## Image Compression by Linear Splines over Adaptive Triangulations

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Citations: | 37 - 8 self |

### BibTeX

@MISC{Demaret_imagecompression,

author = {Laurent Demaret and Nira Dyn and Armin Iske and Over Adaptive Triangulations},

title = {Image Compression by Linear Splines over Adaptive Triangulations},

year = {}

}

### Years of Citing Articles

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### Abstract

This paper proposes a new method for image compression. The method is based on the approximation of an image, regarded as a function, by a linear spline over an adapted triangulation, D(Y ), which is the Delaunay triangulation of a small set Y of significant pixels. The linear spline minimizes the distance to the image, measured by the mean square error, among all linear splines over D(Y ). The significant pixels in Y are selected by an adaptive thinning algorithm, which recursively removes less significant pixels in a greedy way, using a sophisticated criterion for measuring the significance of a pixel. The proposed compression method combines the approximation scheme with a customized scattered data coding scheme. We demonstrate that our compression method outperforms JPEG2000 on two geometric images and performs competitively with JPEG2000 on three popular test cases of real images.

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Citation Context ... compression, including EBCOT [24, 25], are based on wavelets and related techniques [22, 23, 26], see [6] for a survey on wavelet-based image coding. When working with a dyadic wavelet decomposition =-=[18]-=-, digital images are represented by wavelet coefficients. This representation is a linear decomposition over a fixed orthogonal basis. The non-linearity in the approximation of images by wavelets is i... |

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Citation Context ...es, we refer to our paper [15]. 8s2.7 Minimization of the Mean Square Error in SY In a post-processing to adaptive thinning, we further reduce the mean square error (2) by least squares approximation =-=[1]-=-. More precisely, we compute from the set Y ⊂ X of significant pixels, output by Algorithm 1, and from the luminance values at the pixels in X the unique best approximation L ∗ (Y, I) ∈ SY satisfying ... |

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Citation Context ...avec le standard de compression JPEG2000 sur deux images géométriques et trois images naturelles classiques. 1 Introduction Many of the well-established methods for image compression, including EBCOT =-=[24, 25]-=-, are based on wavelets and related techniques [22, 23, 26], see [6] for a survey on wavelet-based image coding. When working with a dyadic wavelet decomposition [18], digital images are represented b... |

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Citation Context ...s géométriques et trois images naturelles classiques. 1 Introduction Many of the well-established methods for image compression, including EBCOT [24, 25], are based on wavelets and related techniques =-=[22, 23, 26]-=-, see [6] for a survey on wavelet-based image coding. When working with a dyadic wavelet decomposition [18], digital images are represented by wavelet coefficients. This representation is a linear dec... |

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Citation Context ...avec le standard de compression JPEG2000 sur deux images géométriques et trois images naturelles classiques. 1 Introduction Many of the well-established methods for image compression, including EBCOT =-=[24, 25]-=-, are based on wavelets and related techniques [22, 23, 26], see [6] for a survey on wavelet-based image coding. When working with a dyadic wavelet decomposition [18], digital images are represented b... |

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Citation Context ... of approximation can be viewed as mildly nonlinear. Recently, several highly nonlinear methods for capturing the geometry of images were developed, such as bandelets [17], curvelets [2], contourlets =-=[13]-=-, wedgelets [14, 21], surflets [3], as well as edge-adapted nonlinear multiresolution [19] and geometric spline approximation [7]. This paper proposes a conceptually new highly nonlinear image compres... |

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Citation Context ...t coefficients [4, 12]. This type of approximation can be viewed as mildly nonlinear. Recently, several highly nonlinear methods for capturing the geometry of images were developed, such as bandelets =-=[17]-=-, curvelets [2], contourlets [13], wedgelets [14, 21], surflets [3], as well as edge-adapted nonlinear multiresolution [19] and geometric spline approximation [7]. This paper proposes a conceptually n... |

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Citation Context ...n can be viewed as mildly nonlinear. Recently, several highly nonlinear methods for capturing the geometry of images were developed, such as bandelets [17], curvelets [2], contourlets [13], wedgelets =-=[14, 21]-=-, surflets [3], as well as edge-adapted nonlinear multiresolution [19] and geometric spline approximation [7]. This paper proposes a conceptually new highly nonlinear image compression method. The ima... |

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Citation Context ...s géométriques et trois images naturelles classiques. 1 Introduction Many of the well-established methods for image compression, including EBCOT [24, 25], are based on wavelets and related techniques =-=[22, 23, 26]-=-, see [6] for a survey on wavelet-based image coding. When working with a dyadic wavelet decomposition [18], digital images are represented by wavelet coefficients. This representation is a linear dec... |

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Citation Context ...This representation is a linear decomposition over a fixed orthogonal basis. The non-linearity in the approximation of images by wavelets is introduced by the thresholding of the wavelet coefficients =-=[4, 12]-=-. This type of approximation can be viewed as mildly nonlinear. Recently, several highly nonlinear methods for capturing the geometry of images were developed, such as bandelets [17], curvelets [2], c... |

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Citation Context ...evant for applications such as zooming, rescaling and conversion between different image representations. The idea to approximate an image by first identifying significant pixels is not new (see e.g. =-=[16]-=-). In this paper, we go further and obtain a complete compression method. Our compression method combines an efficient algorithm for the selection of a set of significant pixels, adaptive thinning, wi... |

