Sampling, Resampling, and WarpingDigital Image ProcessingDigital Image ProcessingSampling and ReconstructionSampling and ReconstructionSampling and ReconstructionSampling TheorySampling TheorySampling TheorySampling TheorySampling TheorySampling TheorySampling TheorySampling TheorySampling TheorySpectral AnalysisFourier TransformFourier TransformSampling TheoremImage ProcessingImage ProcessingSampling TheoremAliasingSpatial AliasingSpatial AliasingTemporal AliasingTemporal AliasingTemporal AliasingTemporal AliasingAntialiasingImage ProcessingImage ProcessingImage ProcessingImage ProcessingImage ProcessingImage ProcessingImage ProcessingImage ProcessingIdeal Bandlimiting FilterPractical Image ProcessingExample: ScalingGeneral Image WarpingGeneral Image WarpingGeneral Image WarpingGeneral Image WarpingTwo OptionsMappingParametric MappingsParametric MappingsParametric MappingsOther Parametric MappingsCOS426 ExamplesMore COS426 ExamplesPoint Correspondence MappingsLine Correspondence MappingsImage WarpingImage WarpingPoint SamplingPoint SamplingPoint SamplingImage Resampling PipelineImage Resampling PipelineResampling with FilterImage ResamplingImage ResamplingImage ResamplingGaussian FilteringImage ResamplingImage Resampling (with width < 1)Image Resampling (with width < 1)Putting it All TogetherPutting it All TogetherPutting it All TogetherSampling Method ComparisonForward vs. Reverse MappingForward vs. Reverse MappingForward vs. Reverse MappingForward vs. Reverse MappingForward vs. Reverse MappingForward vs. Reverse MappingForward vs. Reverse MappingSummaryNext Time…Sampling, Resampling, and Warping COS 426Digital Image Processing • Changing intensity/color Linear: scale, offset, etc. Nonlinear: gamma, saturation, etc. Add random noise • Filtering over neighborhoods Blur Detect edges Sharpen Emboss Median • Moving image locations Scale Rotate Warp • Combining images Composite Morph • Quantization • Spatial / intensity tradeoff DitheringDigital Image Processing When implementing operations that move pixels, must account for the fact that digital images are sampled versions of continuous onesSampling and Reconstruction Sampling Continuous function Discrete samplesSampling and Reconstruction Sampling Reconstruction Continuous function Discrete samples Continuous functionSampling and Reconstruction Figure 19.9 FvDFHSampling Theory How many samples are enough? How many samples are required to represent a given signal without loss of information? What signals can be reconstructed without loss for a given sampling rate? Reconstructed function Original functionSampling Theory What happens when we use too few samples? Aliasing: high frequencies masquerade as low ones Figure 14.17 FvDFHSampling Theory What happens when we use too few samples? Aliasing: high frequencies masquerade as low onesSampling Theory What happens when we use too few samples? Aliasing: high frequencies masquerade as low ones (Barely) adequate sampling Inadequate samplingSampling Theory How many samples are enough to avoid aliasing? How many samples are required to represent a given signal without loss of information? What signals can be reconstructed without loss for a given sampling rate?Sampling Theory How many samples are enough to avoid aliasing? How many samples are required to represent a given signal without loss of information? What signals can be reconstructed without loss for a given sampling rate?Sampling Theory How many samples are enough to avoid aliasing? How many samples are required to represent a given signal without loss of information? What signals can be reconstructed without loss for a given sampling rate?Sampling Theory How many samples are enough to avoid aliasing? How many samples are required to represent a given signal without loss of information? What signals can be reconstructed without loss for a given sampling rate?Sampling Theory How many samples are enough to avoid aliasing? How many samples are required to represent a given signal without loss of information? What signals can be reconstructed without loss for a given sampling rate?Spectral Analysis • Spatial domain: Function: f(x) Filtering: convolution • Frequency domain: o Function: F(u) o Filtering: multiplication Any signal can be written as a sum of periodic functions.Fourier Transform Figure 2.6 WolbergFourier Transform • Fourier transform: • Inverse Fourier transform:Sampling Theorem • A signal can be reconstructed from its samples, iff the original signal has no content >= 1/2 the sampling frequency - Shannon • The minimum sampling rate for bandlimited function is called the “Nyquist rate” A signal is bandlimited if its highest frequency is bounded. The frequency is called the bandwidth.Image Processing • Consider reducing the image resolution Original image 1/4 resolutionImage Processing Resampling • Image processing is a resampling problemSampling Theorem • A signal can be reconstructed from its samples, iff the original signal has no content >= 1/2 the sampling frequency - Shannon Figure 14.17 FvDFH Under-sampling Aliasing will occur if the signal is under-sampledAliasing • In general: Artifacts due to under-sampling or poor reconstruction • Specifically, in graphics: Spatial aliasing Temporal aliasing Figure 14.17 FvDFH Under-samplingSpatial Aliasing Artifacts due to limited spatial resolutionSpatial Aliasing Artifacts due to limited spatial resolution “Jaggies”Temporal Aliasing Artifacts due to limited temporal resolution Strobing FlickeringTemporal Aliasing Artifacts due to limited temporal resolution Strobing FlickeringTemporal Aliasing Artifacts due to limited temporal resolution Strobing FlickeringTemporal Aliasing Artifacts due to limited temporal resolution Strobing FlickeringAntialiasing • Sample at higher rate Not always possible Doesn’t always solve the problem • Pre-filter to form bandlimited signal Use low-pass filter to limit signal to < 1/2 sampling rate Trades blurring for aliasingImage Processing Sample Real world Reconstruct Discrete samples (pixels) Transform Reconstructed function Filter Transformed function Sample Bandlimited function Reconstruct Discrete samples (pixels) DisplayImage Processing Sample Real world Reconstruct Discrete samples (pixels)
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