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Image Transforms Transforming images to images

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Image Processing and Computer Vision: 22 Classification of Image Transforms Point transforms modify individual pixels modify pixels’ locations Local transforms output derived from neighbourhood Global transforms whole image contributes to each output value

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Image Processing and Computer Vision: 23 Point Transforms Manipulating individual pixel values Brightness adjustment Contrast adjustment Histogram manipulation equalisation Image magnification

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Image Processing and Computer Vision: 24 Grey Scale Manipulation Brightness modifications Contrast modifications Histogram manipulation

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Image Processing and Computer Vision: 25 Brightness Adjustment Add a constant to all values g’ = g + k (k = 50)

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Image Processing and Computer Vision: 26 Contrast Adjustment Scale all values by a constant g’ = g*k (k = 1.5)

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Image Processing and Computer Vision: 27 Image Histogram Measure frequency of occurrence of each grey/colour value

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Image Processing and Computer Vision: 28 Histogram Manipulation Modify distribution of grey values to achieve some effect

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Image Processing and Computer Vision: 29 Equalisation/Adaptive Equalisation Specifically to make histogram uniform

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Image Processing and Computer Vision: 210 Equalisation Transform Equalised image has n x m/l pixels per grey level Cumulative to level j jnm/l pixels Equate to a value in input cumulative histogram C[i] C[i] = jnm/l j = C[i]l/nm Modifications to prevent mapping to –1.

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Image Processing and Computer Vision: 211 Thresholding Transform grey/colour image to binary if f(x, y) > T output = 1 else 0 How to find T?

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Image Processing and Computer Vision: 212 Threshold Value Manual User defines a threshold P-Tile Mode Other automatic methods

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Image Processing and Computer Vision: 213 P-Tile If we know the proportion of the image that is object Threshold the image to select this proportion of pixels

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Image Processing and Computer Vision: 214 Mode Threshold at the minimum between the histogram’s peaks.

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Image Processing and Computer Vision: 215 Automated Methods Find a threshold such that (Start at = 0 and work upwards.) Average l Average h

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Image Processing and Computer Vision: 216 Image Magnification Reducing new value is weighted sum of nearest neighbours new value equals nearest neighbour Enlarging new value is weighted sum of nearest neighbours add noise to obscure pixelation

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Image Processing and Computer Vision: 217 Local Transforms Convolution Applications smoothing sharpening matching

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Image Processing and Computer Vision: 218 Convolution Definition Place template on image Multiply overlapping values in image and template Sum products and normalise (Templates usually small)

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Image Processing and Computer Vision: 219 Example ImageTemplateResult … … … … … … … … … … … … … … … … … Divide by template sum

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Image Processing and Computer Vision: 220 Separable Templates Convolve with n x n template n 2 multiplications and additions Convolve with two n x 1 templates 2n multiplications and additions

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Image Processing and Computer Vision: 221 Example Laplacian template Separated kernels 0 – –1 0 –

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Image Processing and Computer Vision: 222 Composite Filters Convolution is distributive Can create a composite filter and do a single convolution Not convolve image with one filter and convolve result with second. Efficiency gain

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Image Processing and Computer Vision: 223 Applications Usefulness of convolution is the effects generated by changing templates Smoothing Noise reduction Sharpening Edge enhancement Template matching A later lecture

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Image Processing and Computer Vision: 224 Smoothing Aim is to reduce noise What is “noise”? How is it reduced Addition Adaptively Weighted

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Image Processing and Computer Vision: 225 Noise Definition Noise is deviation of a value from its expected value Random changes x x + n Salt and pepper x {max, min}

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Image Processing and Computer Vision: 226 Noise Reduction By smoothing (x + n) = (x) + (n) = (x) Since noise is random and zero mean Smooth locally or temporally Local smoothing Removes detail Introduces ringing

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Image Processing and Computer Vision: 227 Adaptive Smoothing Compute smoothed value, s Output = s if |s – x| > T x otherwise

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Image Processing and Computer Vision: 228 Median Smoothing Median is one value in an ordered set: median = median = 4.5

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Image Processing and Computer Vision: 229 OriginalSmoothed Median Smoothing

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Image Processing and Computer Vision: 230 Gaussian Smoothing To reduce ringing Weighted smoothing Numbers from Gaussian (normal) distribution are weights.

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Image Processing and Computer Vision: 231 Sharpening What is it? Enhancing discontinuities Edge detection Why do it? Perceptually important Computationally important

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Image Processing and Computer Vision: 232 Edge Definition An edge is a significant local change in image intensity.

