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Image Processing IB Paper 8 – Part A Ognjen Arandjelović Ognjen Arandjelović http://mi.eng.cam.ac.uk/~oa214/

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Lecture Roadmap Face geometry Lecture 1: Geometric image transformations Lecture 2: Colour and brightness enhancement Lecture 3: Denoising and image filtering Lecture 4: Cross-section through out-of-syllabus techniques

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Filter Design – Matched Filters Consider the convolution sum of a discrete signal with a particular filter: When is the filter response maximal? …234233228240241 … 122100122100 228+ 480+482+ 241 + …

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Filter Design – Matched Filters The summation is the same as for vector dot product: The response is thus maximal when the two vectors are parallel i.e. when the filter matches the local patch it overlaps. …234233228240241 … 122100

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Filter Design – Intensity Discontinuities Using the observation that maximal filter response is exhibited when the filter matches the overlapping signal, we can start designing more complex filters: Kernel with maximal response to intensity edges 0.50.0-0.5

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Filter Design – Intensity Discontinuities Better yet, perform Gaussian smoothing to suppress noise first: Noise suppressing kernel with high response to intensity edges Gaussian kernel

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Unsharp Masking Enhancement The main principle of unsharp masking is to extract high frequency information and add it onto the original image to enhance edges: image HPF + output Original edge Enhanced

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Laplacian of Gaussian (LoG) Filter The Laplacian of Gaussian is an isotropic kernel that responds maximally to changes in the 2 nd derivative: 1D Laplacian of Gaussian2D LoG as a surface2D LoG as an image 2D Laplacian of Gaussian:

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Laplacian of Gaussian (LoG) Filter The response of the 1D Laplacian of Gaussian filter to an edge: - Signal (edge)LoG filterFilter output

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Unsharp Masking Enhancement Unsharp mask filtering performs noise reduction and edge enhancement in one go, by combining a Gaussian LPF with a Laplacian of Gaussian kernel: Gaussian smoothingConvolution with –ve Laplacian of Gaussian += Result

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Unsharp Masking – Example Consider the following synthetic example: Gaussian smoothed then corrupted with Gaussian noise

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Unsharp Masking – Example After unsharp masking: Gaussian smoothed then corrupted with Gaussian noise

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– Distance Transform –

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Motivation – Toy Problem The problem: produce output image with higher pixel value indicating higher level of belief that a roughly square polygon of edge 225 is centered at it: 225

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Matched Filtering May Be? Given the material covered in the previous lecture, you may be tempted to create a matched filter: 225

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Matched Filtering May Be? Here is the output of convolving the filter with the example image: A rather ugly looking result with too sharp discontinuities (i.e. low robustness to small deformations in shape, angle or thickness)

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Distance Transform Rather, compute the distance transformed image – each pixel value indicates the minimal distance of that pixel to the nearest edge in the original image: Original image Distance transformed

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The result is far better looking! Result after Distance Transform Consider now the result of convolving our matched filter with the distance transformed image:

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– Nonlinear Denoising –

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Salt and Pepper Noise Consider an image synthetically corrupted with salt and pepper noise: “Salt” (bright) “Pepper” (dark)

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Salt and Pepper Noise Here is the result of denoising attempt using a Gaussian low-pass filter:

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Median Filter Median filter replaces the old pixel value by the median of its neighbourhood: 0 91 92 93 93 97 108 ± 3 pixel neighbourhood (sorted) MedianOriginal value

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Median Filter – 1D Example The result of applying the median filter (with neighbourhood of size 7) on the corrupted 1D signal:

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Median Filter – 2D Example The result of applying the median filter (mask size 3 х 3) on the synthetically corrupted image: No edge smoothing Noise virtually entirely removed

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Filter Comparison The advantages of the median filter are easily seen when considering the difference to the ground truth: Gaussian denoising Median filtering RMS difference: 15 (from 20)RMS difference: 5 (from 20)

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– De-Convolution –

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An Example Problem Consider an image of a car plate acquired by a speed control camera: The plate is entirely unreadable due to motion blur Is it possible to somehow enhance this image to the level that the plate number can be read off?

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Image Formation Model Assuming constant car velocity* the motion blur is caused by simple spatial averaging in the direction of apparent velocity. As before, this is equivalent to convolving the original, sharp image with a simple pulse function. * A reasonable assumption, given that the exposure is relatively short.

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Recovery Algorithm Given an estimate of the car velocity and our image formation model suggests the following algorithm: 2D Fourier Transform 2D Fourier Transform / De-blurred result Degradation model

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The Result Using the pulse width of 15 pixels produces the following de-blurred result: RE03 TGZ

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– Super-Resolution –

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What is Image Super-Resolution Given one or more low-resolution (LR) images, produce an enhanced, high-resolution (HR) image. Observation model: Observed LR image “True” image Noise Transformation (geometric, photometric…)

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SR via Non-Uniform Interpolation One of the simplest forms of super-resolution takes on the form of interpolation from non-aligned samples: Not quite the same

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Example – 2x Sampling Frequency Non-uniform interpolation Simple scaling

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– That is All for Today –

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