Applications of Wavelets

Image and Video Processing
The wavelet frame-based models and algorithms developed from image denoising [4,6], deblurring [3,12,15], and inpainting [10] were modified to adapt to many other tasks in image processing, such as phase retrieval in [1],   X-ray computed tomography reconstructions in [2, 8], 3D reconstruction in electron microscopy in  [5], recovering over/under-exposed regions in photographs in [7], image demosaicing in [9], restoration of chopped and nodded images in infrared astronomy in [11], and blind deblurring. They were also modified for applications in video processing in [13,14] and analysis. (see also e.g. survey papers.)

    1. Tongyao Pang, Qingna Li, Zaiwen Wen, Zuowei Shen, Phase retrieval: a data-driven wavelet frame based approach, Applied and Computational Harmonic Analysis, 49 (2020), 971-1000.  PDF
    2. Jia Li, Chuang Miao, Zuowei Shen, Ge Wang, Hengyong Yu, Robust Frame Based X-Ray CT Reconstruction, Journal of Computational Mathematics, 34(6), (2016), 683-704. PDF
    3. Yuhui Quan, Hui Ji, Zuowei Shen, Data-driven multi-scale non-local wavelet frame construction and image recovery, Journal of Scientific Computing, 63(2) (2015), 307-329. PDF
    4. Jia Li, Zuowei Shen,Rujie Yin, Xiaoqun Zhang, A reweighted l2 method for image restoration with poisson and mixed poisson-gaussian noise, Inverse Problem and Imaging 9(3), (2015), 875-894. PDF
    5. Ming Li, Zhitao Fan, Hui Ji, Zuowei Shen, Wavelet frame based algorithm for 3D reconstruction in electron microscopy, SIAM Journal on Scientific Computing, 36(1) (2014), 45-69. PDF
    6. Zheng Gong, Zuowei Shen, Kim Chuan Toh, Image restoration with mixed or unknown noises, Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 12(2) (2014), 458-487. PDF
    7. Likun Hou, Hui Ji, Zuowei Shen, Recovering over/under-exposed regions in photographs, SIAM Journal on Imaging Sciences, 6 (4) (2013), 2213-2235. PDF
    8. Bin Dong, Jia Li, Zuowei Shen, X-ray CT image reconstruction via wavelet frame based regularization and radon domain inpainting, Journal of Scientific Computing, 54(2-3) (2013), 333-349. PDF
    9. Jingwei Liang, Jia Li, Zuowei Shen, Xiaoqun Zhang, Wavelet frame based color image demosaicing, Inverse Problem and Imaging 7 (3), (2013), 777-794. PDF
    10. Jianfeng Cai, Raymond Chan, Lixing Shen, Zuowei Shen, Simultaneously inpainting in image and transformed domains, Numerische Mathematik, 112(4) (2009), 509-533. PDF
    11. Jianfeng Cai, Raymond Chan, Lixin Shen, Zuowei Shen, Restoration of chopped and nodded images by framelets, SIAM Journal on Scientific Computing, 30(3) (2008) 1205-1227. PDF
    12. Anwei Chai, Zuowei Shen, Deconvolution: A wavelet frame approach, Numerische Mathematik, 106 (2007), 529-587. deconvolution.pdf
    13. Raymond Chan, Zuowei Shen, Tao Xia, A framelet algorithm for enhancing video stills, Applied and Computational Harmonic Analysis, 23(2) (2007), 153-170. PDF
    14. Cho-Chun Cheng, Wen-Liang Hwang, Zuowei Shen, Tao Xia, Subjective and objective comparison of advanced motion compensation methods for block artifact reduction in a 3-D wavelet coding system, IEEE Transactions on Circuits and Systems for Video Technology, 16 (9) (2006), 1134-1141. 3D.pdf
    15. Raymond Chan, S. D. Riemenschneider, Lixin Shen, Zuowei Shen, Tight Frame: An efficient way for high-resolution image reconstruction, Applied and Computational Harmonic Analysis, 17 (2004), 91-115. crss1.pdf

Surface Fitting and Segmentation
We combined the level set methods for surface fitting and segmentation with wavelet frame methods in image processing to obtain efficient algorithms for surface fitting and segmentation. The level set method catches global smooth components and the wavelet method extracts local features. We were inspired by the understanding of both wavelet and PDE approaches gained from some of our early work that connects the wavelet frame-based model with various partial differential models.


