ROBUST REORIENTATION OF 2D SHAPES USING THE ORIENTATION INDICATOR INDEX Victor H. S. Ha∗ Jos´e M. F. Moura Digital Media Solutions Lab Department of ECE Samsung Information Systems America Carnegie Mellon University Irvine, CA 92612 Pittsburgh, PA 15213 {[email protected]} {[email protected]} ABSTRACT In section 2, we briefly review some of the past research efforts on shape reorientation. In section 3, we introduce Shape reorientation is a critical step in many image process- three orientation indicator indices that we use to remove the ing and computer vision applications such as registration, orientation ambiguity in the shape: the orientation indicator detection, identification, and classification. Shape reorien- index (OII), the variable-size window OII (∆-OII), and the tation is a needed step to restore the correct orientation of extended OII (X-OII). We illustrate the good performance a shape when its image is subject to an arbitrary rotation of these new measures of orientation with experimental ex- and reflection. In this paper, we present a robust method amples. Section 4 summarizes the paper. to determine the standard “normalized” orientation of two- dimensional (2D) shapes in a blind manner, i.e., without any other information other than the given input shape. We in- 2. SHAPE ORIENTATION troduce a set of orientation indicator indices (OII) that use low order central moments of the shape to monitor the ori- There are an extensive number of published efforts to deter- entational characteristics of the shape. Because these OII’s mine the orientation of shapes. A good set of existing tech- use only low (up to third) order moments, they are robust niques are reviewed in [3]; these techniques are shown to to noise and errors. We show with examples how we bring fail for some test shapes. The Shen-Ip symmetries detector consistently a given shape with an unknown arbitrary orien- [4] that survives these tests is concerned with the problem tation to its standard normalized orientation using the OII. of detecting both the reflectional and rotational symmetry axes of a shape. The algorithm requires the computation of many high order generalized complex (GC) moments of the 1. INTRODUCTION shapes. In [1], we introduce the concept of intrinsic shape of In image processing and computer vision applications, the an object as an invariant to the class of affine and permu- shapes of objects are obtained and analyzed for registration, tation distortions of a shape, and summarize a blind algo- detection, identification, and classification. Many research rithm to recover the intrinsic shape—BLAISER. The key efforts are directed to finding the correct orientation of the step in BLAISER is the reorientation of the shape, which is input shape prior to feeding it into such further processing. solved with the point-based reorientation algorithm (PRA). We have developed the point-based reorientation algo- The PRA provides a complete solution to the shape reori- rithm (PRA) [1] and the variable-size window orientation entation problem. The PRA is also an efficient algorithm indicator index (∆-OII) [2] that remove the orientational whose complexity is O(N log N) where N is the number ambiguity from any shape distorted by an arbitrary orien- of feature points (pixels) in the given shape (binary tem- tation. The two algorithms are combined in a complete plate of the object). In practice, the locations of the feature and efficient solution to the shape orientation problem, the points may be measured inaccurately due to the finite reso- PRA-∆-OII algorithm [2]. In this paper, we introduce the lution of the input device and background noise. The PRA extended OII (X-OII). This new OII is resilient to error may be sensitive to such disturbances. and noise, especially when there is an unexpectedly large In the next section, we introduce a set of orientation in- amount of error or noise added to the shape. dicator indices (OII) that are moment-based measures of the ∗The first author performed the work while at Carnegie Mellon Univer- shape orientation. We use these OII’s to develop a robust sity. This work was partially supported by ONR Grant #14-0-1-0593 shape orientation algorithm. 