Noise Model. Important Noise Probability Density Functions (Cont.) Important Noise Probability Density Functions

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1 Others -- Noise Removal Techniques -- Edge Detection Techniques -- Geometric Operations -- Color Image Processing -- Color Spaces Xiaojun Qi Noise Model The principal sources of noise in digital images arise during image acquisition (digitization) and/or transmission. Assumptions in Spatial and Frequency Properties of Noise: Noise is independent of spatial coordinates and it is uncorrelated with respect to the image itself. The Fourier spectrum of noise is constant (The noise usually is called white noise) Important Noise Probability Density Functions Gaussian Noise: It has mathematical tractability in both the spatial and frequency domains. Used in situations in which they are marginally applicable at best. ( z µ ) / σ p( z) e πσ Approximately 7% of the gray level values will be in the range [( µ σ ),( µ + σ )] About 95% will be in the range [( µ σ ),( µ + σ )] 3 Important Noise Probability Density Functions (Cont.) Rayleigh Noise: The displacement from the origin The basic shape of this density is skewed to the right. The Rayleigh density can be quite useful for approximating skewed histograms. ( z a) / b ( z a) e for z a p( z) b for z < a The mean and variance of thisdensity is given by : µ a + πb / 4 b(4 π ) 4 and σ 4 Important Noise Probability Density Functions (Cont.) Erlang (Gamma) Noise: Exponential Noise: It is a special case of Erlang PDF with b. Uniform Noise: Impulse (salt-and-pepper) Noise: 5 6

2 Applications of Noise Probability Density Functions. Gaussian Noise arises in images due to electronic circuit noise and sensor noise.. Rayleigh density is helpful in characterizing noise phenomena in range imaging. 3. The exponential and gamma densities find application in laser imaging. 4. Impulse noise is found in situations where quick transients take place during imaging. 5. Uniform density is useful as the basis for numerous random number generators that are used in simulations. 7 8 Estimation of Noise Parameters. Parameters of periodic noise typically are estimated by inspection of the Fourier spectrum of the image.. When only images already generated by the sensor are available, frequently it is possible to estimate the parameters of the PDF (Probability Density Functions) from small patches of reasonably constant gray level. 9 Restoration in the Presence of Noise Only-Spatial Filtering When the only degradation present in an image is noise, the degradation image is given by: g( f ( + η( G ( F( + N( Estimate N( from the spectrum G(. Spatial filtering is the method of choice in situations when only additive noise is present. Noise Only-Spatial Filtering -- Mean Filters Arithmetic Mean Filter: Noise is reduced as a result of blurring. g( mn ( Geometric Mean Filter: Achieve smoothing comparable to the Arithmetic Mean Filter, but tend to lose less image detail. g( ( S xy mn Noise Only-Spatial Filtering -- Mean Filters (Cont.) Harmonic Mean Filter: Work well for salt noise and Gaussian noise, but fail for pepper noise. mn g( Contraharmonic Mean Filter: Eliminate salt (Negative Q) or pepper (Positive Q) noise. ( t ) ( S xy ( S xy g( g( Q+ Q

