Article

# Splines in Higher Order TV Regularization.

International Journal of Computer Vision (Impact Factor: 3.62). 12/2006; 70:241-255. DOI: 10.1007/s11263-006-8066-7

Source: DBLP

- [Show abstract] [Hide abstract]

**ABSTRACT:**We are interested in minimizing functionals with ℓ2 data and gradient fitting term and ℓ1 regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s algorithm to solve the minimization problem with the ℓ1 norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ2 gradient fitting term can be used to avoid both edge blurring and staircasing effects.Advances in Computational Mathematics 01/2009; 30(1):79-99. · 1.47 Impact Factor -
##### Conference Paper: Two Step Variational Method for Subpixel Optical Flow Computation.

[Show abstract] [Hide abstract]

**ABSTRACT:**We develop an algorithm for the super-resolution optical flow computation by combining variational super-resolution and the variational optical flow computation. Our method first computes the gradient and the spatial difference of a high resolution images from these of low resolution images directly, without computing any high resolution images. Second the algorithm computes optical flow of high resolution image using the results of the first step.Advances in Visual Computing, 5th International Symposium, ISVC 2009, Las Vegas, NV, USA, November 30 - December 2, 2009, Proceedings, Part II; 01/2009 - [Show abstract] [Hide abstract]

**ABSTRACT:**This is an overview of recent research of the authors on the application of variational methods with higher–order derivatives in image processing. We focus on gray-valued and matrix-valued images and deal with a purely discrete setting. We show that regu-larization methods with second–order derivatives can be successfully applied to the denoising of gray–value images. In 1D the solutions of the corresponding minimization problems are discrete polynomial splines (sometimes with higher defects) and inf-convolution splines with certain knots. The proposed methods can be transferred to ma-trix fields. Due to the operator structure of matrices, new tasks like the preservation of positive definiteness and the meaningful coupling of the matrix components come into play.

Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.