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Related Concept Videos

Residuals and Least-Squares Property01:11

Residuals and Least-Squares Property

The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates the actual data value for y.
The process of fitting the best-fit...
Gaussian Elimination: Problem Solving01:30

Gaussian Elimination: Problem Solving

Systems of linear equations in several variables are pivotal in modeling complex scenarios involving multiple unknowns and constraints. Such systems are widely used in various fields to represent relationships where several conditions must be simultaneously satisfied. Each variable in the system corresponds to an unknown quantity, while each equation imposes a linear constraint, leading to a structured approach for analyzing and solving real-world problems.A system of three equations with three...
Linearization and Approximation01:26

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
Quadratic Models01:23

Quadratic Models

Quadratic models are mathematical representations used to describe relationships in which the rate of change changes at a constant rate. These models appear in a wide variety of natural and engineered systems, especially those involving motion, forces, and optimization. One common application is analyzing the vertical motion of objects influenced by gravity, such as a ball thrown into the air.In such scenarios, the object's height changes over time in a curved pattern, rising to a maximum point...
Application of Linearization and Approximation01:29

Application of Linearization and Approximation

A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...
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Application of Nonlinear Inequalities

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Related Experiment Videos

Hessian Schatten-norm regularization for linear inverse problems.

Stamatios Lefkimmiatis1, John Paul Ward, Michael Unser

  • 1Biomedical Imaging Group, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland. stamatis.lefkimmiatis@epfl.ch

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|January 11, 2013
PubMed
Summary
This summary is machine-generated.

We developed new image regularization methods using Schatten norms to solve inverse imaging problems. These methods improve upon total-variation by reducing artifacts like staircasing.

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Area of Science:

  • * Applied Mathematics
  • * Image Processing
  • * Computational Imaging

Background:

  • * Ill-posed linear inverse imaging problems are common in scientific and medical applications.
  • * Total-variation (TV) regularization is widely used but can introduce staircase artifacts.
  • * Existing methods struggle with noise and artifact reduction in complex image reconstructions.

Purpose of the Study:

  • * To introduce novel invariant, convex, and non-quadratic functionals for regularized solutions.
  • * To develop second-order regularization methods that overcome limitations of TV semi-norms.
  • * To enhance the quality and reduce artifacts in reconstructed images.

Main Methods:

  • * Employed invariant, convex, and non-quadratic functionals based on Schatten norms of the Hessian matrix.
  • * Developed a primal-dual algorithm for solving the optimization problems.
  • * Established an efficient matrix projection method onto Schatten norm balls.

Main Results:

  • * Proposed regularizers act as second-order extensions of TV semi-norms, preserving invariance properties.
  • * Demonstrated avoidance of staircase artifacts common in TV-based reconstructions.
  • * Showcased effective performance across various inverse imaging problems with real and simulated data.

Conclusions:

  • * The novel Schatten norm-based regularizers offer a powerful alternative for inverse imaging.
  • * The proposed algorithm and projection methods provide efficient solutions.
  • * The approach effectively reduces artifacts and improves image reconstruction quality.