Image Noise and Filtering CMPE 264: Image Analysis and Computer Vision Hai Tao

Department of Computer Engineering University of California at Santa Cruz

Estimating acquisition noise Q Q

Noise introduced by imaging system Simple model • Assumption: noise at each pixel is independent, characterized by its mean and standard deviation • Estimating the mean and the standard deviation σ for each pixel For each i, j = 0,..., N − 1, let 1 N −1 I (i, j ) = ∑ I k (i, j ) N k =0 1/ 2

⎛ 1 N −1 2⎞ σ (i, j ) = ⎜ ∑ ( I k (i, j ) − I (i, j )) ⎟ ⎝ N − 1 k =0 ⎠

• For most imaging system, σ ≈ 2.5

Department of Computer Engineering University of California at Santa Cruz

Estimating acquisition noise Q

Estimating auto-covariance • In reality, the noise in neighboring pixels is not independent • The correlation is described by auto-covariance • If we assume auto-covariance of noise is the same everywhere in the image, then Let N i ' = N − i '−1, N j ' = N − j '−1, for each i ' , j ' = 0,..., N − 1, compute 1 N N CII (i ' , j ' ) = 2 ∑ ∑ ( I k (i, j ) − I (i, j ))( I k (i + i ' , j + j ' ) − I (i + i ' , j + j ' )) N i =0 j =0 i'

j'

• Example: How to compute CII (2,1) for a 10x10 image ? N i ' = 7, N j ' = 8, for each i ' = 2, j ' = 1, compute 1 7 8 CII (2,1) = 2 ∑ ∑ ( I k (i, j ) − I (i, j ))( I k (i + i ' , j + j ' ) − I (i + i ' , j + j ' )) 10 i =0 j =0

Department of Computer Engineering University of California at Santa Cruz

Estimating acquisition noise Q

Auto-covariance for a typical imaging system. Notice that the covariance along the horizontal direction, which is a characteristic often observed in CCD cameras

Department of Computer Engineering University of California at Santa Cruz

Modeling image noise Q

Additive noise model Random noise n(i, j ) added to pixel value I (i, j ) Iˆ(i, j ) = I (i, j ) + n(i, j )

Q

Signal-to-noise ratio (SNR), often expressed in decibel SNR =

σs σn

σs SNRdB = 10 log10 ( ) σn Q

σs = 100 20 dB means σn

Department of Computer Engineering University of California at Santa Cruz

Modeling image noise Q

Gaussian noise -white Gaussian, zero-mean stochastic process • White – n(i,j) independent in both space and time • Zero-mean – I (i, j ) = 0 • Gaussian - n(i,j) is random variable with distribution p ( x) =

Q

1 e σ 2π

−

x2 2σ 2

Impulsive noise – also called peak, spot, or salt and pepper noise, caused by transmission errors, faulty CCD sites, etc. ⎧ I (i, j ) I sp (i, j ) = ⎨ ⎩ I min + y ( I max − I min )

x