Data smoothing by splines with free knots

Dikoussar, N.D.; Török, Cs.

doi:10.1134/s1547477108030394

Cited by 3 publications

(4 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This optimisation problem is also called free knot problem and has been studied for more than fifty years. We refer to [61,76] for similar considerations as in our work and to [11,37,39,73] and the references therein for more general approaches. Note that it is quite common to relax the interpolation condition and to generalise the problem to explicitly allow approximating functions.…”

Section: Optimal Knots For Interpolating Convex Functionsmentioning

confidence: 99%

2. Optimizing spatial and tonal data for PDE-based inpainting

Hoeltgen¹,

Mainberger²,

Hoffmann³

et al. 2016

Variational Methods

View full text Add to dashboard Cite

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. Apart from these specific contributions, we also give an extensive literature survey on PDE-based image compression methods. Summary and ConclusionsThe contributions of our work are twofold: On the one hand we gave a comprehensive survey on the state-of-the-art in PDE-based image compression.

show abstract

Section: Optimal Knots For Interpolating Convex Functionsmentioning

confidence: 99%

2. Optimizing spatial and tonal data for PDE-based inpainting

Hoeltgen¹,

Mainberger²,

Hoffmann³

et al. 2016

Variational Methods

View full text Add to dashboard Cite

show abstract

“…5, 6); the papers [20] and [21] represent successful smoothing of strongly noising data by cubic splines with free knots based on the model (18), etc.…”

Section: Discussionmentioning

confidence: 99%

“…The estimateθ is determined by the recursive least squares procedure, with the ampliˇca-tion factor K n (α, β) ∼ O(n −3 ), where α, β are smoothing parameters; the 9-point model for surface smoothing [17]; in [13,18] the DPT of polynomials and the assessment of the polynomial degree were studied; the piecewise-cubic algorithm with auto detection of knots [11,21] (Figs. 5, 6); the papers [20] and [21] represent successful smoothing of strongly noising data by cubic splines with free knots based on the model (18), etc.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Four-point transformation methods in approximation and the smoothing problems

Dikoussar

2008

Phys. Part. Nuclei Lett.

Self Cite

View full text Add to dashboard Cite

The main goal of the adaptive local strategy consists in reducing the complexity of computational problems. We propose a new approach to curve approximation and smoothing based on 4-point transformations or Discrete Projective Transforms (DPT). In the framework of DPT the variable point is related to three data points (accompanying points). The variable y ordinate is expressed via the convolution of accompanying y ordinates and weight functions that are deˇned as cross-ratio functions of four x coordinates. DPT has some attractive properties (natural norming, scale invariance, threefold symmetry, ®4-point¯orthogonality), which are useful in designing new algorithms. Diverse methods and algorithms based on DPT have been developed. ¸´μ¢´ Ö Í¥²Ó ¤ ¶É¨¢´μ°²μ± ²Ó´μ°¸É· É¥£¨¨¸μ¸Éμ¨É ¢ ¶μ´¨¦¥´¨¨¢ÒÎ¨¸²¨É¥²Ó´μ°¸²μ¦-μ¸É¨ ²£μ·¨É³μ¢. ‚ · ¡μÉ¥ ¶·¥¤² £ ¥É¸Ö´μ¢Ò° ¶μ¤Ìμ¤ ± ¶ ¶·μ±¸¨³ Í¨¨¨¸£² ¦¨¢ ´¨Õ ±·¨-¢ÒÌ, μ¸´μ¢ ´´Ò°´ 4-ÉμÎ¥Î´ÒÌ ¶·¥μ¡· §μ¢ ´¨ÖÌ¨²¨¤¨¸±·¥É´ÒÌ ¶·μ¥±É¨¢´ÒÌ ¶·¥μ¡· §μ¢ ´¨ÖÌ ("). ‚ · ³± Ì " É¥±ÊÐ Ö ÉμÎ± ¸¢Ö §Ò¢ ¥É¸Ö¸É·¥³Ö ÉμÎ± ³¨¤ ´´ÒÌ (¸μ ¶·μ¢μ¦¤ ÕÐ¨³É μÎ± ³¨). '¥±ÊÐ Ö μ·¤¨´ É y ¢Ò· ¦ ¥É¸Ö Î¥·¥ §¸¢¥·É±Ê¸μ ¶·μ¢μ¦¤ ÕÐ¨Ì y-μ·¤¨´ É¸¢¥¸μ¢Ò³Ë Ê´±Í¨Ö³¨, μ ¶·¥¤¥²Ö¥³Ò³¨¸²μ¦´Ò³ μÉ´μÏ¥´¨¥³ Î¥ÉÒ·¥Ì x-±μμ·¤¨´ É. " μ¡² ¤ ¥É ·Ö¤μ³ ¶·¨¢²¥± É¥²Ó´ÒÌ¸¢μ°¸É¢ (¥¸É¥¸É¢¥´´ Ö´μ·³¨·μ¢± , ³ ¸ÏÉ ¡´ Ö¨´¢ ·¨ ´É´μ¸ÉÓ, É·μ°´ Ö¸¨³³¥-É·¨Ö, ®4-ÉμÎ¥Î´ Ö¯μ·Éμ£μ´ ²Ó´μ¸ÉÓ), ±μÉμ·Ò¥ ¶μ²¥ §´Ò ¶·¨¸μ §¤ ´¨¨´μ¢ÒÌ ²£μ·¨É³μ¢. μ¸´μ¢¥ " ¡Ò² · §· ¡μÉ ´·Ö¤ · §´μμ¡· §´ÒÌ ³¥Éμ¤μ¢¨ ²£μ·¨É³μ¢.

show abstract

The basic element method

Dikusar

2011

Math Models Comput Simul

View full text Add to dashboard Cite

Data smoothing by splines with free knots

Cited by 3 publications

References 5 publications

2. Optimizing spatial and tonal data for PDE-based inpainting

2. Optimizing spatial and tonal data for PDE-based inpainting

Four-point transformation methods in approximation and the smoothing problems

The basic element method

Contact Info

Product

Resources

About