0-7803-8874-7/05/$20.00 ©2005 IEEEII - 777 ICASSP 2005 1 4000 3. ORIENTATION INDICATOR INDEX 0.8 0.6 3500 0.4 0.2 3000 0 We now introduce the orientation indicator index (OII) that Ŧ0.2 2500 Ŧ0.4 monitors the orientation of the shape. The OII uses a series Ŧ0.6 2000 Ŧ0.8 Ŧ1 1500 of low order central moments of the input shape. Moment- Ŧ1 Ŧ0.8 Ŧ0.6 Ŧ0.4 Ŧ0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 based techniques are more robust to errors than point-based (a) Test Shape X (b) OII plot of Test Shape techniques such as the PRA. The OII is computed from up 1 4000 0.8 to third order central moments of the shape. We have chosen 0.6 3500 0.4 0.2 3000 the third order moments because higher order moments are 0 Ŧ0.2 2500 generally more sensitive to noise and other sources of error. Ŧ0.4 Ŧ0.6 2000 We start in subsection 3.1 with a definition of the OII Ŧ0.8 Ŧ1 1500 Ŧ1 Ŧ0.8 Ŧ0.6 Ŧ0.4 Ŧ0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 using the third order central moments. Then, in subsec- ◦ (c) Shape Rotated by 45 (d) OII plot of Rotated Shape tion 3.2, we introduce the ∆-OII, the variable-size window OII. Finally in subsection 3.3, we generalize the concept and introduce the extended orientation indicator index, X- Fig. 1. Examples of OII OII. We illustrate with examples how these OII’s apply to the shape orientation problem. other about the x–axis, their OII plots are identical. The second property requires a lengthy algebraic derivation 3.1. OII to prove. After the derivation, we find that the OII plot is a θ We represent a 2D shape X as the collection of x- and y- cosinusoidal function of the rotation angle with a period T π/ coordinates, (x, y), falling inside the shape. The represen- = 2.Thatis, tation can be either continuous or discrete. We define the 3 OIIofshapeX as OIIXθ = ρ cos (4θ − λ)+τ (4) 4 m2 m2 where OIIX = 30 + 03 (1) 2 (M1 − M3) 2 where mpq is the (p + q)th order central moment of X.If ρ = + M2 the shape X is rotated about its center of mass by an angle 4 2M2 θ,wedefine the OII of the rotated shape Xθ by λ =arctan M1 − M3 m2 m2 5M1 +3M3 OIIXθ = ¯ 30 +¯03 (2) τ = 8 where and N 2 2 3 M1 = m30 + m03 m¯ 30 = (cosθ · xk +sinθ · yk) (3) k=1 M2 =2(m21m30 − m12m03) 2 2 N M3 =2(m12m30 + m21m03)+3(m21 + m12). 3 m¯ 03 = ( − sin θ · xk +cosθ · yk) , k=1 The OII plot represented by Equation (4) has four peaks over the range θ ∈ [0, 2π), which of course simply confirms using the notation of a discrete shape representation consist- Properties 1 and 2. ing of N feature points. An OII plot is generated by rotating We illustrate the usage of the OII with simple examples. the shape by an angle θ and computing the OII at each ro- We first choose the test shape X to be the binary image of tated position over the rotation range of θ ∈ [0, 2π). From an airplane as shown in Fig. 1 (a). The test shape is rotated (1) and (2), we deduce the following properties of the OII. over the rotation range θ ∈ [0, 2π). At each rotated position, Property 1: The OII plot is periodic in θ with period T = the OII value is computed and plotted. The OII plot that π/ ∀ θ ∈ , π 2. That is OIIXθ = OIIX(θ+ π ) for [0 2 ). 2 results after the full rotation is shown in Fig. 1 (b). As we Property 2: The OII plot has four peaks in the rotation expect, the plot is a cosinusoidal function of the rotation range of θ ∈ [0, 2π). angle θ with a period T = π/2. Fig. 1 (c) shows the same Property 3: If two shapes differ from each other by a rota- test shape rotated by an angle π/4 rad. The OII plot of this tion angle of φ, their OII plots are circularly shifted versions rotated shape is shown in Fig. 1 (d). Observe that this plot of each other by the same angular displacement φ. is circularly shifted by the angle π/4 with respect to the OII Property 4: If two shapes are reflected versions of each plot shown in (b). II - 778 0.034 The reorientation of a shape is achieved by first gener- 0.033 ating the OII plot. According to (4), there exist four peaks 0.032 in the OII plot that represent four unique rotational orienta- 0.031 tions of the shape. We can bring any shape to one of its four 0.03 unique orientations by bringing any peak in its OII plot to 0.029 θ 0.028 the origin =0. 0 100 200 300 400 500 600 700 800 900 1000 There are two major weaknesses in the method described (a) Shape 1 (b) ∆-OII plot of (a) above.