Probability, statistical distributions, descriptive statistics

DI&Applied ML

Normal Distributions

  • Normally distributed random variable with mean μ and variance σ²:

  • All higher order moments are given in terms of μ and variance σ².

  • Easily manipulated:

    • If x ~ N(0, σx²) and y ~ N(0, σy²), then x + y ~ N(0, σx² + σy²)

  • Central Limit Theorem:

    • The sum of a large number of IID random variables (with finite mean and variance) is normally distributed.

  • For linear models:

    • Normal distributions are preserved by the principle of superposition.
    • Normally distributed forecast errors: Maximum likelihood gives least squares.
    • Useful for calculating prediction intervals.

  • Problems for nonlinear systems:

    • Use of normal distributions neglects the possibility of asymmetric distributions.
    • Fat-tailed distributions imply larger probability of worse-case scenarios (risk management).

CV_Features

Computer Vision

1. Blobs as Interest Points

  • Characteristic scale

    • 2nd derivative of Gaussian.

  • Normalize LOG

    • Basic Laplacian detector = local maxima of characteristic scale.
    • LOG is good for finding scale.

  • Use difference

2. Combine the Harris and Laplace

  • Combine methods to enhance detection and reliability.

3. SIFT

  • Scale-Invariant Feature Transform (SIFT).

CV_Filtering and Sampling

Computer Vision
  1. Correlation Filtering

Convolution

If H[-u,-v]=H[u,v], then correlation = convolution

1)Gaussian kernel

Gaussian smoothing:

Variance determines extent of smoothing

set filter half-width to about 3*variance

In Matlab:

hsize=10;

sigma=5;

h=fspecial(‘gaussian’ hsize,sigma);

mesh(h);imagesc(h);

outim=imfilter(im,h);

imshow(outim);

2)Oriented Gaussian Filters

3）Difference of Gaussian

DOG

Laplacian of Gaussian can be approximated by the difference between two different Gaussians

3)Derivative of Gaussian

LOG Laplacian of Gaussian

3.Sampling

wagen whell effect

Nyquist theorem: In order to recover a certain frequency f, we need to sample with at least 2f.

Representation of scale

The Gaussian pyramid

blur+subsample

The Laplacian Pyramid

a band pass representation vice a low pass representation of Gaussian

4.Detectors:Harris

Bolbs and conners

auto-correlation matrix

f(x+triangelex)=f(x)+trianglex*f'(x)+ trianglex^2*f”(x2)

5.Gaboe Wavelet

小波变换教程（1）：基本原理