## Adaptively quadratic (AQua) image interpolation (2004)

Venue: | IEEE Transactions on Image Processing |

Citations: | 23 - 1 self |

### BibTeX

@ARTICLE{Muresan04adaptivelyquadratic,

author = {D. Darian Muresan and Thomas W. Parks and D. Darian and Muresan Digital and Multi Media Design},

title = {Adaptively quadratic (AQua) image interpolation},

journal = {IEEE Transactions on Image Processing},

year = {2004},

volume = {13},

pages = {690--698}

}

### OpenURL

### Abstract

Image interpolation is a key aspect of digital image processing. This paper presents a novel interpolation method based on optimal recovery and adaptively determining the quadratic signal class from the local image behavior. The advantages of the new interpolation method are the ability to interpolate directly by any factor and to model properties of the data acquisition system into the algorithm itself. Through comparisons with other algorithms it is shown that the new interpolation is not only mathematically optimal with respect to the underlying image model, but visually it is very efficient at reducing jagged edges, a place where most other interpolation algorithms fail. Index Terms image modeling, quadratic classes, interpolation I.

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Citation Context ...rection step for edge directed interpolation and claim that this produces sharper images. A second class of image interpolation algorithms are those based on multi-resolution analysis. The authors of =-=[10]-=- model the wavelet coefficients of a dyadic (non-decimated) wavelet transform using exponential decay. In particular, they show that the local maximum of the dyadic wavelet coefficients decreases expo... |

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Citation Context ...If an exponential decay is detected, an estimate of the fine scale wavelet coefficient is made based on the estimated decay, otherwise the estimate is zero. The super-resolution image models of [13], =-=[14]-=- can also be described as interpolation methods based on adding image details in the wavelet domain, although the details are added based on training data and not wavelet decay. Using high resolution ... |

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Citation Context ...echnique which results in the optimization of convex functionals. Their nonlinear image expansion technique sharpens edges but the interpolation technique does not eliminate jagged edges. The work of =-=[20]-=- also uses a MAP framework for estimating a high resolution image from a sequence of under-sampled images. Their expansion method could be used for a single image by assuming only one frame. Applying ... |

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Citation Context ... n th degree polynomial. Whether separable or non-separable, these methods can be efficiently implemented using an up-sampler followed by a filter. Examples of such interpolation algorithms are cubic =-=[3]-=- and other spline based methods [4]. These polynomial image models have the advantage of being fast but tend to introduce December 5, 2003 DRAFTsIEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. Y, ... |

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Citation Context ...mated) wavelet transform using exponential decay. In particular, they show that the local maximum of the dyadic wavelet coefficients decreases exponentially from coarse to fine scales. The authors of =-=[11]-=-, [12] apply the wavelet exponential decay model to image interpolation by posing the interpolation problem as one of estimating the fine detail wavelet coefficients. Exponential decay is estimated fr... |

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Citation Context ...erpolated sample, the filter coefficients can be changed in order to sharpen edges. It is not clear how, or even if, this method removes the staircase affect in curved edges. The image models of [6], =-=[7]-=- use splines for image resizing. Their methods are especially useful for down-sampling, while up-sampling is similar to spline interpolation. Other attempts at modifying the polynomial image model in ... |

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Citation Context ...December 5, 2003 DRAFT (7) (8) (9)sIEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. XX, NO. Y, MONTH 2002 21 Then F (ū) is the best approximation to the value of F (x). ThatisF (ū) is the Chebyshev center =-=[29]-=- ofF (x) on Cf: sup |F (ū) − F (x)| = inf sup |F (u) − F (x)| (11) x∈Cf u∈Cf x∈Cf Next, let ¯y be the unit norm element in F for which the functional F attains its least upper bound: F (¯y) = sup |F (... |

10 |
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Citation Context ...re than the image itself. The image models of [15] are similar in nature. The authors of [16] model the wavelet coefficients using Gaussian mixtures and apply their models to image denoising [17]. In =-=[18]-=- the image model of [16] is extended to image interpolation. For this interpolation method the interpolation results are comparable with bi-cubic interpolation. One particular feature of this approach... |

8 |
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Citation Context ...d the local weights. This method works for interpolation by factors of two and performs well around edges, but performs poorly in high frequency regions, sometimes introducing undesired artifacts. In =-=[22]-=- it is shown how this solution can be reformulated using optimal recovery which allows for additional assumptions about the local derivatives in addition to the known local pixels to be used in the in... |

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Citation Context ... under-sampled images. Their expansion method could be used for a single image by assuming only one frame. Applying the algorithm to a single frame results in images very similar to those of [19]. In =-=[21]-=- the authors present a least squares edge directed interpolation method. The method assumes that each pixel is a linear combination of its neighboring pixels. Further, it is assumed that locally the w... |

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Citation Context ...ifacts more than the image itself. The image models of [15] are similar in nature. The authors of [16] model the wavelet coefficients using Gaussian mixtures and apply their models to image denoising =-=[17]-=-. In [18] the image model of [16] is extended to image interpolation. For this interpolation method the interpolation results are comparable with bi-cubic interpolation. One particular feature of this... |

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Citation Context ...from the local neighborhood. A second approach is to use patches from other high density images. This works well when interpolating images that belong to a certain predetermined class. For example in =-=[24]-=- AQua interpolation is applied to face interpolation and the training set is determined from other high resolution faces. A third approach is to adaptively search for training patches in other high re... |

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Citation Context ...ndrill, andpeppers. Therings image is 256 × 256 and consists of concentric circles that get closer and closer to each other as they move outward, away from the origin. The rings image is suggested by =-=[26]-=- for visualizing the results of applying different interpolation filters. Images barbra, lena, mandrill, andpeppers are 512 × 512 gray scale images available from [27]. All images are gray scale image... |

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Citation Context ...erpolation against four other interpolation techniques. Section V concludes with final remarks and future research work in this area. Finally, the Appendix reviews the theory of optimal recovery [1], =-=[2]-=-, which is key to the problem of interpolating missing samples in a quadratic signal class. II. REVIEW In the area of image interpolation by far the most well known and widely used techniques are thos... |

1 |
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Citation Context ...n be reformulated using optimal recovery which allows for additional assumptions about the local derivatives in addition to the known local pixels to be used in the interpolation process. The work of =-=[23]-=- is related to our image interpolation approach. The authors pose the image interpolation problem as one where the image belongs to a fixed quadratic image class. To solve the interpolation problem th... |

1 |
denoising using wavelet-domain hidden Markov trees
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(Show Context)
Citation Context ...ifacts more than the image itself. The image models of [15] are similar in nature. The authors of [16] model the wavelet coefficients using Gaussian mixtures and apply their models to image denoising =-=[17]-=-. In [18] the image model of [16] is extended to image interpolation. For this interpolation method the interpolation results are comparable with bi-cubic interpolation. One particular feature of this... |