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Wavelets and PDE Techniques in Image Processing, A Quick Tour of

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Encyclopedia of Complexity and Systems Science

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Explosive information has dominated nearly all aspects of modern society, science and technology. Visualization is one of the most direct and preferable ways to observe information carried by data, which are often massive in size and uncertain in data quality. To better reveal the information, especially when it is hidden, implicit, or corrupted, data must first be properly processed. In achieving this, image processing, which includes many different tasks such as compression , restoration , inpainting , segmentation , pattern recognition and registration , has played a critical role. Historically, it has been viewed as a branch of signal processing, and many classical methods are adopted from traditional Fourier‐based signal processing algorithms. In the past couple of decades, numerous new...

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Abbreviations

Wavelets :

Wavelets are selected functions that generate orthonormal bases of the square integrable function space L 2 (or more generally, frames of L p spaces) by using dilations and translations. The basis functions have certain locality, such as compact support or fast decay property. And they are usually organized according to different scales or resolutions, which are called Multi‐Resolution Analysis (MRA) . Fast wavelet transforms are filtering procedures that compute the projection of any given function onto a wavelet basis.

Digital images :

Digital images usually refer to n dimensional data arrays recorded by optical or other imaging devices, such as digital cameras, Radar, Computed Tomography (CT), and Magnetic Resonance Imaging (MRI). They can also be generated by computer graphics software. Most digital images in the literature are 2- or 3‑dimensional.

Image restoration :

Image restoration rebuilds high‐quality images from given images that are corrupted or polluted during acquisition or transmission processes. The most commonly seen restoration tasks are denoising and deblurring . Denoising is to remove random perturbations to individual pixel values. Deblurring is to remove the unwanted correlation between nearby pixels and to recover the original clear images.

Image compression :

Compression converts images from n dimensional data arrays into “0” and “1” bit streams so that they can be stored or transmitted more efficiently. There are two types of compression, lossy and lossless, depending on whether information is permanently lost or recoverable, respectively. Many of the commonly used compression algorithms, such as the ones used by international image compression standards JPEG and JPEG2000, are transform‐based compression, which consists of three basic steps: transform pixel values into frequency coefficients, quantization of the frequency coefficients, and coding to convert them into bit streams.

Image segmentation :

Segmentation partitions images into subregions (segments), on which images share similar features. Each region often corresponds to the image of an individual object in the 3‑dimensional world.

Image inpainting :

Inpainting is an artistic word referring to filling in missing image information on damaged regions, e. g., scratches and damages in precious old photos, old Hollywood films, and ancient paintings. The objective of digital image inpainting is to fill in the missing information automatically and meaningfully.

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Acknowledgments

Research supported in part by grants ONR-N00014-06-1-0345, NSF CCF-0430077, CCF-0528583, DMS-0610079, DMS-0410062 and CAREER Award DMS-0645266, NIH U54 RR021813, and STTR Program from TechFinity Inc.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, USA

    Hao-Min Zhou

  2. Department of Mathematics, University of California, Los Angeles, USA

    Tony F. Chan

  3. Barclays Capital, New York, USA

    Jianhong Shen

Authors
  1. Hao-Min Zhou
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  2. Tony F. Chan
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Zhou, HM., Chan, T.F., Shen, J. (2009). Wavelets and PDE Techniques in Image Processing, A Quick Tour of. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_589

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