wavelet Transform Technology. Sensors, 18(10), 3323. doi:10.3390/s18103323Gradolewski, D., Magenes, G., Johansson, S., & Kulesza, W. (2019). A Wavelet Transform-Based Neural Network Denoising Algorithm for Mobile Phonocardiography. Sensors, 19(4), 957. doi:10.3390/s19040957Shikhsarmast, F., Lyu, T., Liang, X., Zhang, H., & Gulliver, T. (2018). Random-Noise Denoising and Clutter Elimination of Human Respiration Movements Based on an Improved Time Window Selection Algorithm Using Wavelet Transform. Sensors, 19(1), 95. doi:10.3390/s19010095Shensa, M. J. (1992). The discrete wavelet transform: wedding the a trous and Mallat algorithms. IEEE Transactions on Signal Processing, 40(10), 2464-2482. doi:10.1109/78.157290Li, M., & Ghosal, S. (2015). Fast Translation Invariant Multiscale Image Denoising. IEEE Transactions on Image Processing, 24(12), 4876-4887. doi:10.1109/tip.2015.2470601Hazarika, D., Nath, V. K., & Bhuyan, M. (2016). SAR Image Despeckling Based on a Mixture of Gaussian Distributions with Local Parameters and Multiscale Edge Detection in Lapped Transform Domain. Sensing and Imaging, 17(1). doi:10.1007/s11220-016-0141-8Sakhaee, E., & Entezari, A. (2017). Joint Inverse Problems for Signal Reconstruction via Dictionary Splitting. IEEE Signal Processing Letters, 24(8), 1203-1207. doi:10.1109/lsp.2017.2701815Ong, F., Uecker, M., Tariq, U., Hsiao, A., Alley, M. T., Vasanawala, S. S., & Lustig, M. (2014). Robust 4D flow denoising using divergence-free wavelet transform. Magnetic Resonance in Medicine, 73(2), 828-842. doi:10.1002/mrm.25176Rehman, N. ur, Abbas, S. Z., Asif, A., Javed, A., Naveed, K., & Mandic, D. P. (2017). Translation invariant multi-scale signal denoising based on goodness-of-fit tests. Signal Processing, 131, 220-234. doi:10.1016/j.sigpro.2016.08.019Mota, H. de O., Vasconcelos, F. H., & de Castro, C. L. (2016). A comparison of cycle spinning versus stationary wavelet transform for the extraction of features of partial discharge signals. IEEE Transactions on Dielectrics and Electrical Insulation, 23(2), 1106-1118. doi:10.1109/tdei.2015.005300Li, D., Wang, Y., Lin, J., Yu, S., & Ji, Y. (2016). Electromagnetic noise reduction in grounded electrical‐source airborne transient electromagnetic signal using a stationarywavelet‐based denoising algorithm. Near Surface Geophysics, 15(2), 163-173. doi:10.3997/1873-0604.2017003San Emeterio, J. L., & Rodriguez-Hernandez, M. A. (2014). Wavelet Cycle Spinning Denoising of NDE Ultrasonic Signals Using a Random Selection of Shifts. Journal of Nondestructive Evaluation, 34(1). doi:10.1007/s10921-014-0270-8Rodriguez-Hernandez, M. A., & Emeterio, J. L. S. (2015). Noise reduction using wavelet cycle spinning: analysis of useful periodicities in the z-transform domain. Signal, Image and Video Processing, 10(3), 519-526. doi:10.1007/s11760-015-0762-8Rodriguez-Hernandez, M. A. (2016). Shift selection influence in partial cycle spinning denoising of biomedical signals. Biomedical Signal Processing and Control, 26, 64-68. doi:10.1016/j.bspc.2015.12.002Beylkin, G., Coifman, R., & Rokhlin, V. (1991). Fast wavelet transforms and numerical algorithms I. Communications on Pure and Applied Mathematics, 44(2), 141-183. doi:10.1002/cpa.3160440202Beylkin, G. (1992). On the Representation of Operators in Bases of Compactly Supported Wavelets. SIAM Journal on Numerical Analysis, 29(6), 1716-1740. doi:10.1137/0729097Donoho, D. L., & Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3), 425-455. doi:10.1093/biomet/81.3.425Donoho, D. L., & Johnstone, I. M. (1995). Adapting to Unknown Smoothness via Wavelet Shrinkage. Journal of the American Statistical Association, 90(432), 1200-1224. doi:10.1080/01621459.1995.10476626Johnstone, I. M., & Silverman, B. W. (1997). Wavelet Threshold Estimators for Data with Correlated Noise. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 59(2), 319-351. doi:10.1111/1467-9868.00071Pardo, E., San Emeterio, J. L., Rodriguez, M. A., & Ramos, A. (2006). Noise reduction in ultrasonic NDT using undecimated wavelet transforms. Ultrasonics, 44, e1063-e1067. doi:10.1016/j.ultras.2006.05.101Donoho, D. L. (1995). De-noising by soft-thresholding. IEEE Transactions on Information Theory, 41(3), 613-627. doi:10.1109/18.382009Lázaro, J. C., San Emeterio, J. L., Ramos, A., & Fernández-Marrón, J. L. (2002). Influence of thresholding procedures in ultrasonic grain noise reduction using wavelets. Ultrasonics, 40(1-8), 263-267. doi:10.1016/s0041-624x(02)00149-xKarpur, P., Shankar, P. M., Rose, J. L., & Newhouse, V. L. (1987). Split spectrum processing: optimizing the processing parameters using minimization. Ultrasonics, 25(4), 204-208. doi:10.1016/0041-624x(87)90034-5Pardo, E., Emeterio, S. J. L., Rodriguez, M. A., & Ramos, A. (2008). Shift Invariant Wavelet Denoising of Ultrasonic Traces. Acta Acustica united with Acustica, 94(5), 685-693. doi:10.3813/aaa.91808