    1. Jiulong Liu, Xiaoqun Zhang, Bin Dong, Zuowei Shen, Lixu Gu, A wavelet frame method with shape prior for ultrasound video segmentation, SIAM Journal on Imaging Sciences, 9(2) (2016), 495-519. PDF .
    2. Bin Dong, Qingtang Jiang, Chaoqian Liu, Zuowei Shen, Multiscale representation of surfaces by tight wavelet frames with applications to denoising, Applied and Computational Harmonic Analysis, 41(2), (2016), 561-589. PDF
    3. Cheng Tai, Xiaoqun Zhang, Zuowei Shen, Wavelet frame based Multi-phase image segmentation; SIAM Journal on Imaging Sciences, 6 (4) (2013), 2521-2546. PDF
    4. Bin Dong, Zuowei Shen, Wavelet frame based surface reconstruction from unorganized points, Journal of Computational Physics, 230(22), (2011), 8247-8255. PDF
    5. Bin Dong, Aichi Chien, Zuowei Shen, Stanley Osher, A new multiscale representation for shapes and its application to blood vessel recovery, SIAM Journal on Scientific Computing, 32(4) (2010), 1724-1739. PDF
    6. Bin Dong, Aichi Chien, Zuowei Shen, Frame based segmentation for medical images, Communications in Mathematical Sciences, 32(4), (2010), 1724-1739. PDF

Applications in Science and Engineering
The wavelet frame-based models and algorithms developed for image, video, and surface processing can be modified to adapt to many other applications in science and engineering.

    1. Chenglong Bao, George Barbastathis, Hui Ji, Zuowei Shen and Zhengyun Zhang, Coherence retrieval via trace regularization, SIAM Journal on Imaging Sciences, 11(1), (2018), 679-706. PDF
    2. Giuanhua Zhu, Wei Liu, Chenglong Bao, Dudu Tong, Hui Ji, Zuowei Shen, Daiwei Yang and Lanyuan Lu. Investigating energy-based pool structure selection in the structure ensemble modeling with experimental distance constraints: the example from a multi-domain protein Pub 1, Proteins: Structure, Function, and Bioinformatics 86(5),(2018), 501-514. PDF
    3. Jianbin Yang, Guanhua Zhu, Dudu Tong, Lanyuan Lu, Zuowei Shen, B-spline tight frame based force matching method, Journal of Computational Physics, 362, (2018), 208-219. PDF
    4. Zhengyun Zhang, Chenglong Bao, Hui Ji, Zuowei Shen, George Barbastathi, Apparent coherence loss in phase space tomography Journal of the Optical Society of America A., 34(11), (2017), 2025-2033. PDF
    5. Hui Ji, Zuowei Shen, Yuhong Xu, Wavelet frame based scene reconstruction from range data, Journal of Computational Physics, 229(6), (2010), 2093-2108. PDF
    6. Herwig Wendt, Patrice Abry, Stephane Jaffard, Hui Ji, Zuowei Shen, Wavelet leader multifractal analysis for texture classification, IEEE International Conference on Image Processing (ICIP) Cairo, Egypt, 2009. PDF
    7. Bao Liu, S. D. Riemenschneider, Zuowei Shen, A Fast adaptive time-frequency analysis method and its application in vibration analysis, Journal of Vibration and Acoustics of ASME(Trans. ASME) 129 (2), (2007), 169-178. lrs.pdf