1. Biologically Inspired Algorithms for Computer Vision: Motion Estimation from Steerable Wavelet Construction D. Conte A, J. Ng B, E. Grisan A, A. Ruggeri A A Department of Information Engineering, University of Padova, Padova, Italy B Department of Bioengineering, Imperial College London, London, UK Introduction Since structural information is mostly carried by phase, image reconstruction from a phase spectrum gives better results than using magnitude spectrum [4]. At variance with traditional Fourier analysis, using wavelets provides image features that are localized both in space and in frequency, decomposing the image in local amplitude and local phase information, within a narrow frequency band. Bharath and Ng in [1], designed a complex orientation steerable framework, applying it to image denoising. Here, we use this framework to perform motion estimation analysis. Given the steered filters output, we are able to compute the optical flow from a measure of local phase difference, and then to obtain an estimation of the displacement field between two images. Materials and Methods Correspondence Davide Conte, University of Padova and University of Verona, Italy. E-mail: bioengineering@email.it Jeffrey Ng, Imperial College London, UK. E-mail: jeffrey.ng@imperial.ac.ukAcknowledgements The authors would like to thank Dr. Anil Bharath, head of the Bioengineering Vision Research Group at Imperial College London. References [1] A. A. Bharath and J. Ng. A steerable complex wavelet construction and its application to image denoising. IEEE Transactions on Image Processing, 14(7): 948–959, 2005. [2] P. Dayan and L. F. Abbott. Theoretical Neuroscience, MIT Press, 2001. [3]M. Hubener et al., Spatial Relationships among Three Columnar Systems in Cat Area 17, J.of Neuroscience, 17(23): 9270-9284, 1997. [4] A. V. Oppenheim and J. S. Lim. The importance of phase in signals. Proceedings of the IEEE, 69(5): 529–541, 1981. [5] M. Felsberg. Optical flow estimation from monogenic phase. 1 st International Workshop on Complex Motion, 3417: 1–13, 2006. [6] E. P. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. Heeger. Shiftable multiscale transforms. IEEE Transactions on Information Theory, 38(2): 587–607, 1992. [7] D. Conte. Biologically Inspired Algorithms for Computer Vision: Phase-Based motion Estimation, M.Sc. Thesis, University of Padova, April 2008. [8] S. S. Beauchemin and J. L. Barron. The computation of optical flow. ACM Computing Surveys, 27(3): 433–467, 1995. [9] H. Knutsson and M. Andersson. Morphons: Paint on priors and elastic canvas for segmentation and registration. SCIA 2005, LNCS 3540: 292–301, 2005. Results and Discussion Spatial filter kernels were built along 4 different orientations for each frequency band. Each filter provides a complex response and the steerability property [6] allows us to compute a single filter response along the estimated dominant orientation θ x. If the signal is locally 1-D around the point x, we can define a local image model as: where local amplitude A(x) is supposed to be constant around x, and I 0 represents the mean intensity value of the image. The local phase at point x is therefore extracted as argument of the complex steered filter response:  (−π, π] Considering orientation and phase information as a whole, both can be combined in the phase vector: Considering two frames of a video, we can assume that the new frame has been obtained from the first one by a local displacement d(x), and the same condition stands also for the phase vectors: Since the restriction to a narrow band of frequency components allows the approximation of the phase vector with a Taylor series around x [5],[7], if the displacement is sufficiently small with respect to the considered level of resolution of the image, remembering that the local frequency ω x is defined as and that n x n x T projects the displacement vector d along the dominant orientation, we obtain: The concept of displacement from phase is not new, and applications of the method in biomedical image processing have successfully been done [9]. However, motion estimation from steerable filters differs from traditional phase-based methods, because we computed the phase difference along the dominant orientation only, instead of interpolating phase difference values from different directions. The method has been succesfully applied on a synthetic pattern and on a registration problem of IR retinal blood vessels images, but there is certainly the need of a deep comparison with other motion estimation methods, and, before that, the need of improving estimates precision. Orientation-steerable wavelet-based filters allow a 2-D generalization of the concept of analytic signal defined in a 1-D context by means of the oriented in-quadrature filters. Moreover, the filters emulate the properties of receptive fields in biological vision: The computational model derived from the properties of neurons in the Primary Visual Cortex [2], was used as inspiring model to design a system that permits the decomposition of image information at different scales (i.e. different spatial frequencies) and orientations with different phase-symmetry behaviour. Symmetric and antisymmetric orientented in- quadrature filters V1 neurons selectivity to orientation (a) and spatial frequencies (b) of visual stimuli [3]. In figure (b) white areas represent neurons that are sensitive to high spatial frequencies, while gray areas represent neurons that a re sensitive to low spatial frequencies. This can be seen as the analogous for the phase to the Optical Flow Constraint Equation [8], and indeed it suffers from the same ill-positioning, since only the projection of d along n x is determinable from the difference of the phase vectors (the local 1-dimensionality of the signal leads to the so called aperture problem [8]). To overcome this limit and try to compute the true displacement vector, we followed a weighted least squares approach (WLS) integrating information over a small region Ω around x. θxθx Finally, a multi-resolution approach can be used to integrate displacement estimates from different scales. Optical flow estimation on synthetic pattern, with plot of absolute error for constant motion No smoothing has been applied after the WLS, that demonstrates of being able to handle the aperture problem. (a) Original first frame, an IR image of retinal blood vessels; (b) Normalized image subtraction of the two frames (gray means zero); (c) Normalized image subtraction after using the estimated displacement field to warp the first frame. (a) (b) (c) Local phase measured along dominant orientation, with the signal represented as a sinusoidal function given by the local image model. Illustration of the aperture problem (a) (b)