Stochastic Image Denoising using Minimum Mean Squared Error (Wiener) Filtering

ECEN 670 MINI-CONFERENCE PROJECT

BRANDON CARROLLLAITH SAHAWNEH

ECEN 670 CLASSSTOCHASTIC PROCESSES

Stochastic Image Denoising using Minimum Mean

Squared Error (Wiener) Filtering

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Outline

IntroductionTheory: Wiener Filter DerivationResults & AnalysisConclusion

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Introduction

o A digital image is generally encoded as a matrix of gray-level or color values.o An image may be defined as a two-dimensional function, x[u,v], where u, v

are spatial (plane) coordinates.o In the case of color images, x[u,v] is a triplet of values for the red, green, and

blue components

Digital Images (Very brief introduction):

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Example: This is how images represented in computer

Color Images

Introduction

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1. Image denoising is one of the fundamental challenges in the field of image processing.

2. Employed using variety of configurations in a wide variety of applications: namely: Object recognition, photo enhancement, and image restoration.

3. The objective of image restoration is to improve a given image in some predefined sense. Image restoration attempts to reconstruct or recover a degraded image by using a priori knowledge of the degradation phenomenon.

Introduction

Image Denoising and Restoration:

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Introduction

o The purpose of image restoration is to restore a degraded/distorted image to its original content and quality.

o Image restoration assumes a degradation model that is known or can be estimated.

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Linear, space-invariant degradation model: Apply the inverse process to recover the original image ??

Degradation Function h ∑

Restoration Filter (Wiener

Filter)

y(u, v)x(u, v) x(u, v)

Degradation Process with additive noise Denoising and Restoration Process

Noise n(u, v)

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Introduction

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hfg ,

What h will give us g = f ?Dirac Delta Function (Unit Impulse)

x2

1

0

Convolution Kernel – Impulse Response

Point spread function (PSF) :

f gh

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Introduction

OpticalSystemscene image

• An ideal optical system that does not degrade the image at all would have a Dirac delta function as its PSF

x xPSFOpticalSystem

point source point spread function

• However, optical systems are never ideal.

Point spread function (PSF) :

• The PSF is the response of the system to a Dirac delta function input.

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Introduction

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Point spread function (PSF) : The impulse is a point (or pixel) of light The impulse response is commonly referred to as the

PSF

),( vuH

Impulse of Light Image (degraded) impulse

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Introduction

Noise modelso Most types of noise are modeled as probability density

functions (PDFs)o Noise model is decided based on understanding of the

physics of the sources of noise. Gaussian: poor illumination Rayleigh: range image Gamma, exp: laser imaging Impulse: faulty switch during imaging, Uniform: quantization.

o Parameters can be estimated based on histogram on small flat area of an image.

De-noising o Spatial filtering (Mean filters, Median, Max, Min …)o Frequency domain filtering ( Inverse Filter, Wiener Filter,

…)

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Introduction

Estimation of noise parameters:

1. Gaussian Noise

2. Uniform Noise

3. Impulse (Salt& Pepper) Noise

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Introduction

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o The parameters of noise in the spatial domain may be known from the sensors specification or a priori-knowledge of noise distribution.

o In most cases it is necessary to estimate these parameters to specify the corresponding noise PDF from sample images being denoised.

Estimation of noise parameters:

How to estimate the noise parameters?1. The common approach is to select a region of interest (ROI) in an

image with as featureless a background as possible, so that the variability of intensity values in the region will be due primarily to noise!

{ Let zi be a discrete random variable that denotes intensity levels in an image, and p(zi), i = 1, 2, . . .,L - 1 be the corresponding normalized histogram, where, L is thenumber of possible intensity values. The histogram component p(zi) is an estimate of the probability of occurrence of intensity value zi.}

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Estimation of noise parameters:

2. Find the histogram of ROI.

3. Normalized histogram of (ROI) which can be viewed as an approximation of the intensity PDF!

4. Describe the shape of noise PDF via its central moments (estimate the mean and variance and then compute a and b variables if needed).

