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Reversible, Fast, and High-Quality Grid Conversions Laurent Condat, Dimitri Van De Ville, Brigitte Forster-Heinlein

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Reversible, Fast, and High-Quality Grid Conversions Laurent Condat, Dimitri Van De Ville, Brigitte Forster-Heinlein Reversible, Fast, and High-Quality Grid Conversions Laurent Condat, Dimitri van de Ville, Brigitte Forster-Heinlein To cite this version: Laurent Condat, Dimitri van de Ville, Brigitte Forster-Heinlein. Reversible, Fast, and High-Quality Grid Conversions. IEEE Transactions on Image Processing, Institute of Electrical and Electronics Engineers, 2008, 17 (5), pp. 679-693. 10.1109/TIP.2008.919361. hal-00377101 HAL Id: hal-00377101 https://hal.archives-ouvertes.fr/hal-00377101 Submitted on 3 Jun 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 5, PP. 679–693, MAY 2008 1 Reversible, Fast, and High-Quality Grid Conversions Laurent Condat, Member, IEEE, Dimitri Van De Ville, Member, IEEE and Brigitte Forster-Heinlein, Member, IEEE Abstract— A new grid conversion method is proposed to re- of its samples f(Rk). Specifically, a generator function at sample between two 2-D periodic lattices with the same sampling every lattice site is weighted by a coefficient, after which density. The main feature of our approach is the symmetric the reconstruction is sampled on the target lattice. On the reversibility, which means that when using the same algorithm for the converse operation, then the initial data is recovered exactly. Cartesian lattice, generator functions are typically obtained as To that purpose, we decompose the lattice conversion process into tensor-product extensions of 1-D versions. On non-Cartesian (at most) three successive shear operations. The translations along lattices, such as the hexagonal one, intrinsically 2-D generator the shear directions are implemented by 1-D fractional delay functions are deployed; e.g., hex-splines [4] or box-splines [5]. operators, which revert to simple 1-D convolutions, with appro- The use of high-quality versions of these methods (say, beyond priate filters that yield the property of symmetric reversibility. We show that the method is fast and provides high-quality resampled linear interpolation) requires a prefiltering step that can be images. Applications of our approach can be found in various computationally expensive. We highlight two main properties settings, such as grid conversion between the hexagonal and the of these approaches: (1) successive resampling operations Cartesian lattice, or fast implementation of affine transformations increasingly degrade the image quality; (2) the process is not such as rotations. invertible by a similar procedure. The latter point is related to Index Terms— 2-D lattices, resampling, shears, fractional delay the fact that the generator function is intricately linked to the filters, hexagonal grid, rotation. source lattice, only. For instance, when using a linear tensor- product B-spline for Cartesian-to-hexagonal resampling, one I. INTRODUCTION would consider the three-directional linear box-spline [5]– [7] as an “equivalent” way to return from the hexagonal to IGITAL images are almost exclusively available on the Cartesian lattice. However, the original image will not be Cartesian lattice, despite the existence of other periodic D recovered after cascading both operations. lattices with attractive properties. Probably, the main reasons In this paper, we propose an alternative approach that is behind this omnipresence are the trivial correspondence be- driven by the property of “symmetric reversibility”; i.e., the tween pixel position and pixel index, and the availability of conversion between two arbitrary lattices with the same sam- display devices with Cartesian geometry. pling density has an exact inverse operation, that is achievable Resampling procedures come into play when converting with the same algorithm. Formally, we pursue from one lattice to another. Mathematically, a 2-D lattice [1]– ′ [3] is a regular set of points of the plane, characterized by R′→R R→R′ = d, R, R . (1) r r R2 C ◦C I ∀ two linearly independant vectors 1, 2 , conveniently Our method is based on two fundamental observations: (1) grouped in a matrix R r r ∈. Lattice sites have 2 2 = [ 1 2] a lattice can be turned into another one by at most three Rk×k Z2 coordinates , , and the lattice sampling density successive shear operations; (2) shear operations can be carried can be obtained as ∈ R (lattice sites per unit surface). 