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Related Concept Videos

Reconstruction of Signal using Interpolation01:10

Reconstruction of Signal using Interpolation

Signal processing techniques are essential for accurately converting continuous signals to digital formats and vice versa. When a continuous signal is sampled with a period T, the resulting sampled signal exhibits replicas of the original spectrum in the frequency domain, spaced at intervals equal to the sampling frequency. To handle this sampled signal, a zero-order hold method can be applied, which creates a piecewise constant signal by retaining each sample's value until the next sampling...
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For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
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Linear Approximation in Frequency Domain

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Upsampling01:22

Upsampling

Managing signal sampling rates is essential in digital signal processing to maintain signal integrity. A decimated signal, characterized by a reduced frequency range due to its lower sampling rate, can be upsampled by inserting zeros between each sample. This upsampling process expands the original spectrum and introduces repeated spectral replicas at intervals dictated by the new Nyquist frequency. To refine this zero-inserted sequence, it is passed through a lowpass filter with a cutoff...
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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...
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Linear Approximation in Time Domain

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Related Experiment Video

Updated: Jun 16, 2026

Meso-Scale Particle Image Velocimetry Studies of Neurovascular Flows In Vitro
08:00

Meso-Scale Particle Image Velocimetry Studies of Neurovascular Flows In Vitro

Published on: December 3, 2018

Regularized interpolation for noisy images.

Sathish Ramani1, Philippe Thevenaz, Michael Unser

  • 1Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 USA. sramani@umich.edu

IEEE Transactions on Medical Imaging
|February 5, 2010
PubMed
Summary
This summary is machine-generated.

This study introduces a novel interpolation method for noisy medical imaging data. It balances data fidelity with noise robustness using variational principles and regularization, outperforming exact fits and Tikhonov methods.

Related Experiment Videos

Last Updated: Jun 16, 2026

Meso-Scale Particle Image Velocimetry Studies of Neurovascular Flows In Vitro
08:00

Meso-Scale Particle Image Velocimetry Studies of Neurovascular Flows In Vitro

Published on: December 3, 2018

Area of Science:

  • Medical Imaging
  • Image Processing
  • Applied Mathematics

Background:

  • Interpolation fits continuous models to discrete data.
  • Exact fitting is ideal for noise-free data but impractical in noisy medical imaging.
  • High-quality medical imaging necessitates robust interpolation techniques.

Purpose of the Study:

  • To develop an interpolation scheme balancing data fidelity and noise robustness for medical imaging.
  • To explore variational principles for imposing smoothness constraints on models with noisy data.
  • To identify optimal regularization methods for invariant interpolation models.

Main Methods:

  • Utilized variational principles to enforce smoothness constraints on interpolated models.
  • Investigated L(p)-norm of vector derivatives for regularization, including Tikhonov and total-variation (TV) regularization.
  • Developed algorithms for model recovery from noisy samples and a data-driven scheme for optimal regularization determination.

Main Results:

  • Proposed a scheme that effectively trades off data fidelity and noise robustness.
  • Demonstrated the superiority of the proposed method over exact fitting in numerical examples.
  • Showcased the benefits of edge-preserving total-variation-like regularization over Tikhonov-like quadratic regularization.

Conclusions:

  • The proposed variational approach with appropriate regularization enhances medical image interpolation quality.
  • Shift-, rotation-, and scale-invariant requirements guide the selection of regularization norms.
  • Edge-preserving TV-like regularization offers advantages over traditional quadratic methods for noisy medical data.