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Image Processing

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Overview ImagesPixel Filters Neighborhood Filters Dithering

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Image as a Function We can think of an image as a function, f, f: R 2 R –f (x, y) gives the intensity at position (x, y) –Realistically, we expect the image only to be defined over a rectangle, with a finite range: f: [a,b] x [c,d] [0,1] A color image is just three functions pasted together. We can write this as a “ vector- valued ” function:

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Image as a Function

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Image Processing Define a new image g in terms of an existing image f –We can transform either the domain or the range of f Range transformation: What kinds of operations can this perform?

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Some operations preserve the range but change the domain of f : What kinds of operations can this perform? Still other operations operate on both the domain and the range of f. Image Processing

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Point Operations

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Point Processing Original Darken Invert Lighten Lower Contrast Raise Contrast Nonlinear Raise Contrast Nonlinear Lower Contrast

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Point Processing Original Darken Invert Lighten Lower Contrast Raise Contrast Nonlinear Raise Contrast Nonlinear Lower Contrast x + 128x * 2 255 - x ((x / 255.0) ^2) * 255.0 x - 128x / 2x ((x / 255.0) ^ 0.33) * 255.0

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Gamma correction Γ = 1.0; f(v) = v Γ = 2.5; f(v) = v 1/2.5 = v 0.4 Monitors have a intensity to voltage response curve which is roughly a 2.5 power function Send v actually display a pixel which has intensity equal to v 2.5

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Neighborhood Operations

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Convolution 0.20.1 0.3 0.0 0.9 0.10.3

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Properties of Convolution Commutative Associative Cascade system

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Convolution LSIS is doing convolution; convolution is linear and shift invariant kernel h

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Convolution - Example Eric Weinstein’s Math World

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12-2 1 1 1 1 1 Convolution - Example

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Point Spread Function Optical System sceneimage Ideally, the optical system should be a Dirac delta function. Optical System point sourcepoint spread function However, optical systems are never ideal. Point spread function of Human Eyes

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Point Spread Function normal visionmyopiahyperopia Images by Richmond Eye Associates astigmatism

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Original Image

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Shifted Image

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Original Image

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Blurred Image

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Gaussian Smoothing http://www.michaelbach.de/ot/cog_blureffects/index.html by Charles Allen Gillbert by Harmon & Julesz

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Gaussian Smoothing http://www.michaelbach.de/ot/cog_blureffects/index.html

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Original Image

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Sharpened Image

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Original Image

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Noise

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Blurred Noise

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Median Filter Smoothing is averaging (a) Blurs edges (b) Sensitive to outliers (a) (b) –Sort values around the pixel –Select middle value (median) –Non-linear (Cannot be implemented with convolution) Median filtering sortmedian

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Median Filter Can this be described as a convolution?

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Original Image

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Example: Noise Reduction Image with noiseMedian filter (5x5)

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3x3 5x5 7x7 Salt and pepper noise Gaussian noise

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Example: Noise Reduction Original image Image with noiseMedian filter (5x5)

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Original Image

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X-Edge Detection

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Y-Edge Detection

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General Edge Detection Can this be described as a convolution?

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Some operations preserve the range but change the domain of f : What kinds of operations can this perform? Still other operations operate on both the domain and the range of f. Image Processing

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Aliasing

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Alias: n., an assumed name Picket fence receding into the distance will produce aliasing… Input signal: x = 0:.05:5; imagesc(sin((2.^x).*x)) Matlab output: WHY? Alias! Not enough samples

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Image Scaling This image is too big to fit on the screen. How can we reduce it? How to generate a half- sized version?

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Image Sub-Sampling Throw away every other row and column to create a 1/2 size image - called image sub-sampling 1/4 1/8

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Image Sub-Sampling 1/4 (2x zoom) 1/8 (4x zoom) 1/2

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Good and Bad Sampling Good sampling: Sample often or, Sample wisely Bad sampling: see aliasing in action!

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Really bad in video

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Sub-Sampling with Gaussian Pre-Filtering G 1/4 G 1/8 Gaussian 1/2 Solution: filter the image, then subsample –Filter size should double for each ½ size reduction. Why?

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G 1/4G 1/8Gaussian 1/2 Sub-Sampling with Gaussian Pre-Filtering

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Compare with... 1/4 (2x zoom) 1/8 (4x zoom) 1/2

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Aliasing

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Canon D60 (w/ anti-alias filter)Sigma SD9 (w/o anti-alias filter) From Rick Matthews website, images by Dave Etchells

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Figure from David Forsyth

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Original Image

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Warped Image

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=+ origvector fieldwarped how?

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Advection (just like a fluid)

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