Simple cells modeling through a sparse overcomplete Gabor wavelet representation based on local inhibition and facilitation Rafael Redondo1, Sylvain Fischer1, Laurent Perrinet2 and Gabriel Crist´obal1 1 Instituto de Optica (CSIC), Serrano 121, 28006 Madrid, Spain 2INCM/CNRS, 31 ch. J. Aiguier, 13402 Marseille Cedex 20, France {rafa, sylvain, gabriel }@optica.csic.es Laurent.Perrinet@incm.cnrs-mrs.fr 1 Receptive ﬁelds: log-Gabor ﬁlters space (real part) space (imag. part) Fourier domain Figure 1: log-Gabor ﬁlters Model for the Simple cells receptive ﬁelds: • log-Gabor ﬁlters [1, 2]. • frequency bandwidth = 1.85 octave. • half-orientation bandwidth = 25 degrees for a 6 orien- tations scheme (18.5 degrees for 8 orientations). log-Gabor wavelets implementation : • 5 scales separated by 1 octave, 6 or 8 orientations. • Same ﬁlters both for the wavelet transform (W) and for the inverse (WT). • exact reconstruction (∀x ∈RN, WTWx = x). • overcompleteness: factor of 28 using 6 orientations (factor of 37 using 8 orientations). space (real part) space (imag. part) Fourier (contours) Figure 2: log-Gabor wavelets on 5 scales, 8 orient. 2 Non-linear lateral interactions: sparse coding and edge extraction 2.1 Need of sparse coding • The linear transform induces redundancy. • There is non-uniqueness of the representation. • Sparse representations are available. ⇒Necessity of a sparse coding algorithm [3, 4]. ⇒Sparse coding algorithms represent adaptively the sig- nals through a selection of coding functions taken from an overcomplete set. image linear log-Gabor sparse representation 2.2 Local competitions and cooperations Non-linear interactions between neighboring cells Selection of coefﬁcients inspired from the architecture of V1 cortical columns: • Inhibition of non-local maxima (or competition [5]). • Enhancement of colinear and coaligned coefﬁcients (or cooperation [6]). ⇒Efﬁcient for edge and contour extraction [5, 6, 7] (and Fig. 4) and for sparse coding [8, 9]. response to a ridge response to an edge Figure 4: complex-valued log-Gabor response to ridges and edges 2.3 Algorithm for edge selection and back- projection of the residual input image linear Gabor transform (3 adjacent channels) remaining coefﬁcients after the competition Cooperation: chain length calculation and amplitude propagation • Thresholds of selection on the chain length and on the maximum of amplitude along the chain. • Non-selected coefﬁcients r are projected on the trans- form space by the WWTr operation. • Exact reconstruction (WTWWTr = WTr). • Repeating several times the projection (e.g. 30 times) enhances selected coefﬁcients (typically by a factor of 7), while close non-selected coefﬁcients vanish. Figure 6: Amplitude growing of a selected coefﬁcient through iterative projection of the non-selected ones. Vanishing of the neighboring non-selected coefﬁcients. 2.4 Contour representation by chain coding Figure 7: Chain coding scheme. Coefﬁcients are quantized in module and phase (8 phases). Specially adapted to Gabor wavelets features, our chain coding [10] represents every chain by its end-points (head and tail) and the elementary movements necessary to walk through the chain from head to tail. As a result, redun- dancy of the transform domain is further diminished. • Head position are referred to the previous coded head. • Movement scheme depends on channel orientation. • Quantized coefﬁcient (module and phase) is predicted according to the previous coded link. Figure 8: Schemes of movement coding. Absence of movement means end-of-chain. Huffman codes compress head positions and movements and arithmetic coding does to coefﬁcient amplitude. Residual low-pass channel is coded by a predictive codec adapted from FELICS [11]. 3 Evaluation We apply the model to tasks the Human Visual System (HVS) is supposed to deal with: edge and contour extrac- tion, noise segregation, sparse representation. 3.1 Edge extraction Using complex-valued Gabor ﬁlters both edges and lines