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Image Processing Basics

Image Processing Basics. 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

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Image Processing Basics

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  1. Image Processing Basics

  2. What are images? • An image is a 2-d rectilinear array of pixels

  3. Pixels as samples • A pixel is a sample of a continuous function

  4. Images are Ubiquitous • Input • Optical photoreceptors • Digital camera CCD array • Rays in virtual camera (why?) • Output • TVs • Computer monitors • Printers

  5. 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

  6. Image errors • Spatial aliasing • Not enough spatial resolution • Intensity quantization • Not enough intensity resolution

  7. Two issues • Sampling and reconstruction • Creating and displaying images while reducing spatial aliasing errors • Halftoning techniques • Dealing with intensity quantization

  8. Sampling and reconstruction

  9. Aliasing • Artifacts caused by too low sampling frequency (undersampling) or improper reconstruction • Undersampling rate determined by Nyquist limit (Shannon’s sampling theorem)

  10. Aliasing in computer graphics • In graphics, two major types • Spatial aliasing • Problems in individual images • Temporal aliasing • Problems in image sequences (motion)

  11. Spatial Aliasing • “Jaggies”

  12. Spatial aliasing Ref: SIGGRAPH aliasing tutorial

  13. Spatial aliasing • Texture disintegration Ref: SIGGRAPH aliasing tutorial

  14. Temporal aliasing • Strobing • Stagecoach wheels in movies • Flickering • Monitor refresh too slow • Frame update rate too slow • CRTs seen on other video screens

  15. 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

  16. 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?

  17. Spatial (time) domain: Frequency domain: Spectral Analysis Any (spatial, time) domain signal (function) can be written as a sum of periodic functions (Fourier)

  18. Fourier Transform

  19. Fourier Transform • Fourier transform: • Inverse Fourier transform:

  20. 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.

  21. 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

  22. Filtering, sampling and image processing • Many image processing operations basically involve filtering and resampling. • Blurring • Edge detection • Scaling • Rotation • Warping

  23. Resampling • Consider reducing the image resolution:

  24. 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

  25. Ideal low pass filter • Frequency domain: • Spatial (time) domain:

  26. 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)

  27. Discrete Convolution

  28. Finite low pass filters • Triangle filter

  29. Finite low pass filters • Gaussian filter

  30. Edge Detection • Convolve image with a filter that finds differences between neighboring pixels

  31. Scaling • Resample with a gaussian or triangle filter

  32. Image processing • Some other filters

  33. 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

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