| Summary: | Geometric invariants play a vital role in the field of object recognition where the objects of interest are affected by a group of transformations. However, designing robust algorithms that are tolerant to noise and image occlusion remains still an open problem. In particular, numerically invariant signature curves in terms of joint invariants, as an approximation to differential invariant signature curves, suffer from instability, bias, noise and indeterminacy in the resulting signatures. The expression presented in the previous works to approximate the affine arc length along the given mesh is only in terms of the approximating ellipse and, in consequence, the current formulae cannot be correctly demonstrated on data containing non-elliptical boundaries. We also prove the current formulation depends on the viewpoint that may change the curve-orientation and the signature-direction, and results in different numerical signatures for congruent ordinary meshes - in other words, Signature Theorem is not correct in Mesh group-planes. In addition, we show that the current expressions do not support a numerical version of Signature-Inverse Theorem- in other words, non-congruent approximating meshes may have the same numerically invariant signatures. Prior to addressing above mentioned issues (except Signature-Inverse Theorem that is not in the scope of this thesis), we first undertake to modify Calabi et al's numerically invariant scheme and refine the current methodology by adding new terminology which improves the clarity and extends the methodology to planar ordinary meshes. To address the issue of stability in the Euclidean formulae Heron's formula is replaced by the accurate area that improves the numerical stability and, in terms of mean square error (MSE) results in a closer approximation in comparison with the current formulation. In the affine geometry, we will introduce a general formulation for the full conic sections to find a numerically invariant approximation of the equiaffine arc length, measured along the given plane curve and between each two points. In addition, closer numerical expressions will be presented that also need fewer points of the given mesh of points to approximate the first order derivative of the affine-curvature compared with the current formulae. Next, we will introduce the first order finite n-difference quotients in both Euclidean and affine signature calculus which not only approximate the first derivatives of the corresponding differential curvatures, but can be also used to minimize the effects of noise and indeterminacy in the resulting outputs. The arising numerical biases in the resulting numerical signatures will be classified as Bias Type-1 and Bias Type-2 and it will be showed how they can be removed. To parameterize numerically invariant signatures independently of the viewpoint, we will introduce an orientation-free version which results in all congruent planar ordinary meshes have the same orientation-free numerically invariant signature, and therefore the Signature Theorem will be correct then in Mesh and Digital group-planes. Finally, we will bring up our experimental results. First the results of applying the proposed scheme to generate numerically invariant signatures will be presented. In this experiment the sensitivity of the parameters used in the algorithm will be examined. These parameters include mesh regularity factors as well as the effect of selecting different mesh resolution to represent the boundary of the object of interest. To reduce noise in the resulting numerical signatures, the n-difference technique and the m-mean signature method will be introduced and will be illustrated that these methods are capable of noise reduction by more than 90%. The n-difference technique will be also applied to eliminate indeterminacy in the resulting outputs. Next, the potential applications of the proposed scheme in object description will be presented by two manners: applying the plots of numerically invariant group-signatures for discriminating between objects of interest by visual judgment, and, quantifying them by introducing the associated group-signature energy. This numerically invariant energy will be demonstrated by using a medical image example.
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