1/ det( ) out by 1-D fractional delay operators that ensure the “sym- For resampling 2-D data| from| one lattice, with matrix R, metric reversibility” property. By construction, this approach R′ onto another lattice, with matrix , typical approaches are will be separable and simple to implement; i.e., through based on reconstruction: a continuous-domain representation 1-D operations along selected directions. Additionally, 1-D lying in a shift-invariant function space is constructed, that translators of arbitrary high order can be designed to guarantee x estimates the underlying (unknown) function f( ) by means high quality results. Manuscript received September 7,2007; revised July 27,2008. This work Many applications can benefit from this type of grid con- was supported by the Marie Curie Excellence Team Grant MEXT-CT-2004- version; in particular we propose the following settings: 013477, Acronym MAMEBIA, funded by the European Commission. The associate editor coordinating the review of this manuscript and approving it Hexagonal-to-Cartesian Resampling for publication was Dr. Peyman Milanfar. Laurent Condat and Brigitte Forster-Heinlein are with the Institute The hexagonal lattice of the so-called first type [8], depicted of Biomathematics and Biometry, part of the Helmholtz Zentrum in Fig. 1, is characterized by the matrix M¨unchen — German Research Center for Environmental Health, 85764 Neuherberg, Germany (e-mail: [email protected], 2 1 1/2 [email protected]). Rhex = . (2) ´ √3 0 √3/2 Dimitri Van De Ville is with the Biomedical Imaging Group (BIG), Ecole s Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland (e-mail: dimitri.vandeville@epfl.ch). Hexagonal sampling has numerous attractive properties, such Digital Object Identifier 10.1109/TIP.2008.919361 as more efficient representation of isotropic band-limited 2-D IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 5, PP. 679–693, MAY 2008 2 For the particular setting of image rotation, our decomposition of the conversion process into three shears has been reported already in the literature—see [26] and references therein. How- ever, our specific choice of fractional delay operations allows our method to satisfy the original and appealing property of symmetric reversibility: = d, θ R. (4) R−θ ◦ Rθ I ∀ ∈ Fig. 1. The regular hexagonal lattice (a) and the square lattice (b), with the It is satisfying to know that one can rotate an image without same sampling density equal to 1. losing any information and that the same operation applied in opposite sense will exactly recover the initial image. signals [1], [9]. Recent technical progress has brought hexag- Outline onal sampling into the domain of consumer electronics and This work is organized as follows. We present in Section II there now exist imaging sensors that acquire on the hexagonal our decomposition in shears. This suggests an easy and fast lattice [10], [11]. This is likely to foster a renewed interest for algorithm, detailed in Section III. We stress the fact that the image processing on this lattice. A non-Cartesian acquisition proposed method is essentially discrete and only relies on lattice may also be constrained for technical reasons, e.g. in 1-D shifts along rows or columns of the image. There is medical imaging devices. However, data sampled on such no underlying continuous model fitted on the image, as is a lattice generally have, at some stage of their processing the case with interpolation methods. On the other hand, our chain, to be resampled on the Cartesian lattice, typically for approach is limited to the conversion between lattices having visualization. the same sampling density, because we only use operations Hexagonal-to-Cartesian resampling can be performed fast that leave the sampling density invariant. In Section IV, and efficiently with our new approach using 1-D filtering only. we present an in-depth analysis of our approach, including The reversibility property ensures that no loss of information the “frequency shuffle” theorem that demonstrates how high- is introduced during this conversion operation. One particular frequency components are affected by our method. Finally, in useful application of the reversibility is when hexagonally- Section V, we show the experimental results illustrating the sampled data needs to be stored or transmitted on a Cartesian reversibility property, the efficiency of the implementation, and lattice. Our approach will produce a high-quality intermediate the quality compared to interpolation techniques. representation of the data (that can be directly visualized), which can be reconverted into the original data on the hexago- II. LATTICE CONVERSION: DECOMPOSITION INTO nal lattice at the receiver
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