Browsing by Author "Chan, Tony F."

Now showing 1 - 6 of 6
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    Euler's elastica and curvature based inpaintings
    (2001-04) Chan, Tony F.; Kang, Sung Ha; Shen, Jianhong
    Image inpainting is a special image restoration problem for which image prior models play a crucial role. Euler's elastica was first introduced by Mumford [21] to computer vision as a prior curve model. By functionalizaing the elastica energy, Masnou and Morel [19] proposed an elastica based variational inpainting model. The current paper is intended to contribute to the development of its mathematical foundation, and the study of its properties and connections to the earlier works of Bertalmio, Sapiro, Caselles, and Ballester [2] and Chan and Shen [6,7]. A computational scheme based on numerical PDEs is presented, which allows the handling of topologically complex inpainting domains.
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    A good image model eases restoration - on the contribution of Rudin-Osher-Fatmi's BV image model
    (2002-02) Chan, Tony F.; Shen, Jianhong
    What we believe images are determines how we take actions in image and low-level vision analysis. In the Bayesian framework, it is known as the importance of a good image prior model. This paper intends to give a concise overview on the vision foundation, mathematical theory, computational algorithms, and various classical as well as unexpected new applications of the BV (bounded variation) image model, first introduced into image processing by Rudin, Osher, and Fatemi in 1992 [Physica D, 60:259-268].
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    Morphologically invariant PDE inpaintings
    (2001-04) Chan, Tony F.; Shen, Jianhong
    This paper studies the PDE method for image inpaintings. Image inpainting is essentially an image interpolation problem, with wide applications in film and photo restoration, text removal, special effects in movies, disocclusion, digital zoom-in, and edge-based image compression and coding. Bertalmio, Sapiro, Caselles, and Ballester (2000) [3] first innovatively introduced the PDE method for the inpainting problem. Ever since, the authors of the present paper have worked along this line and developed the PDE method, mostly inspired by the Bayesian and variational method (especially by good image {\rm prior} models). The current paper has two major goals. First, by surveying all the recent PDE inpainting techniques, we intend to develop a unified viewpoint based on two infinitesimal mechanisms: transportation and curvature driven diffusions (CDD). Furthermore, based this knowledge, we construct a new class of third order inpainting PDEs, which is derived from the set of axioms (or principles) refined from the existing works: morphological invariance, rotational invariance, stability principle, and linearity principle.
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    Non-texture inpainting by curvature-driven diffusions (CDD)
    (2001-02) Chan, Tony F.; Shen, Jianhong
    Inpainting refers to the practice of artists of restoring ancient paintings. Simply speaking, inpainting is to complete a painting by filling in the missing informa tion on prescribed domains. On such domains, the original painting has been damaged due to aging, scratching, or some other factors. Inpainting and disocclusion in vision analysis are closely connected but also clearly different. Both try to recover the missing visual information from a given 2-D image, and mathematically, can be classified into the same category of inverse problems. The difference lies in both their goals and approaches. The main goal of disocclusion is to model how human vision works to complete occluded objects in a given 2-D scene, and understand their physical or ders in the direction perpendicular to the imaging plane, and thus reconstruct approximately a meaningful 3-dimensional world (Nitzberg, Mumford, and Shiota [14]). The outputs from disocclusion are complete objects, and their relative orders or depth. Inpainting, on the other hand, is to complete a 2-D image which have certain regions missing. The output is still a 2-D image. (In applications, a missing region can indeed be the 2-D projection of a real object, such as the female statue in Figure 9.) Therefore, from the vision point of view, inpainting is a lower level process compared to disocclusion. This fundamental difference naturally influences the approaches. The main approach for disocclusion is to segment the regions in a 2-D image, and then logically connect those which belong to the projection of a same physical object, and finally generate the order or depth for all the completed objects. Edge completion is one crucial step during the whole process. Disocclusion also often uses some high level information about objects (such as the near symmetry of human faces). For inpainting, an ideal scheme should be able to reconstruct an incomplete 2-D image in every detail so that it looks "complete" and "natural." More specifically, to inpaint, is not only to complete the broken edges, but also to connect each broken isophote (or level-line), so that the 2-D objects completed in such a way show their natural variation in intensity (or color for color images) [3, 6,11]. This comparison helps us understand better the real nature of the inpainting problem in a broader context. The terminology of digital inpainting was first introduced by Bertalmio, Sapiro, Caselles, and Ballester [3]. Inspired by the real inpainting process of artists, the authors invented a successful digital inpainting scheme (referred to below as the BSCB inpainting scheme for convenience) based on the PDE method. The authors also deepened the interest in digital inpainting by demon strating its broad applications in text removal, restoring old photos, and creating special effects such as object disappearance from a scene. Though a qualitative understanding based on the transportation mechanism can be well established, rigorous mathematical analysis on the BSCB scheme appears to be much more difficult. This has encouraged Chan and Shen [6] to develop a new inpainting model which is founded on the variational principle. Since the energy function is based on the total variational (TV) norm [6], the model is called TV inpainting. The TV inpainting scheme is surprisingly a close variation of the well known restoration model of Rudin, Osher and Fatemi (16, 17].
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    Variational image inpainting
    (2002-07) Chan, Tony F.; Shen, Jianhong
    Inpainting is an image interpolation problem, with broad applications in image and vision analysis. This paper presents our recent efforts in developing inpainting models based on the Bayesian and variational principles. We discuss several geometric image models, their role in the construction of variational inpainting models, and the associated Euler-Lagrange PDEs and their numerical computation.
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    Variational PDE models in image processing
    (2002-08) Chan, Tony F.; Shen, Jianhong; Vese, Luminita
    In this article, we intend to give a broad picture of mathematical image processing through one of the most recent and very successful approaches -- the variational PDE method. We first discuss two crucial ingredients for image processing: image modeling or representation, and processor modeling. We then focus on the variational PDE method. The backbone of this article consists of two major problems in image processing -- inpainting and segmentation.