3 Noise Only-Spatial Filtering -- Order-Statistics Filters Median Filter: Reduce certain types of random noise and both bipolar and unipolar impulse noise. median{ g( } ( Max Filter: Find the brightest points in the image and eliminate pepper noise. max { g( } ( Min Filter: Find the darkest points in the image and eliminate salt noise. { } Noise Only-Spatial Filtering -- Order-Statistics Filters (Cont.) Midpoint Filter: Work best for randomly distributed noise, like Gaussian or uniform noise. max ( S { g( } + min { g( } xy ( Alpha-trimmed Mean Filter: Useful in situations involving multiple types of noise, such as a combination of salt-and-pepper and Gaussian noise. Here we delete the d/ lowest and the d/ highest gray-level values of g( in the neighborhood Sxy. min g( 3 r 4 ( mn d ( g ( Linear, Position-Invariant Degradations Additivity: If H is a linear operator, the response to a sum of two inputs is equal to the sum of the two responses. Homogeneity: The response to a constant multiple of any input is equal to the response to that input multiplied by the same constant. Position (or space) Invariant: The response at any point in the image depends only on the value of the input at that point, not on its position. 5 Linear, Position-Invariant Degradations Knowing the impulse response (i.e., the response to a specific signal) of a linear system allows us to compute its response, g, to any input f. The result is simply the convolution of the impulse response and the input function. A linear, spatially-invariant degradation system with additive noise can be modeled in the spatial domain as the convolution of the degradation (point spread) function with an image, followed by the addition of noise. g( h( f ( + η( G ( H ( F( + N( Image deconvolution is used frequently to signify linear image restoration. The filters used in the restoration process often are called deconvolution filters. 6 Estimating the Degradation Function -- Observation Gather information from the image itself. Look for areas of strong signal content. Assume that the effect of noise is negligible because of our choice of a strong-signal area, it follows: Gs ( H s ( Fˆ s ( From the characteristics of this function, we deduce the complete function H( by making use of the fact that we are assuming position invariance. 7 Estimating the Degradation Function -- Experimentation If equipment similar to the equipment used to acquire the degraded image is available, it is possible in principle to obtain an accurate estimate of the degradation. Images similar to the degraded image can be acquired with various system settings until they are degraded as closely as possible to the image we wish to restore. The idea is to obtain the impulse response of the degradation by imaging an impulse (small dot of ligh using the same system settings. Since the Fourier transform of an impulse is a constant, it follows: G( H ( 8 A

4 Estimating the Degradation Function -- Modeling Take into account environmental conditions that cause degradation. Derive a mathematical model starting from basic principles. One scenario is to suppose that an image f( undergoes planar motion and that x ( and y ( are the time varying components of motion in the x- and y- direction respectively. Derive the expression in the similar form of the following: G ( H ( F( 9 Minimum Mean Square Error (Wiener) Filtering Incorporate both the degradation function and statistical characteristics of noise into the restoration process. Find an estimate fˆ of the uncorrupted image f such that the mean square error between them is minimized. This error measure is given by: e E{( f fˆ) } The simplified restoration expression is: ˆ H ( F( G( H ( H ( K + Minimum Mean Square Error (Wiener) Filtering (Cont.) The Wiener filter consists of the terms inside the brackets. It is commonly referred to as the minimum mean square error filter or the least square error filter. ˆ H ( F( G( H ( H ( s ( / s f ( + η The restored image in the spatial domain is given by the inverse Fourier transform of the frequency-domain estimate F ˆ (. Edge Detection -- Discontinuity Based Approach Edge: It is a set of connected pixels that lie on the boundary between two regions. In general, an edge is where the intensity of an image moves from a low value to a high value or vice versa. Ideal edge: It is located at an orthogonal (i.e., perpendicular to) step transition in gray level Ramp edge: The slope of the ramp is inversely proportional to the degree of blurring in the edge. Edge Examples Edge Examples Related to Derivatives roof edge line edge step edge ramp edge 3 4

5 Common Edge Detectors st Order Derivative for Edge Detection Sobel Prewitt Roberts Laplacian of Gaussian Zero-Crossing Canny Refer to edge command in Matlab for details. 5 An approach used frequently is to approximate the gradient by absolute values: f G x + Gy where Gx and Gy are convolution results with the two mask respectively 6 First Order Derivative Ex Original Image Apply the Sobel Edge Detector, yields: Sobel Horizontal Edge Detector: Sobel Vertical Edge Detector: Gx Gy Gx Gy Normally, we need Threshold at to choose a threshold to find strong edges. Here we get rid of the boundary of Gx and Gy 8 First Order Derivative Ex nd Order Derivative for Edge Detection 9 Laplacian derivative including the horizontal and vertical neighbors is given by: f f f + 4z5 ( z + z4 + z6 + z8) x y Laplacian derivative including the diagonal neighbors is given by: f f f + + x y 8z ( ) 3 5 z + z + z3 + z4 + z6 + z7 + z8 z9

