The 2D orientation is unique through Principal Moments Analysis

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1 The 2D orientation is unique through Principal Moments Analysis João F. P. Crespo Pedro M. Q. Aguiar Institute for Systems and Robotics / IST Lisboa, Portugal September, 2010

2 Outline 1 Motivation 2 Moments for shape representation 3 Shape normalization through Principal Moments Analysis 4 Extension to gray-level images 5 Experiments 6 Conclusion

3 Motivation Representing 2D shapes for classification Shape diversity: For us, shapes are arbitrary sets of 2D points

4 Motivation Shape normalization Normalizing w.r.t. translation and scale is simple: Sampling density and unknown point labels also tratable: Problem: unknown shape orientation

5 Motivation PCA-based orientation Only the principal axis is determined, not its direction: Not a serious problem for elongated shapes. However...

For rotationally symmetric shapes, PCA can not be

6 Motivation Ambiguity in PCA-based orientation Many shapes do not have a stable principal axis For rotationally symmetric shapes, PCA can not be used at all: This the problem we address in the paper

7 Motivation Previous work Seeking a reasonable geometric orientation : mirror-symmetry axes [Atallah,1985][Marola,1989] universal principal axis [Lin, 1993] generalized principal axis [Tsai and Chou, 1991][Zunic et al, 2006] Detection of symmetry and fold number [Lin et al, 1994] [Shen and Ip, 1999] [Derrode and Ghorbel, 2004] [Prasad and Yegnanarayana, 2004] Requiring the exhaustive search for an angle maximizing a given measure (without any guarantee of uniqueness) [Ha and Moura, 2005] More theoretically sustained methods are based on moments: CMs [Abu-Mostafa and Psaltis, 1985][Teh and Chin,1988] GCMs [Shen and Ip, 1997][Shen and Ip, 1999] An insightful recent review: [Flusser, Suk, and Zitova, 2009] This work: overcoming limitations of previous works

8 Moments for shape representation Problem definition 2D shape given by an N-dimensional complex vector z = [ z 1 z 2 z N ] T, (zn = x n + jy n ) We seek an orientation angle θ(z) s.t. any shape z is brought to its normalized version j θ(z) w(z) = ze To guarantee the desired invariance, i.e., w ( ze jφ) = w(z), it suffices that ( φ, θ ze jφ) = θ(z) + φ (add prop the orientation angle of a rotated shape is equal to the sum of the orientation angle of the original shape with the rotation angle)

9 Moments for shape representation Shape moments We define and compute θ(z) using power sums: µ k (z) = z1 k + z2 k + + zn k, k {1, 2, 3,...} {µ 1, µ 2,..., µ N } describes univocally z [Kanatani, 1990], thus forms a complete set of invariants over the permutation group, or, equivalently, is maximally invariant to permutations A much smaller set suffices to discriminate in practice, having been called Principal Moments (PMs) [Crespo and Aguiar, 2009] We thus call our approach Principal Moments Analysis (PMA) Translation, scale, and sampling invariance is obtained by working with N (z z) / z z, rather than directly with z

10 Shape normalization through Principal Moments Analysis Arguments of moments of rotated shapes Shape rotation propagates to the moments in a nice way: µ k (ze jφ ) = µ k (z)e jkφ The choice θ(z) = arg µ 1 (z) satisfies the add prop θ(ze jφ ) = arg µ 1 (ze jφ ) = arg µ 1 (z) + φ = θ(z) + φ but it is useless in practice when µ 1 = 0 Equivalent to zeroing the argument of the first-order moment of the rotationally normalized shape Our approach is to generalize by doing the same to the k th : arg µ k (ze j θ(z)) = 0

11 Shape normalization through Principal Moments Analysis Zeroing arguments of moments Imposing arg µ 2 ( ze j θ(z) ) = 0 : arg n z 2 n e 2jθ(z) = 0 arg µ 2 (z) 2θ(z) = 0 θ(z) = arg µ 2(z) 2 Imposing arg µ k ( ze j θ(z) ) = 0 : θ(z) = arg µ 2(z) 2 + π (PCA) θ(z) = arg µ k(z) k + 2π k l, l {0, 1,..., k 1} Ambiguity: there are k distinct values of θ(z) that annihilate the argument of µ k (ze j θ(z) ) Pick a solution that satisfies the add prop

