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Digital Image Filters. 26 th Nov ember 2012. Outline. Definitions Linear, shift invariant filters: convolution. Examples of convolutional filters Smoothing Filters ( denoising ) Differentiating Filters (edge detector) Examples of Nonlinear Filtering. Systems.

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Digital Image Filters


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  1. Digital Image Filters 26thNovember 2012

  2. Outline • Definitions • Linear, shift invariant filters: convolution. • Examples of convolutional filters • Smoothing Filters (denoising) • Differentiating Filters (edge detector) • Examples of Nonlinear Filtering

  3. Systems • A System H isconsideredasa black box thatprocesses some input signal (f) and gives the output (Hf) • In our case, f isa digital image or a 1-D digitalsignal, but in principlecould be analogic and n-D signalaswellas a scalar

  4. Linearity • A systemissaid to be lineari.i.f. and thishold for all and for aribrarysignals • itis the classicaldefinition of linearity

  5. Signal and System - Property • A systemissaid to be time (or shift) – invarianti.i.f. • Systems that are Linear and TranslationInvariant are represented by a convolutoin. • LTI systems are characterizedentirely by a singlefunction called the system's impulse response. • The impulse response corresponds to the filter associated to the system.

  6. Convolution - LTI System • Let usconsider a signal and a filter • Theirconvolutionis a signal . • For continuous-domain 1D signals and filters i.e., • For discrete signals and filters

  7. Convolution LIT Systems on Images • are discrete 2D signals = Point-wiseproduct sum * • z5 = h9y1+ h8y2 + h7y3 + h6y4 + h5y5 + h5y6 + h3y7 + h2y8 + h1y9

  8. Convolution LIT Systems on Images • are discrete 2D signals = Point-wiseproduct sum * • z5 = h9y1+ h8y2 + h7y3 + h6y4 + h5y5 + h5y6 + h3y7 + h2y8 + h1y9

  9. Convolution LIT Systems on Images • are discrete 2D signals = Point-wiseproduct sum * • z5 = h9y1+ h8y2 + h7y3 + h6y4 + h5y5 + h5y6 + h3y7 + h2y8 + h1y9

  10. Convolution LIT Systems on Images • are discrete 2D signals = Point-wiseproduct sum * • z5 = h9y1+ h8y2 + h7y3 + h6y4 + h5y5 + h5y6 + h3y7 + h2y8 + h1y9

  11. Linear Filters for Digital Images • Linear Filters are particular systems where the output at each pixel (sample) is given by a linear combination of neighboring pixels (samples) Original Signal Filtered Output

  12. Filters or Kernels • The coefficients used in the linear combination are given by the kernel = Image Filter Output Kernel

  13. A well-known Test Image - Lena

  14. Trivial example

  15. Transalation

  16. Linear Filtering -local averaging

  17. The original Lena

  18. Filtered Lena

  19. Linear Filtering - local averages

  20. What about normalization?

  21. … convolutionis linear

  22. …what about

  23. Convolution Properties • It is a linear System; it enjoys every LTI properties • Is it Commutative ? but it depends on the patting used. In continuous domain it holds as well as on periodic signals • It is also associative • and dissociative

  24. 2D Gaussian Filter Continuous Function Discrete Function

  25. Linear Filtering(Gaussian Filter)-weighted local averages-

  26. Linear Filtering(Gaussian Filter)-weighted local averages-

  27. Gaussian Vs Average Gaussian Smoothing Support 7x7 Smoothing by Averaging On 7x7 window

  28. Denoising: The Issue • A Noisy Detail in Camera Raw Data.

  29. Denoising: The Issue • Denoised

  30. Denoising: The Issue • A Noisy Detail in Camera Raw Data.

  31. Denoising: The Issue • Denoised

  32. Image Formation Model • Observation model is • The goal is to obtain , a reliable estimate of , given and the distribution of . • For the sake of simplicity it is often assumed and independent . • The noise standard deviation is assumed as known. Sensed image Pixel index Original (unknown) image noise

  33. Image Formation Model • Observation model is • Thus we can pursue a “regression-approach”,

  34. Image Formation Model • Observation model is • Thuswe can pursue a “regression-approach”, but on images itmaynot be convenient to assume a parametricexpressionof on

  35. Image Formation Model • Observation model is • Thuswe can pursue a “regression-approach”, but on images itmaynot be convenient to assume a parametricexpressionof on

  36. Denoising Approaches • ParametricApproaches • Transform Domain Filtering, they assume the noisy-free signalissomehow sparse in a suitable domain (e.g Fourier, DCT, Wavelet) or w.r.t. some dictionarybaseddecomposition)

  37. Denoising Approaches • ParametricApproaches • Transform Domain Filtering, they assume the noisy-free signalissomehow sparse in a suitable domain (e.g Fourier, DCT, Wavelet) or w.r.t. some dictionarybaseddecomposition) • Non ParametricApproaches • Local Smoothing / Local Approximation • Non Local Methods

  38. Denoising Approaches • ParametricApproaches • Transform Domain Filtering, they assume the noisy-free signalissomehow sparse in a suitable domain (e.g Fourier, DCT, Wavelet) or w.r.t. some dictionarybaseddecomposition) • Non ParametricApproaches • Local Smoothing / Local Approximation • Non Local Methods

  39. Denoising Approaches • ParametricApproaches • Transform Domain Filtering, they assume the noisy-free signalissomehow sparse in a suitable domain (e.g Fourier, DCT, Wavelet) or w.r.t. some dictionarybaseddecomposition) • Non ParametricApproaches • Local Smoothing / Local Approximation • Non Local Methods Estimating from can be statistically treated as regression of on

  40. Fitting and Convolution • One can prove that the leastsquarefit of polynomial of 0-th order (i.econstant) isgiven by where and thus

  41. Denoising Approaches • Non Parametric Approaches: there are no global model for and basically • Local Methods: define, in each image pixel, the “best neighborhood” where a simple parametric model can be enforced to perform regression is described by a (polynomial) model

  42. Ideal neighborhood – an illustrative example • Ideal in the sense that it defines the support of a pointwise Least Square Estimator of the reference point. • Typically, even in simple images, every point has its own different ideal neighborhood. • For practical reasons, the ideal neighborhood is assumedstarshaped Furtherdetailsat LASIP c/o Tampere University of Technology http://www.cs.tut.fi/~lasip/

  43. Neighborhood discretization • A suitablediscretization of thisneighborhoodisobtained by using a set of directional LPA kernels wheredetermines the orientation of the kernelsupport, andwhere controls by the scale of kernel support. Directional kernels Discrete Adaptive Neighborhood Ideal Neighborhood

  44. Neighborhood discretization • A suitablediscretization of thisneighborhoodisobtained by using a set of directional LPA kernels wheredetermines the orientation of the kernelsupport, andwhere controls by the scale of kernel support. • The initial shape optimization problem can be solved by using standard easy-to-implement varying-scale kernel techniques, such as the ICI rule.  Directional kernels Discrete Adaptive Neighborhood Ideal Neighborhood

  45. Ideal neighborhood – an illustrative example • Ideal in the sense that it defines the support of pointwise Least Square Estimator of the reference point.

  46. Examples of Adaptively Selected Neighorhoods • Define, , the “ideal” neigborhood • Compute the denoised estimate at , “using”

  47. Examples of adaptively selected neighorhoods • Adaptively selected neighborhoods selected using the LPA-ICI rule