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Related Concept Videos

Convolution: Math, Graphics, and Discrete Signals01:24

Convolution: Math, Graphics, and Discrete Signals

In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t0 indicating the initial time.
To simplify the convolution integral, it is assumed that both the input signal and impulse response are zero for negative time values. The graphical convolution process...
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...
Convolution Properties II01:17

Convolution Properties II

The important convolution properties include width, area, differentiation, and integration properties.
The width property indicates that if the durations of input signals are T1 and T2, then the width of the output response equals the sum of both durations, irrespective of the shapes of the two functions. For instance, convolving two rectangular pulses with durations of 2 seconds and 1 second results in a function with a width of 3 seconds.
The area property asserts that the area under the...
Convolution Properties I01:20

Convolution Properties I

Convolution computations can be simplified by utilizing their inherent properties.
The commutative property reveals that the input and the impulse response of an LTI (Linear Time-Invariant) system can be interchanged without affecting the output:
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...
Convergence of Fourier Series01:21

Convergence of Fourier Series

The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...

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Related Experiment Videos

Approximating large convolutions in digital images.

D M Mount1, T Kanungo, N S Netanyahu

  • 1Dept. of Comput. Sci., Maryland Univ., College Park, MD 20740, USA. mount@cs.umd.edu

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

We developed a new algorithm for computing binary convolutions in image processing. This method efficiently calculates convolutions using convex polygonal kernels, achieving O(kmn) time complexity for k-sided kernels.

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Area of Science:

  • Computer Vision
  • Image Processing
  • Computational Geometry

Background:

  • Binary convolutions are crucial in image processing and mathematical morphology.
  • Computing these convolutions can be computationally expensive, particularly with large kernels.

Purpose of the Study:

  • To present an efficient algorithm for computing binary convolutions with convex polygonal kernels.
  • To demonstrate that valid binary convolutions can be computed more efficiently than standard methods for large kernels.

Main Methods:

  • The algorithm utilizes a convex polygonal kernel, allowing flexibility in digitization.
  • It employs Bresenham's line-drawing algorithm and prefix-sums for incremental updates.
  • The convolution is computed in O(kmn) time for an m x n image and a k-sided kernel.

Main Results:

  • A novel algorithm for valid binary convolutions with convex polygonal kernels is presented.
  • The algorithm achieves a running time of O(kmn), independent of kernel area or perimeter.
  • The method avoids the need for Fast Fourier Transforms.

Conclusions:

  • The proposed algorithm offers a significant efficiency improvement for computing binary convolutions with convex polygonal kernels.
  • This approach provides a practical solution for image processing tasks involving large kernels.
  • The technique combines geometric properties of kernels with efficient computational methods.