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Image Processing and Computer Vision: 233 Edge Types Step edge Line edge Roof edge Real edges

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Image Processing and Computer Vision: 234 First Derivative, Gradient Edge Detection If an edge is a discontinuity Can detect it by differencing

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Image Processing and Computer Vision: 235 Roberts Cross Edge Detector Simplest edge detector Inaccurate localisation

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Image Processing and Computer Vision: 236 Prewitt/Sobel Edge Detector

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Image Processing and Computer Vision: 237 Edge Detection Combine horizontal and vertical edge estimates

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Image Processing and Computer Vision: 238 Problems Enhanced edges are noise sensitive Scale What is “local”?

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Image Processing and Computer Vision: 239 Canny/Deriche Edge Detector Require edges to be detected accurate localisation single response to an edge Solution Convolve image with Difference of Gaussian (DoG)

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Image Processing and Computer Vision: 240 Example Results

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Image Processing and Computer Vision: 241 Second Derivative Operators Zero Crossing Model HVS Locate edge to subpixel accuracy Convolve image with Laplacian of Gaussian (LoG) Edge location at crossing of zero axis

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Image Processing and Computer Vision: 242 Example Results

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Image Processing and Computer Vision: 243

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Image Processing and Computer Vision: 244 Global Transforms Computing a new value for a pixel using the whole image as input Cosine and Sine transforms Fourier transform Frequency domain processing Hough transform Karhunen-Loeve transform Wavelet transform

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Image Processing and Computer Vision: 245 Cosine/Sine A halfway solution to the Fourier Transform Used in image coding

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Image Processing and Computer Vision: 246 Fourier All periodic signals can be represented by a sum of appropriately weighted sine/cosine waves

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Image Processing and Computer Vision: 247 Transformed section of BT Building image.

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Image Processing and Computer Vision: 248 Frequency Domain Filtering Convolution Theorem: Convolution in spatial domain is equivalent to Multiplication in frequency domain

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Image Processing and Computer Vision: 249 Smoothing Suppress high frequency components

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Image Processing and Computer Vision: 250 Sharpening Suppress low frequency components

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Image Processing and Computer Vision: 251 Example

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Image Processing and Computer Vision: 252 Wavelet A hierarchical representation detail

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Image Processing and Computer Vision: 253 Hough Transform To detect curves analytically Example straight lines

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Image Processing and Computer Vision: 254 Straight Line y = mx + c gradient m, intercept c c = -mx + y gradient -x, intercept y ALL points in (x, y) transform to a straight line in (c, m) Can therefore detect collinear points

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Image Processing and Computer Vision: 255 Analytic Curve Finding Alternative representation to avoid infinities Other curves higher dimensional accumulators

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Image Processing and Computer Vision: 256 Performance Improvement Techniques Look at pairs of points Use edge orientation

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Image Processing and Computer Vision: 257 Karhunen-Loeve (Principal Component) A compact method of representing variation in a set of images PCs define a co-ordinate system 1st PC records most of variation 2nd PC records most of remainder etc

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Image Processing and Computer Vision: 258 Method Take a set of typical images Compute mean image and subtract from each sample Transform images into columns Group images into a matrix Compute covariance matrix Compute eigenvectors these are the PCs eigenvalues show their importance

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Image Processing and Computer Vision: 259 Uses Compact representation of variable data Object recognition

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Image Processing and Computer Vision: 260 Uses Hierarchical representation Multiresolution processing Coding

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Image Processing and Computer Vision: 261 Geometric Transformations Definitions Affine and non-affine transforms Applications Manipulating image shapes

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Image Processing and Computer Vision: 262 Affine Transforms Scale, Shear, Rotate, Translate Change values of transform matrix elements according to desired effect. a, e scaling b, d shearing a, b, d, e rotation c, f translation = x’ y’ 1 [] a b c d e f g h i [] xy1xy1 [] Length and areas preserved.

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Image Processing and Computer Vision: 263 Affine Transform Examples

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Image Processing and Computer Vision: 264 Warping Example Ansell Adams’ Aspens

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Image Processing and Computer Vision: 265 Image Resampling Moving source to destination pixels x’ and y’ could be non-integer Round result can create holes in image Manipulate in reverse where did warped pixel come from source is non-integer interpolate nearest neighbours

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Image Processing and Computer Vision: 266 You Should Know… Point transforms scaling, histogram manipulation,thresholding Local transforms edge detection, smoothing Global transforms Fourier, Hough, Principal Component, Wavelet Geometrical transforms

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