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Citation Context ...trois images naturelles classiques. 1 Introduction Many of the well-established methods for image compression, including EBCOT [24, 25], are based on wavelets and related techniques [22, 23, 26], see =-=[6]-=- for a survey on wavelet-based image coding. When working with a dyadic wavelet decomposition [18], digital images are represented by wavelet coefficients. This representation is a linear decompositio... |

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Citation Context ...4, 12]. This type of approximation can be viewed as mildly nonlinear. Recently, several highly nonlinear methods for capturing the geometry of images were developed, such as bandelets [17], curvelets =-=[2]-=-, contourlets [13], wedgelets [14, 21], surflets [3], as well as edge-adapted nonlinear multiresolution [19] and geometric spline approximation [7]. This paper proposes a conceptually new highly nonli... |

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Citation Context ...thods for capturing the geometry of images were developed, such as bandelets [17], curvelets [2], contourlets [13], wedgelets [14, 21], surflets [3], as well as edge-adapted nonlinear multiresolution =-=[19]-=- and geometric spline approximation [7]. This paper proposes a conceptually new highly nonlinear image compression method. The image, viewed as a function, is approximated by a linear spline over the ... |

28 |
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Citation Context ...s géométriques et trois images naturelles classiques. 1 Introduction Many of the well-established methods for image compression, including EBCOT [24, 25], are based on wavelets and related techniques =-=[22, 23, 26]-=-, see [6] for a survey on wavelet-based image coding. When working with a dyadic wavelet decomposition [18], digital images are represented by wavelet coefficients. This representation is a linear dec... |

25 | Multiscale wedgelet image analysis: fast decompositions and modeling
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Citation Context ...n can be viewed as mildly nonlinear. Recently, several highly nonlinear methods for capturing the geometry of images were developed, such as bandelets [17], curvelets [2], contourlets [13], wedgelets =-=[14, 21]-=-, surflets [3], as well as edge-adapted nonlinear multiresolution [19] and geometric spline approximation [7]. This paper proposes a conceptually new highly nonlinear image compression method. The ima... |

24 | Multiresolution Methods in Scattered Data Modelling - Iske - 2004 |

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Citation Context ...eδ of edges in D(Y ). Each priority queue has a least significant element (pixel or pixel pair) at its head, and is updated after each pixel removal. The resulting algorithm has complexity O(N log N) =-=[15]-=-. For more details concerning the efficient maintenance of such priority queues, we refer to our paper [15]. 8s2.7 Minimization of the Mean Square Error in SY In a post-processing to adaptive thinning... |

13 | Iske: Adaptive thinning for terrain modelling and image compression
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Citation Context ...icant pixels, i.e., whose corresponding linear spline over their Delaunay triangulation approximates the image well, requires a suitable measure for the significance of pixels. In our previous survey =-=[10]-=- a first prototype of an image compression method, here referred to as AT − , was presented, where some ideas from our earlier papers [8, 9] were included. For the prototype AT − a suitable measure fo... |

11 | Scattered data coding in digital image compression
- Demaret, Iske
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(Show Context)
Citation Context ...easure for the significance of pixels. In our previous survey [10] a first prototype of an image compression method, here referred to as AT − , was presented, where some ideas from our earlier papers =-=[8, 9]-=- were included. For the prototype AT − a suitable measure for a pixel significance, based on the error incurred by the removal of one pixel, was developed and studied. In this paper, the compression m... |

11 | Advances in digital image compression by adaptive thinning
- Demaret, Iske
(Show Context)
Citation Context ...easure for the significance of pixels. In our previous survey [10] a first prototype of an image compression method, here referred to as AT − , was presented, where some ideas from our earlier papers =-=[8, 9]-=- were included. For the prototype AT − a suitable measure for a pixel significance, based on the error incurred by the removal of one pixel, was developed and studied. In this paper, the compression m... |

7 | Compression of higher dimensional functions containing smooth discontinuities
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(Show Context)
Citation Context ...mildly nonlinear. Recently, several highly nonlinear methods for capturing the geometry of images were developed, such as bandelets [17], curvelets [2], contourlets [13], wedgelets [14, 21], surflets =-=[3]-=-, as well as edge-adapted nonlinear multiresolution [19] and geometric spline approximation [7]. This paper proposes a conceptually new highly nonlinear image compression method. The image, viewed as ... |

4 | Applied and computational aspects of nonlinear wavelet approximation”, Multivariate Approximation and Applications
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(Show Context)
Citation Context ...This representation is a linear decomposition over a fixed orthogonal basis. The non-linearity in the approximation of images by wavelets is introduced by the thresholding of the wavelet coefficients =-=[4, 12]-=-. This type of approximation can be viewed as mildly nonlinear. Recently, several highly nonlinear methods for capturing the geometry of images were developed, such as bandelets [17], curvelets [2], c... |

2 |
Low bit-rate image coding using adaptive geometric splines approximation”, in preparation
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Citation Context ...es were developed, such as bandelets [17], curvelets [2], contourlets [13], wedgelets [14, 21], surflets [3], as well as edge-adapted nonlinear multiresolution [19] and geometric spline approximation =-=[7]-=-. This paper proposes a conceptually new highly nonlinear image compression method. The image, viewed as a function, is approximated by a linear spline over the Delaunay triangulation of a small adapt... |