Matching pursuit analysis of STRFs with gabor atoms

As described above, Gabor filters have been proposed as models for STRFs. Explicitly describing the STRFs found in the present work in terms of the Gabor filter bank underscores this choice. This was done by implementing a two-dimensional matching pursuit (MP) algorithm (Mallat and Zhang, 1993). MP is a greedy reconstruction algorithm of signals by a dictionary of given “atoms.” In our case, the target “signals” were the STRFs, and the atoms were defined by the elements of the Gabor filter bank. MP computes the overlap γ between signal and each atom in the dictionary by correlation. At each iteration step i, γ_ijk was defined as the maximal correlation coefficient, computed from two-dimensional correlation between the j-th Gabor atom and the k-th STRF. Let CC denote two-dimensional cross-correlation, g the Gabor function as defined above, and S the STRF:

γ was computed for all Gabor atoms, the projection of the atom with the largest correlation coefficient was subtracted from the signal, and the process was repeated on the residual signal. The correlation coefficient θ between original and reconstructed STRFs increases monotonically through MP iterations. For most tasks, we chose θ = 0.8 as termination criterion. As described above, the Gabors are complex-valued filters; for MP, we used the real (symmetric) and imaginary (antisymmetric) parts as independent, real-valued Gabor atoms. Re-synthesized STRFs were in turn used for feature extraction and subsequent classification. The effect of the MP re-synthesis on these features was two-fold: first, the filter shapes are approximated by smooth Gabor shapes, i.e., irrelevant noise and artifacts in the STRF patterns may be removed when the reconstruction is sufficiently coarse. Second, from an information point of view, approximating filters incompletely results in a loss of information.

For further analysis, we defined a measure κ of “spectro-temporality” of the reconstructed filters, which is computed as follows: the vector of modulation frequencies of a Gabor atom, ω, was L2-normalized, i.e., it was projected to the unit circle in modulation space. κ was then computed as its L1-norm (cf. Figure ​Figure3).3). Let ω = (ω_t, ω_s) be the vector of temporal and spectral modulation frequency of a Gabor atom. For each element of ω: π/2 ≤ ω_t|s ≤ π/2. Then the spectro-temporality measure κ_ω of this Gabor atom is defined as the L1-norm of the L2-normalized frequency vector:

Computation of the measure of spectro-temporality κ_ω. The modulation vector ω = (ω_t, ω_s) is normalized to unit length; κ_ω equals the L1-norm of the normalized vector.

κ ranges from 1 (purely spectral or purely temporal) to 2 (diagonal). In the MP task, the value κ assigned to the MP reconstruction of an STRF over N iterations was the average of the κs of the Gabor atoms involved: