Invariants to convolution I(f) = I(f*h) for any admissible h g( x, y ) = ( f * h )( x, y ) + n( x, y )
Motivation
Two approaches Traditional approach: Image restoration (blind deconvolution) Proposed approach: Invariants to convolution
f(x,y)… image function h(x,y)…shift invariant PSF of a linear imaging system g(x,y)…blurred image g(x,y) = (f h) (x,y) The moments under convolution *
Our assumption: PSF is centrosymmetric Assumptions on the PSF
Invariants to convolution PSF is centrosymmetric where (p + q) is odd
Invariants to convolution PSF is centrosymmetric where (p + q) is odd PSF is circularly symmetric where p > q
Face recognition – simulated example
Template matching
Our assumption: PSF has N-fold rotation symmetry, N > 1 The set of invariants depends on N. The bigger N, the more invariants. Parametric shape of the PSF. Other assumptions on the PSF
; Combined moment invariants Invariants to convolution and rotation I(f) = I(R(f*h)) for any admissible h and rotation R
Robustness of the invariants
Satellite image registration by moment invariants
( v1 1, v2 1, v3 1, … ) ( v1 2, v2 2, v3 2, … ) min distance(( v1 k, v2 k, v3 k, … ), ( v1 m, v2 m, v3 m, … )) k,m Control points
Point matching
Registration result
Camera motion estimation
Combined blur-affine invariants Let I(μ00,…, μPQ) be an affine moment invariant. Then I(C(0,0),…,C(P,Q)), where C(p,q) are blur invariants, is a combined blur-affine invariant.
Examples
Digit Recognition by Combined Invariants
AMI [%]Comb [%]
AMI [%]Comb [%]
AMI [%]Comb [%]
AMI [%]Comb [%]
noiseAMI [%]Comb [%] σ= σ=
Combined blur-affine invariants
Affine invariants
Real Data
Invariants to convolution PSF is centrosymmetric where (p + q) is odd The more we know about the PSF, the more invariants and the higher discriminability we get
Discrimination power The null-space of the blur invariants Intuitive meaning of the invariants The number of the invariants Uniqueness theorem
Convolution invariants in FT domain
Relationship between FT and moment invariants