Computer Vision Lecture 4: Filtering and Noise Reduction, Lecture notes of Computer Vision

Download Computer Vision Lecture 4: Filtering and Noise Reduction and more Lecture notes Computer Vision in PDF only on Docsity!

Intro to Computer Vision

Lecture 4

Greg Shakhnarovich

April 13, 2010

Review: filters

2D cross-correlation filtering of image I(x, y) with

(2k + 1) × (2k + 1) filter H:

(I ⊗ H)(i, j) =

∑^ k

q=−k

∑k

r=−k

I(x + q, y + r) · H(q, r)

− 2 − (^1 0 1 )

2

1

0

1

− 2

H

Review: filters

2D cross-correlation filtering of image I(x, y) with (2k + 1) × (2k + 1) filter H:

(I ⊗ H)(i, j) =

∑^ k

q=−k

∑k

r=−k

I(x + q, y + r) · H(q, r)

j

i

− 2 − (^1 0 1 )

2

1

0

1

− 2

H

Review: filters

2D cross-correlation filtering of image I(x, y) with (2k + 1) × (2k + 1) filter H:

(I ⊗ H)(i, j) =

∑^ k

q=−k

∑k

r=−k

I(x + q, y + r) · H(q, r)

2D convolution:

(I ∗ H)(i, j) =

∑^ k

q=−k

∑k

r=−k

I(x − q, y − r) · H(q, r)

j

i

− 2 − (^1 0 1 )

2

1

0

1

− 2

H

Another noise model

Gaussian noise model

Additive noise model:

I^ ˜(x, y) = I(x, y) + ν(x, y)

Assumption I: ν(x, y) is a random variable, independent of (x, y)

• When could this assumption be violated?

Assumption II: white noise, i.e., zero mean E[ν(x, y)] = 0

Gaussian noise:

p(ν; σ) =

2 πσ^2

exp

ν 2

2 σ^2

Refresher: the Gaussian distribution

(1D) Gaussian with mean μ and variance σ 2 :

p(x; μ, σ) =

2 πσ^2

exp

2 σ^2

(x − μ) 2

The mean determines location

The variance determines shape

x

p(x)

μ = 0, σ = 1 μ = 2

Refresher: the Gaussian distribution

(1D) Gaussian with mean μ and variance σ 2 :

p(x; μ, σ) =

2 πσ^2

exp

2 σ^2

(x − μ) 2

The mean determines location

The variance determines shape

x

p(x)

μ = − 3 μ = 0, σ = 1 μ = 2

Refresher: the Gaussian distribution

(1D) Gaussian with mean μ and variance σ 2 :

p(x; μ, σ) =

2 πσ^2

exp

2 σ^2

(x − μ) 2

The mean determines location

The variance determines shape

x

p(x)

μ = 0, σ = 1

μ = 2, σ =. 25

σ = 2

Refresher: the Gaussian distribution

(1D) Gaussian with mean μ and variance σ 2 :

p(x; μ, σ) =

2 πσ^2

exp

2 σ^2

(x − μ) 2

The mean determines location

The variance determines shape

x

p(x)

μ = 0, σ = 1

μ = 2, σ =. 25

Extremely widely used. Some reasons:

Removing Gaussian noise

noisy mean, k = 3 median, k = 3

Removing Gaussian noise

noisy mean, k = 3 median, k = 3

Gaussian filter, k = 3, σ = 1:

Gaussian filter - 1D

Mean filtering: convolve signal

with

−k 0 k

1 /(2k + 1)

Idea: instead,convolve with a filter shaped like a Gaussian

Gaussian filter - 1D

Mean filtering: convolve signal

with

−k 0 k

1 /(2k + 1)

Idea: instead,convolve with a filter shaped like a Gaussian

Actual digital filter Gσ is an approximation of the Gaussian function

p(x; 0, σ), sampled at integer coordinates x = −k,... , k.