Extraction and characterisation of pectin from dragon fruit (hylocereus polyrhizus) peels

Pectins are complex carbohydrate molecules that are used in numerous food applications as a gelling agent, thickener, stabiliser, and emulsifier. Dragon fruit (Hylocereus polyrhizus) is one of the tropical fruits that belong to the cactus family, Cactaceae. Since the peels of dragon fruit are often discarded as waste, it would be an advantage to convert it into a value-added product such as pectin. The objective of this study was to investigate the extraction of pectin from dragon fruit peels under different extraction time using hot water extraction method. The dragon fruit peels were extracted using distilled water at 80 °C with different extraction time of 20, 40, 60 and 80 min. The extracted pectin was characterised by its yield, moisture and ash content, degree of esterification and antioxidant activity. Determination of moisture and ash content was conducted using AOAC standard method. The determination of the degree of esterification of pectin was performed using Fourier Transform Infrared Spectroscopy (FTIR). DPPH assay was used to determine the antioxidant activity of the pectin extract. Based on the result, the yield of pectin decreases (20.34 to 16.20 %) with the increase of extraction time, moisture contents were between 4 to 6 % while ash contents were between 7 to 10 %. Pectin from dragon fruit peels was determined as low methoxyl pectin and has high percentage of antioxidant activity with low value of inhibition concentration (IC50) (0.0063 to 0.0080 mg/mL). 60 min extraction sample exhibits the highest antioxidant activity (81.91 % at 40 μg/mL), followed by 80 min extraction (81.68 % at 40 μg/mL), 40 min extraction (81.38 % at 40 μg/mL) and 20 min extraction (81.31 % at 40 μg/mL)

Shift Selection Influence in Partial Cycle Spinning Denoising of Biomedical Signals

Denoising of biomedical signals using wavelet transform is a widely used technique. The use of undecimated wavelet transform (UWT) assures better denoising results but implies a higher complexity than discrete wavelet transform (DWT). Some implementation schemes have been proposed to perform UWT, one of them is Cycle Spinning (CS). CS is performed using the DWT of several circular shifted versions of the signal to analyse. The reduction of the number of shifted versions of the biomedical signal during denoising process used is addressed in the present work. This paper is about a variant of CS with a reduced number of shifts, called Partial Cycle Spinning (PCS), applied to ultrasonic trace denoising. The influence of the choice of PCS shifts in the denoised registers quality is studied. Several shifts selection rules are proposed, compared and evaluated. Denoising results over a set of ultrasonic registers are provided for PCS with different shift selection rules, CS and DWT. The work shows that PCS with the appropriate choice of shifts could be the best option to denoise biomedical ultrasonic traces. (C) 2015 Elsevier Ltd. All rights reserved.This work was partially supported by Spanish Government MEC Project TIN2013-47272-C2-1-R.Rodríguez-Hernández, MA. (2016). Shift Selection Influence in Partial Cycle Spinning Denoising of Biomedical Signals. Biomedical Signal Processing and Control. 26:64-68. https://doi.org/10.1016/j.bspc.2015.12.002S64682

Analysis and Synthesis Prior Greedy Algorithms for Non-linear Sparse Recovery

In this work we address the problem of recovering sparse solutions to non linear inverse problems. We look at two variants of the basic problem, the synthesis prior problem when the solution is sparse and the analysis prior problem where the solution is cosparse in some linear basis. For the first problem, we propose non linear variants of the Orthogonal Matching Pursuit (OMP) and CoSamp algorithms; for the second problem we propose a non linear variant of the Greedy Analysis Pursuit (GAP) algorithm. We empirically test the success rates of our algorithms on exponential and logarithmic functions. We model speckle denoising as a non linear sparse recovery problem and apply our technique to solve it. Results show that our method outperforms state of the art methods in ultrasound speckle denoising

Curvelet Denoising with Improved Thresholds for Application on Ultrasound Images

In medical image processing, image denoising has become a very essential exercise all through the diagnose. Negotiation between the preservation of useful diagnostic information and noise suppression must be treasured in medical images. In case of ultrasonic images a special type of acoustic noise, technically known as speckle noise, is the major factor of image quality degradation. Many denoising techniques have been proposed for effective suppression of speckle noise. Removing noise from the original image or signal is still a challenging problem for researchers. In this paper, a Curvelet transform based denoising with improved thresholds is proposed for ultrasound images