12 Shape normalization through Principal Moments Analysis Principal Moments Analysis Pick l s.t. θ(z) = arg µ k (z) + 2πl satisfies the add prop θ(ze jφ ) = θ(z) + φ; k k rotation propagation: µ k (ze jφ ) = µ k (z)e jkφ The argument of the k th moment of a rotated shape is arg µ k (ze jφ ) = arg µ k (z) + kφ + 2πˆl, where ˆl is s.t. arg µ k (ze jφ ) and arg µ k (z) are in ( π, π] The normalization angle of the rotated shape is then θ(ze jφ ) = arg µ k(z) k + φ + 2π k (l + ˆl) (1) If we choose any fixed l, the add prop would require ˆl = 0, which fails to guarantee that the arguments fall in ( π, π] Solution: select a value for l that depends on the shape, l(z)

13 Shape normalization through Principal Moments Analysis Principal Moments Analysis The normalization angle is θ(z) = arg µ k (z) k + 2πl(z) k with l(z) {0, 1,..., k 1} Consider a supplementary moment µ m, with k and m coprime; the normalization angle θ(z) uses arg µ k (z) and arg µ m (z) The values of arg µ m ( ze j θ(z) ), with l(z) {0, 1,..., k 1}}, are spaced by intervals of length 2π/k PMA: choose l(z) s.t. arg µ m ( ze j θ(z) ) falls within an arbitrary but fixed interval, e.g., [0, 2π/k) The choice for l(z) is unambiguous (uniqueness) The normalization angle θ(z) satisfies the add prop (rotation invariance) All shapes have more than one nonzero moment (universality)

14 Shape normalization through Principal Moments Analysis Rotational symmetry Are there always coprime (k, m) s. t. µ k 0 and µ m 0? No! It may occur γ = gcd{l : µ l 0} > 1 Considering PMs as coefficients of a Fourier series, it is simple to conclude that this is equivalent to a γ-fold rotational symmetry, i.e., that the shape is invariant to rotations of 2π/γ All normalization angles θ + ˆk 2π/γ lead to the same result and θ is computed by using PMA with decimated PMs

15 Extension to gray-level images Extension to gray-level images and improved robustness We generalize the power sums to the corresponding moments of continuous images: + µ k (g) = (x + jy) k g(x, y) dx dy, k {0, 1, 2,...} In what respects to representation, the generalization loses completeness: in opposition to the case of a set of points, the PMs {µ k } do not determine g(x, y) univocally Naturally, the lack of completeness does not impede the usage of PMA to normalize continuous images w.r.t. orientation To improve robustness, we integrate the contributions of several pairs {(k i, m i )} by computing the angle as the (angular) weighted average θ(z) = arg i p i e j θ i (z)

16 The 2D orientation is unique through Principal Moments Analysis Experiments Direction disambiguity and rotational symmetric examples

17 Experiments Normalization angle accuracy

18 The 2D orientation is unique through Principal Moments Analysis Experiments Robustness to sampling density

19 Experiments Normalization of grey-level images

20 Experiments Normalization of grey-level images

21 Experiments Normalization of grey-level images

22 Experiments Failure cases Grey-level images that appear to be rotationally symmetric Must be carefully constructed, e.g., the image f (r, θ) = R(r) (cos θ + cos 2θ), with a particular R(r), has moments µ 0 = 0, µ 1 = 0, µ 2 0, µ 3 = 0, µ 4 = 0, µ 5 = 0, µ 6 = 0,

23 Conclusion Summary The paper presents a new algorithm PMA to normalize 2D shapes (arbitrary sets of 2D points) w.r.t. orientation PMA computes an unambiguous orientation angle for any shape, including rotationally symmetric ones PMA uses a set of moments, or power sums, that fully describe the shape PMA can be applied to grey-level images but the completeness is lost Experiments illustrate robustness

Revisiting Complex Moments For 2D Shape Representation and Image Normalization

IEEE TRANSACTIONS ON IMAGE PROCESSING 1 Revisiting Complex Moments For 2D Shape Representation and Image Normalization João B. F. P. Crespo and Pedro M. Q. Aguiar, Senior Member, IEEE Abstract When comparing