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Invariants to convolution

Invariants to convolution. Motivation – template matching. Image degradation models. Space-variant. Space-invariant  convolution. Two approaches Traditional approach: Image restoration (blind deconvolution) and traditional features Proposed approach: Invariants to convolution.

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Invariants to convolution

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  1. Invariants to convolution

  2. Motivation – template matching

  3. Image degradation models Space-variant Space-invariant  convolution

  4. Two approaches Traditional approach: Image restoration (blind deconvolution) and traditional features Proposed approach: Invariants to convolution

  5. Invariants to convolution for any admissible h

  6. The moments under convolution Geometric/central Complex

  7. Assumptions on the PSF PSF is centrosymmetric (N = 2) and has a unit integral supp(h)

  8. Common PSF’s are centrosymmetric

  9. Invariants to centrosymmetric convolution where (p + q) is odd What is the intuitive meaning of the invariants? “Measure of anti-symmetry”

  10. Invariants to centrosymmetric convolution

  11. Invariants to centrosymmetric convolution

  12. Convolution invariants in FT domain

  13. Convolution invariants in FT domain

  14. Relationship between FTand moment invariants where M(k,j) is the same as C(k,j) but with geometric moments

  15. Template matching sharp image with the templates the templates (close-up)

  16. Template matching a frame where the templates were located successfully the templates (close-up)

  17. Template matching performance

  18. Boundary effect in template matching Valid region

  19. Boundary effect in template matching • Invalid region • invariance is • violated

  20. Boundary effect in template matching zero-padding

  21. Other types of the blurring PSF • N-fold rotation symmetry, N > 2 • Axial symmetry • Circular symmetry • Gaussian PSF • Motion blur PSF The more we know about the PSF, the more invariants and the higher discriminability we get

  22. Discrimination power The null-space of the blur invariants = a set of all images having the same symmetry as the PSF. One cannot distinguish among symmetric objects.

  23. Recommendation for practice It is important to learn as much as possible about the PSF and to use proper invariants for object recognition

  24. Combined invariants Convolution and rotation For any N

  25. Robustness of the invariants

  26. Satellite image registration by combined invariants

  27. Registration algorithm • Independent corner detection in both images • Extraction of salient corner points • Calculating blur-rotation invariants of a circular neighborhood of each extracted corner • Matching corners by means of the invariants • Refining the positions of the matched control points • Estimating the affine transform parameters by a least-square fit • Resampling and transformation of the sensed image

  28. Control point detection ( v11, v21, v31, … ) ( v12, v22, v32, … ) min distance(( v1k, v2k, v3k, … ) , ( v1m, v2m, v3m, … )) k,m

  29. Control point matching

  30. Registration result

  31. Application in MBD

  32. Combined blur-affine invariants • Let I(μ00,…, μPQ) be an affine moment invariant. ThenI(C(0,0),…,C(P,Q)), where C(p,q) are blur invariants, is a combined blur-affine invariant (CBAI).

  33. Digit recognition by CBAI CBAI AMI

  34. Documented applications of convolution and combined invariants • Character/digit/symbol recognition in the presence of vibration, linear motion or out-of-focus blur • Robust image registration (medical, satellite, ...) • Detection of image forgeries

  35. Moment-based focus measure • Odd-order moments blur invariants • Even-order moments blur/focus measure If M(g1) > M(g2)  g2 is less blurred (more focused)

  36. Other focus measures • Gray-level variance • Energy of gradient • Energy of Laplacian • Energy of high-pass bands of WT

  37. Desirable properties of a focus measure • Agreement with a visual assessment • Unimodality • Robustness to noise • Robustness to small image changes

  38. Agreement with a visual assessment

  39. Robustness to noise

  40. Sunspots – atmospheric turbulence

  41. Moments are generally worse than wavelets and other differential focus measures because they are too sensitive to local changes but they are very robust to noise.

  42. Invariants to elastic deformations

  43. How can we recognize objects on curved surfaces ...

  44. Moment matching • Find the best possible fit by minimizing the error and set

  45. The bottle

  46. Orthogonal moments - set of orthogonal polynomials • Motivation for using OG moments • Stable calculation by recurrent relations • Easier and stable image reconstruction

  47. Numerical stability How to avoid numerical problems with high dynamic range of geometric moments?

  48. Standard powers

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