What is the Laplacian?

Laplacian of a function

Gradient as a vector field

The divergence

Divergence of a gradient field

Laplacian as a measure of minimum points

Analogy with second derivatives

Laplacian as an analog of the second derivative

Gradient of a function

Second component of the Laplacian

Evaluating the dot product

Divergence of the gradient field

Conclusion

The Laplacian has meaning

Calculation of the Laplacian

Graph of the Laplacian

Approximating the Laplacian

What about the integral of cos(alpha) from 0 to 2pi?

Simplifying the Laplacian

Does the Laplacian describe a change in shape?

Alternative definition

Conclusion

Laplacian in Cartesian Coordinates

What is the Laplacian or Laplace Operator?

What is the Laplace Operator?

What is the function f?

Direction of steepest descent

Gradient of a scalar function

What is the Del Operator?

The Laplace Operator is the sum of all of the unmixed second order derivatives

Recap

Applications of the Laplace Operator

Summary

Introduction

Laplacian of a function

Graph Analogy

Directional derivative of the gradient

Brownian motion and the Laplace operator

Self-adjoint property of the Laplace operator

Laplacian operator

Gradient of a function

Multiplying first components

Laplacian of a function

Alternate Laplacian formula

Conclusion

Laplace smoothing formula

Why use Laplace smoothing?

Laplace smoothing example

Estimating a conditional distribution

What is smoothing?

Maximum likelihood estimates

Computing estimates

Conclusion

Intro

Logarithm of the posterior distribution

Multivariate Taylor series expansion

History of the Normal Approximation

Algorithm Optimizing

Gaussian latent variable models

Normal approximation

Importance resampling

Normal approximation example

Normal mode vs Normal with VI

The Laplacian Operator

Zero Crossing

What is the Laplacian?

Laplacian Operator Implementation

Laplacian operator on an image

Noise in Edges

Gaussian smoothing

Derivative of the Gaussian

Laplacian of Gaussian Operator

The Canny Edge Detector

What is the phase of a complex number?

Frequency domain function

Time domain function

What is the Laplace Transform?

Sums of exponentials of complex numbers

Adding exponential functions together

Laplace Transform Visualization

Multiplying by a complex number

Finding the Laplace transform of a waveform

Conclusion

Definition of the Laplacian

Laplacian of a scalar function example

Laplacian of a scalar function formula

Applying the Laplacian operator

Second derivative

Second derivative of sine

Simplify second derivative

Eigenfunctions of the Laplacian

Laplacian in one dimension

Evaluate the function at 4 points

First derivative

Evaluating the Laplacian

Change of coordinates example

Example: Evaluating the Laplacian of a function

Laplacian as average change in change of function

Change-of-components example

What is the Laplacian?