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ECE 484 Digital Image Processing Lec 10 - Image Interpolation

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ECE 484 Digital Image Processing Lec 10 - Image Interpolation ECE 484 Digital Image Processing Lec 10 - Image Interpolation Zhu Li Dept of CSEE, UMKC Office: FH560E, Email: [email protected], Ph: x 2346. http://l.web.umkc.edu/lizhu slides created with WPS Office Linux and EqualX equation editor Z. Li, ECE 484 Digital Image Processing, 2019 p.1 Outline Recap of Lec 09 Image Sampling . subsampling . interpolation Summary Z. Li, ECE 484 Digital Image Processing, 2019 p.2 BF - kernel reflects range and value changes Bilateral Filtering process W Ws Ws*Wc * c Output input geometry kernelphotometry kernel bilateral kernel Z. Li, ECE 484 Digital Image Processing, 2019 p.3 Non Local Means (NLM) Filters Baudes et al. (2005) use a weighted average based on similarity . BL filter's spatial kernel is turned off . intensity kernel reflects similarity between self similar patches: pixel j and its neighborhood y(j) is defined as say 5x5 block centered at j. Z. Li, ECE 484 Digital Image Processing, 2019 p.4 CBF - Cross Bilateral Filter Decouple the filter image input, from the images that gives range/value kernel . a guide image I (in a way NLM is a guided filter ) from say short exposure images, while filter input is the long exposure image Z. Li, ECE 484 Digital Image Processing, 2019 p.5 Guided Filter - Definitions Motivated by the Cross Bilateral Filter: . Filter input image pixels {pi} . Guide image {Ii} . Filter output {qi} Output: . weight W is completely derived from the guidance image I. Guided image filter output: where {ak, bk} is totally defined by the neigborhood wk • The filter image output model: qi = pi - ni . The output is the input image subtract some noise Z. Li, ECE 484 Digital Image Processing, 2019 p.6 Guided Filter Operations Denoising . when covariance of I and P is small, then GF is in denoising/smoothing mode Edge Preserving . when covariance of I and P are large, i.e, very similar, then GF is preserving edge. Z. Li, ECE 484 Digital Image Processing, 2019 p.7 Outline Recap of Lec 09 Image Sampling . anti-aliasing subsampling . interpolation Summary Z. Li, ECE 484 Digital Image Processing, 2019 p.8 Image Scaling Need to resample images This image is too big to fit on the screen. How can we reduce it? How to generate a half- sized version? Z. Li, ECE 484 Digital Image Processing, 2019 p.9 Image sub-sampling sub-sample without filtering, what is wrong ? 1/8 1/4 Throw away every other row and column to create a 1/2 size image - called image sub-sampling Z. Li, ECE 484 Digital Image Processing, 2019 p.10 Image sub-sampling Aliasing... 1/2 (2x zoom) 1/4 1/8 (4x zoom) Why does this look so crufty? Z. Li, ECE 484 Digital Image Processing, 2019 p.11 Even worse for synthetic images Aliasing effect Z. Li, ECE 484 Digital Image Processing, 2019 p.12 Sampling and the Nyquist rate Aliasing can arise when you sample a continuous signal or image . occurs when your sampling rate is not high enough to capture the amount of detail in your image . Can give you the wrong signal/image—an alias . formally, the image contains structure at different scales . called “frequencies” in the Fourier domain . the sampling rate must be high enough to capture the highest frequency in the image To avoid aliasing: . sampling rate > 2 * max frequency in the image . This minimum sampling rate is called the Nyquist rate Z. Li, ECE 484 Digital Image Processing, 2019 p.13 Sampling in Time Domain Computer needs a discrete representation of signals • Many signals originate as continuous-time signals, e.g. conventional music or voice • By sampling a continuous-time signal at isolated, equally-spaced points in time, we obtain a sequence of numbers Ts n {…, -2, -1, 0, 1, 2,…} t T Ts is the sampling period. s s(t) Sampled analog impulse waveform train Z. Li, ECE 484 Digital Image Processing, 2019 p.14 Spectrum of several special functions Spatial-Frequency signatures Z. Li, ECE 484 Digital Image Processing, 2019 p.15 Consequence in Freq Domain Multiplication with sampling train function, is convolving in freq domain • Replicates spectrum of continuous-time signal At offsets that are integer multiples of sampling frequency • Fourier series of impulse train where s = 2 fs • Example Modulation by Modulation by cos(2 cos( t) t) F() s G() s -2f 2f max max s s s s Z. Li, ECE 484 Digital Image Processing, 2019 p.16 Shannon/Nyquist Sampling Theorem What is the sampling rate to recover the original continuous signal w/o loss ? • A continuous-time signal x(t) with frequencies no higher than fmax can be reconstructed from its samples x[n] = x(n Ts) if the samples are taken at a rate fs which is greater than 2 fmax. Nyquist rate = 2 fmax Nyquist frequency = fs/2. • What