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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...
Linear Approximations01:23

Linear Approximations

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...
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...
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...
Calibration Curves: Linear Least Squares01:20

Calibration Curves: Linear Least Squares

A calibration curve is a plot of the instrument's response against a series of known concentrations of a substance. This curve is used to set the instrument response levels, using the substance and its concentrations as standards. Alternatively, or additionally, an equation is fitted to the calibration curve plot and subsequently used to calculate the unknown concentrations of other samples reliably.
For data that follow a straight line, the standard method for fitting is the linear...
Downsampling01:20

Downsampling

When considering a sampled sequence with zero values between sampling instants, one can replace it by taking every N-th value of the sequence. At these integer multiples of N, the original and sampled sequences coincide. This process, known as decimation, involves extracting every N-th sample from a sequence, thereby creating a more efficient sequence.
The Fourier transform of the decimated sequence reveals a combination of scaled and shifted versions of the original spectrum. This...

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Related Experiment Video

Updated: Jul 7, 2026

Quantifying Intermembrane Distances with Serial Image Dilations
07:45

Quantifying Intermembrane Distances with Serial Image Dilations

Published on: September 28, 2018

Least-squares image resizing using finite differences.

A Muñoz1, T Blu, M Unser

  • 1Biomed. Imaging Group, Swiss Federal Inst. of Technol., Lausanne. arrate.munoz@epfl.ch

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|February 8, 2008
PubMed
Summary
This summary is machine-generated.

This study introduces an optimal spline-based algorithm for digital image scaling, improving image quality by reducing artifacts and enhancing signal-to-noise ratio. The novel finite difference method offers efficient computation for arbitrary scaling factors.

Related Experiment Videos

Last Updated: Jul 7, 2026

Quantifying Intermembrane Distances with Serial Image Dilations
07:45

Quantifying Intermembrane Distances with Serial Image Dilations

Published on: September 28, 2018

Area of Science:

  • Digital Image Processing
  • Computational Mathematics
  • Computer Vision

Background:

  • Standard image interpolation methods often introduce artifacts like aliasing and blocking.
  • Arbitrary (noninteger) scaling factors pose challenges for traditional algorithms.
  • Improving signal-to-noise ratio (SNR) is crucial for image quality.

Purpose of the Study:

  • To develop an optimal spline-based algorithm for digital image enlargement and reduction.
  • To address the challenge of arbitrary noninteger scaling factors.
  • To improve image quality by minimizing artifacts and enhancing SNR.

Main Methods:

  • A projection-based approach utilizing a novel finite difference method.
  • Computation of inner products with B-spline basis functions of any degree n.
  • Algorithm complexity is independent of the scaling factor, ensuring efficiency.

Main Results:

  • The spline-based algorithm consistently outperforms standard interpolation procedures.
  • Significant reduction in artifacts such as aliasing and blocking.
  • Substantial improvement in the signal-to-noise ratio.

Conclusions:

  • The proposed algorithm provides superior image scaling results compared to existing methods.
  • The method is computationally efficient and adaptable to various piecewise polynomial functions.
  • This approach offers a robust solution for high-quality digital image resizing.