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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

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– Image Denoising and Filtering –

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Image Noise Sources Image noise may be produced by several sources: Quantization Photonic Thermal Electric

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Denoising To effectively perform denoising, we need to consider the following issues: Signal (uncorrupted image) model Typically piece-wise constant or linear Noise model (from the physics of image formation) Additive or multiplicative, Gaussian, white, salt and pepper…

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Salt and Pepper Noise

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Gaussian Noise

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Modelling Noise Most often noise is additive: Observed pixel luminance True luminanceNoise process

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Additive Gaussian Noise – Example A clear original image was corrupted by additive white Gaussian noise: Original, uncorrupted imageAdditive Gaussian noise

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Additive Gaussian Noise – Example A clear original image was corrupted by additive white Gaussian noise: Additive Gaussian noise

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Additive Gaussian Noise – Example Taking a slice through the image can help us visualize the behaviour of noise better:

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Temporal Average for Video Denoising A video feed of a static scene can be easily denoised by temporal averaging, under the assumption of zero-mean additive noise: Pixel luminance estimate Pixel luminance in frame i Average noise energy is reduced by a factor of N:

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Temporal Averaging – Example Consider our noisy CCTV image from the previous lecture and the result of brightness enhancement: Original imageBrightness enhanced image

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Temporal Averaging – Example The effect of temporal averaging over 100 frames is dramatic: But note that moving objects cause blur. The clarity of image detail is much improved.

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Spatial Averaging Although attractive, a static video feed is usually not available. However, a similar technique can be used by noting: Images are mostly smoothly varying Original smoothly varying signal and the signal corrupted with zero mean Gaussian noise

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Simple Spatial Averaging Thus, we can attempt to denoise the signal by simple spatial averaging: The result of averaging each neighbouring 7 (± 3) pixels

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Simple Spatial Averaging – Example Using out synthetically corrupted image: Additive Gaussian noise Spatially averaged using 5 х 5 neighbourhood

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Simple Spatial Averaging – Example Consider the difference between the uncorrupted image and the corrupted and denoised images: Before averaging After averaging RMS difference = 29RMS difference = 12

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Simple Spatial Averaging – Analysis The result of averaging looks good, but a closer inspection reveals some loss of detail: Difference imageMagnified patch

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Simple Spatial Averaging – Analysis To formally analyze the filtering effects, rewrite the original averaging expression: Rectangular pulse Convolution integral

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1D Convolution A quick convolution re-cap: f(x)h(x) Flip and slide over

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Discrete 1D Convolution In dealing with discrete signals: Flip and slide over …234233228240241 … 001221 f(x)h(x) …234233228240241 … 122100122100 228+ 480+ 482+ 241 + …

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2D Convolution The concept of linear filtering as convolution with a filter (or kernel) extends to 2D and the integral becomes: We shall be dealing with separable filters only in which this is equivalent to two 1D convolutions:

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Simple Spatial Averaging – Analysis By considering the effects of convolution in the frequency domain, we can now see why there was loss of detail: Rectangular pulse function Fourier transform The sinc function High frequencies are damped

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White Noise Model This insight allows to devise the denoising filter in a principled way by considering the SNR over different frequencies: Signal frequency spectrum Noise frequency spectrum Frequency Energy

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Frequency Energy White Noise Model This insight allows to devise the denoising filter in a principled way by considering the SNR over different frequencies: Pass Do not pass

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The Ideal LPF Again As when we dealt with reconstructing a signal from a set of samples, we can low-pass filter by convolving with the sinc function in the spatial domain: The key limitation is that the sinc function has a wide spatial support Thus, in practice we often use filters that offer a better trade-off in terms of spatial support and bandwidth

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Gaussian Low Pass Filter The Gaussian LPF is one of the most commonly used LPFs. It possesses the attractive property of minimal space-bandwidth product. 1D Gaussian2D Gaussian as a surface 2D Gaussian as an image

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Gaussian LPF – Toy Example Using the Gaussian filter on our toy 1D example produces a nearly perfect filtering result: RMS error reduction from 0.1 to 0.02

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Gaussian LPF – Example Using out synthetically corrupted image: Additive Gaussian noise LP filtered using a Gaussian with

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Low, Band and High-Pass Filters A quick recap of relevant terminology: Frequency Gain Low-passBand-passHigh-pass

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Low, Band and High-Pass Filters A summary of main uses: Low-pass: denoising High-pass: removal of non-informative low frequency components Band-pass: combination of low-pass and high-pass filtering effects

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Gaussian High-Pass Filter A high pass filter can be simply constructed from the Gaussian LPF: Convolution with the delta function leaves the function unchanged High-pass filterLow-pass filter

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Gaussian HPF – Toy Example Consider the effects of high pass filtering our 1D toy example: Original signalHigh-pass filter output The result is not dependent on the signal mean Maximal responses around discontinuities

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Gaussian HPF – Example Consider the effects of high pass filtering an image: Original imageHigh-pass filtered image Information rich intensity discontinuities are extracted.

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High Frequency Image Content An example of the importance of high-frequency content: ? + Low-pass filterHigh-pass filter

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High Frequency Image Content And the result of the experiment is…

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HPFs in Face Recognition High-pass filters are used in face recognition to achieve quasi-illumination invariance: Original image of a localized face High-pass filtered

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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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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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– That is All for Today –

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