Bernal S D_2010.pdf - University of Plymouth

4.1. HMAX AS A BAYBSIAN NETWORK
iniptement contrast invarlance, mimicking complex cells in striate cortex, hy taking the absolute
value of their SI inputs, Thererore, at each C! location there are 32 CI units, one for each of the
f^ci {= 4) orientations x 8 scale bands. Note that, unlike S1 units, CI units are not computed at
every possihle location but are .sampled every ffi pixels or SI units, where Cfi ranges from 3
pixels to 15 pixels, in steps of 2 pixels, according lo lite CI scale band.
Physiological data on simple and complex RF size, spatial frequency and orientation band­
width are in good agreement with the model S! and CI tuning properties, as well as with the
hyiKtthesis of complex cells performing a imix operation over simple cell afferents (Serre and
Riesenhuber 2004).
S2 layer - The response of each S2 unit depends in a Gaussian-like way on the F.uclidean dis-
lance between the input and previously learned prototypes. More specilically. it implements
a Radial Basis Function (RBF) network, where the prototypes are ihe RBF centres. Dur­
ing the training phase, Ks2 prototypes are learned from ihe CI layer, each one composed of
AA'.s7 X AA'52 X Kc[{= 4) elements. In some HMAX versions (Scrre el al. 2007c) