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## Image Processing Basics

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**What are images?**• An image is a 2-d rectilinear array of pixels**Pixels as samples**• A pixel is a sample of a continuous function**Images are Ubiquitous**• Input • Optical photoreceptors • Digital camera CCD array • Rays in virtual camera (why?) • Output • TVs • Computer monitors • Printers**Properties of Images**• Spatial resolution • Width pixels/width cm and height pixels/ height cm • Intensity resolution • Intensity bits/intensity range (per channel) • Number of channels • RGB is 3 channels, grayscale is one channel**Image errors**• Spatial aliasing • Not enough spatial resolution • Intensity quantization • Not enough intensity resolution**Two issues**• Sampling and reconstruction • Creating and displaying images while reducing spatial aliasing errors • Halftoning techniques • Dealing with intensity quantization**Aliasing**• Artifacts caused by too low sampling frequency (undersampling) or improper reconstruction • Undersampling rate determined by Nyquist limit (Shannon’s sampling theorem)**Aliasing in computer graphics**• In graphics, two major types • Spatial aliasing • Problems in individual images • Temporal aliasing • Problems in image sequences (motion)**Spatial Aliasing**• “Jaggies”**Spatial aliasing**Ref: SIGGRAPH aliasing tutorial**Spatial aliasing**• Texture disintegration Ref: SIGGRAPH aliasing tutorial**Temporal aliasing**• Strobing • Stagecoach wheels in movies • Flickering • Monitor refresh too slow • Frame update rate too slow • CRTs seen on other video screens**Antialiasing**• Sample at a higher rate • What if the signal isn’t bandlimited? • What if we can’t do this, say because the sampling device has a fixed resolution? • Pre-filter to form bandlimited signal • Low pass filter • Trades aliasing for blurring • Non-uniform sampling • Not always possible, done by your visual system, suitable for ray tracing • Trades aliasing for noise**Sampling Theory**• Two issues • What sampling rate suffices to allow a given continuous signal to be reconstructed from a discrete sample without loss of information? • What signals can be reconstructed without loss for a given sampling rate?**Spatial (time) domain:**Frequency domain: Spectral Analysis Any (spatial, time) domain signal (function) can be written as a sum of periodic functions (Fourier)**Fourier Transform**• Fourier transform: • Inverse Fourier transform:**Sampling theorem**• A signal can be reconstructed from its samples if the signal contains no frequencies above ½ the sampling frequency. -Claude Shannon • The minimum sampling rate for a bandlimited signal is called the Nyquist rate • A signal is bandlimited if all frequencies above a given finite bound have 0 coefficients, i.e. it contains no frequencies above this bound.**Filtering and convolution**• Convolution of two functions (= filtering): • Convolution theorem: • Convolution in the frequency domain is the same as multiplication in the spatial (time) domain, and • Convolution in the spatial (time) domain is the same as multiplication in the frequency domain**Filtering, sampling and image processing**• Many image processing operations basically involve filtering and resampling. • Blurring • Edge detection • Scaling • Rotation • Warping**Resampling**• Consider reducing the image resolution:**Resampling**• The problem is to resample the image in such a way as to produce a new image, with a lower resolution, without introducing aliasing. • Strategy- • Low pass filter transformed image by convolution to form bandlimited signal • This will blur the image, but avoid aliasing**Ideal low pass filter**• Frequency domain: • Spatial (time) domain:**Image processing in practice**• Use finite, discrete filters instead of infinite continous filters • Convolution is a summation of a finite number of terms rather than in integral over an infinite domain • A filter can now be represented as an array of discrete terms (the kernel)**Finite low pass filters**• Triangle filter**Finite low pass filters**• Gaussian filter**Edge Detection**• Convolve image with a filter that finds differences between neighboring pixels**Scaling**• Resample with a gaussian or triangle filter**Image processing**• Some other filters**Summary**• Images are discrete objects • Pixels are samples • Images have limited resolution • Sampling and reconstruction • Reduce visual artifacts caused by aliasing • Filter to avoid undersampling • Blurring (and noise) are preferable to aliasing