happens if fs = 2fmax? • Consider a sinusoid sin(2 p fmax t) Use a sampling period of Ts = 1/fs = 1/2fmax. Sketch: sinusoid with zeros at t = 0, 1/2fmax, 1/fmax, … Z. Li, ECE 484 Digital Image Processing, 2019 p.17 Illustration of Sampling Theorem 1-D example sampling pattern w 1/w sampled signal Spatial domain Frequency domain Z. Li, ECE 484 Digital Image Processing, 2019 p.18 Reconstruction Recon via convolving with Sinc function (low pass filtering) . in practice, truncated sinc() function sinc function w 1/w reconstructed signal Spatial domain Frequency domain Z. Li, ECE 484 Digital Image Processing, 2019 p.19 Aliasing from under-sampled reconstruction Aliasing effect (a) f(x) and its spectrum (b) sampling train (c) aliasing from under sampling (a) (d) reconstruction via convolving with sinc function (e) aliasing (b) (c) (d) (e) Z. Li, ECE 484 Digital Image Processing, 2019 p.20 Anti-Aliasing Filtering Apply low pass filtering to limit the band of signal before filtering . ideally need to use sinc filter . infinite taps, not practical . use Gaussian instead or . Low Pass FIR design below Z. Li, ECE 484 Digital Image Processing, 2019 p.21 Image Anti-Aliasing Filtering Images are not bandlimited as we would like 2D sampling effects Z. Li, ECE 484 Digital Image Processing, 2019 p.22 2D image anti aliasing example Pre-Low Pass Filter before subsampling Z. Li, ECE 484 Digital Image Processing, 2019 p.23 Subsampling with Gaussian pre-filtering Pre LP filtering with Gaussian G 1/8 G 1/4 Gaussian 1/2 Solution: filter the image, then subsample • Filter size should double for each ½ size reduction. Why? Z. Li, ECE 484 Digital Image Processing, 2019 p.24 Subsampling with Gaussian pre-filtering Anti-alias filtered sub-sampling Gaussian 1/2 G 1/4 G 1/8 Solution: filter the image, then subsample • Filter size should double for each ½ size reduction. Why? • How can we speed this up? Z. Li, ECE 484 Digital Image Processing, 2019 p.25 Aliasing w/o pre filtering In comparison, w/o anti-aliasing filtering: 1/2 1/4 (2x zoom) 1/8 (4x zoom) Z. Li, ECE 484 Digital Image Processing, 2019 p.26 Outline Recap of Lec 09 Image Sampling . anti-aliasing subsampling . interpolation Summary Z. Li, ECE 484 Digital Image Processing, 2019 p.27 Image Resampling/Interpolation Reconstruct the continuous signal with different filters Z. Li, ECE 484 Digital Image Processing, 2019 p.28 0-th and 1-st Order Interpolation Simple interpolation schemes 1st order: not differentiable 0-th order: introduced discontinuity Z. Li, ECE 484 Digital Image Processing, 2019 p.29 Linear Image resampling/interpolation So what to do if we don’t know • Answer: guess an approximation • Can be done in a principled way: filtering 1 d = 1 in this example 1 2 2.5 3 4 5 Image reconstruction • Convert to a continuous function • Reconstruct by convolution: Z. Li, ECE 484 Digital Image Processing, 2019 p.30 Cubic Interpolation Linear, Quadratic and Cubic Interpolation Cubic is popular, supported in Matlab x = -3:3; y = [-1 -1 -1 0 1 1 1]; xq1 = -3:.01:3; s = spline(x,y,xq1); plot(x,y,'o',xq1,p,'-',xq1,s,'-.') legend('Sample Points','spline','Location','SouthEast') Z. Li, ECE 484 Digital Image Processing, 2019 p.31 2D Interpolation Divide & conquer . 1/2 pel position interpolation Z. Li, ECE 484 Digital Image Processing, 2019 p.32 Bilinear Interpolation 2D linear interpolation Z. Li, ECE 484 Digital Image Processing, 2019 p.33 Interpolation Results NN, Bilinear vs Bicubic Original image: x 10 Nearest-neighbor interpolation Bilinear interpolation Bicubic interpolation Z. Li, ECE 484 Digital Image Processing, 2019 p.34 DCT derived Interpolation Filter DCTIF: Elena & Alex Alshina from Samsung DMC: . JVTVC-F247: https://www.itu.int/wftp3/av-arch/jctvc- site/2011_07_F_Torino/JCTVC-F247-v6.zip Basic Idea: . DCT for signal representation: . then the signal at fractional pixel location can be derived as, Z. Li, ECE 484 Digital Image Processing, 2019 p.35 DCTIF Interpolation Filter DCT Derived interpolation filters (DCTIF, Samsung) . 1/4 pel interpolations . row first then column Z. Li, ECE 484 Digital Image Processing, 2019 p.36 DCTIF Filter Filter derived from DCT transforms . row interpolation . column interpolation Performance . better than bicubicle ! Z. Li, ECE 484 Digital Image Processing, 2019 p.37 Deep Learning Interpolation Filter High performing solution in HEVC for sub-pel motion estimation . half pixel location images: . Deep learning ground truth: . Network: Z. Li, ECE 484 Digital Image Processing, 2019 p.38 Summary Downsampling images: . downsampling is multiplication with the impulse train in time domain . it is convolution with signal spectrum in freq domain . if not sampling at 2fmax, aliasing. Interpolation . Ideal reconstruction via Sinc function, infinite support not practical .
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