Introduction

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Introduction

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Estimation of noise parameters (Example – Our Matlab Code Simulation):The original Image

The original image with additive Gaussian noise N(0.01,0.009

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intensity levels

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f pix

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intensity levelsp(

z) p

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alue

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Original Image

Image with Gaussian Noise

Histogram of ROI Normalized histogram of ROI

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Introduction

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intensity levels

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Uniform Noise Histogram

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intensity levels

p(z)

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Impulse Noise Histogram

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0.14Normalized Histogram-approxmiated PDF of ROI

intensity levels

p(z)

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Wiener Filter

Image Restoration Approaches

Classical approaches

Inverse filter

Weiner filter

Algebraic approaches

The regularization theory

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Unconstrained optimization

Constrained optimization

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Wiener Filter

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],[],[],[

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vu

vuvu

vu

vuvu

ffHffNffX

ffHffYffX

Inverse Filter:

],[],[],[],[),(),(),(],[

vvvv ffNffHffXffYvunvuhvuxvuy

uuuu

Degradation Function h ∑

Restoration Filter (Wiener

Filter)

y(u, v)x(u, v) x(u, v)

Degradation Process with additive noise Denoising and Restoration Process

Noise n(u, v)

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Wiener Filter

Most neighboring pixels are highly correlated, while widely separated pixels are only loosely correlated.

Therefore, the autocorrelation function of typical images generally decreases away from the origin.

Power spectrum = Fourier transform of autocorrelation, therefore the power spectrum of an image generally decreases with frequency.

Typical noise sources have either a flat power spectrum or one that decreases with frequency more slowly than typical image power spectrum.

Therefore, the signal should dominate at low frequencies, while the noise dominates at high frequencies.

Wiener Filter:

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Wiener Filter:Assumptions:1. The noise and the image are uncorrelated.2. Either the noise or the image is zero-mean, and that the

intensities in the estimate are a linear function of the intensities in the noisy image.

3. It also assumes that the power spectrum of both the noise and the original image are known

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Wiener Filter

Degradation model

Theory (Derivation)

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Wiener Filter

The Goal We wish to find an LTI filter with impulse response g[u, v] that gives us the MMSE estimate of x[u, v]:

The MSE is given by:

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Parseval’s theorem states:

Since they are directly proportional, we can minimize the spatial domain mean squared error by minimizing the frequency domain mean squared error:

Using property of the Fourier transform:

Substitute in Y then rearrange the terms

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Wiener Filter

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Using the definition of an absolute square, we get:

Multiplying out theterms and distributing the expected value across the sums

We assume that the noise is independent of the originalimage and that either the noiseor the image is zero-mean:

Where the power spectral densities are defined as:

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Wiener Filter

To find the G that minimizes the MSE, we take the derivative with respect to G(fu, fv):

Solving for G* gives:HH* = │H│2

Finally, we write the equationin terms of the signal-to-noiseRatio:

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Wiener Filter

Wiener Filter Equation:

Arranged intuitively:

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Results & Analysis

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Original image 20 pixel motion blur

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Results & Analysis

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Original image 20 pixel motion blur, inverse filtered

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Results & Analysis

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Original image 5 pixel motion blur, imperceptible Gaussian noise

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Results & Analysis

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Original image 5 pixel motion blur, imperceptible Gaussian noise, inverse filtered

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Results & Analysis

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Original image 5 pixel motion blur, imperceptible Gaussian noise, Wiener filtered

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Results & Analysis

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Original image 15 pixel motion blur, Gaussian noise

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Results & Analysis

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15 pixel motion blur, Gaussian noise

15 pixel motion blur, Gaussian noise, inverse filtered

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Results & Analysis

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15 pixel motion blur, Gaussian noise

15 pixel motion blur, Gaussian noise, Wiener filtered

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Results & Analysis

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3 pixel motion blur, salt and pepper noise

3 pixel motion blur, salt and pepper noise, inverse filtered

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Results & Analysis

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3 pixel motion blur, salt and pepper noise

3 pixel motion blur, salt and pepper noise, Wiener filtered

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Results & Analysis

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Wiener filter using estimated noise spectrum

Wiener filter using actual noise spectrum

Region of Interest

Conclusions

Inverse filter works well with no noiseWiener filter performs much better in the presence

of noise Assumes knowledge of degradation function (a common

requirement for image restoration algorithms) Assumes knowledge of the power spectra of the noise and

original image (less common, makes it less useful)The noise power spectrum can be effectively

estimated by analyzing the histogram of an ROI in the noisy image

Forms the basis of other more robust restoration approaches

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Thank you

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