6 Discontinuity Based Approach -- Summary Discontinuity based approach should yield pixels lying only on edges. In practice, this set of pixels seldom characterizes an edge completely because of noise, breaks in the edge from nonuniform illumination, and other effects that introduce spurious intensity discontinuities. Thus edge detection algorithms typically are followed by linking procedures to assemble edge pixels into meaningful edges. Several basic approaches are: Local Processing Global Processing Via the Hough Transform 3 Global Processing via Graph-Theoretic Technique Geometric Operations One of the most commonly used forms of spatial transformations is the affine transform. The affine transform can be written in matrix form as: t t [ x y ] [ w z ] T [ w z ] t t t t 3 3 This transformation can scale, rotate, translate, or shear a set of point depending on the values chosen for the elements of T. 3 Identity Geometric Operations Scaling Rotation Shear (Horizontal) Shear (Vertical) Translation 33 A linear conformal transformation is a type of affine transformations that preserves the shapes and angles. It consist of a scale factor, a rotation angle, and a translation. The affine transformation matrix in this case has the form: s cosθ s sin θ T ssinθ s cosθ δ x δ y 34 One Matlab Example f checkerboard(5) ; s.8 ; theta pi/6 ; T [s*cos(theta) s*sin(theta) ; -s*sin(theta) s*cos(theta) ; ] ; imcrop tform maketform('affine', T) ; g imtransform(f, tform) ; g imtransform(f, tform, 'nearest') ; g3 imtransform(f, tform, 'FillValue',.5) ; Other Geometric Spatial Transformation Related Commands: tformfwd tforminv imrotate Imresize 35 The purpose of a color model (also called color space or color system) is to facilitate the specification of colors in some standard, generally accepted way. In essence, a color model is a specification of a coordinate system and a subspace within that system where each color is represented by a single point. Most color models in use today are oriented either toward hardware (such as for color monitor and printers) or toward applications where color manipulation is a goal (such as in the creation of color graphics for animation). 36

7 -- RGB Color Model -- RGB Color Model (Cont.) RGB model is a model for color monitors and a broad class of 37 color video cameras CMY Color Model Most devices that deposit colored pigments on paper, such as color printers and copier require CMY data input or perform an RGB to CMY conversion internally. This conversion is performed using the simple operation (Note: All color values have been normalized to the range [, ]) C R M G 39 Y B -- HSI Color Model HSI (Hue, Saturation, and Intensit color model decouples the intensity component from the color-carrying information (hue and saturation) in a color image. Hue is a color attribute that describes a pure color (pure yellow, orange, or red). Saturation gives a measure of the degree to which a pure color is diluted by white light. Intensity is one of the key factors in describing color sensation. As a result, the HSI model is an ideal tool for developing image processing algorithms based on color descriptions that are natural and intuitive to humans HSI Color Model (Cont.) 4 4

8 The HSI color model based on triangular and circular color planes. The triangles and circles are perpendicular to the vertical intensity axis HSI Color Model (Cont.) Given an image in RGB color format, the H component of each RGB pixel is obtained by using the equation: θ if B G H 36 θ if B > G with θ cos [( R G) + ( R B) ] ( R G) + ( R B)( G B) How about the denominator is zero? [ ] / HSI Color Model (Cont.) The saturation component is given by 3 S min( R, G, B R + G + B ) ( ) [ ] The intensity component is given by I R + G + B 3 ( ) Here the RGB values have been normalized to the range [, ] and that the angle theta is measured with respect to the red axis of the HSI space Other Y: Luminance; I: Hue; Q: Saturation 47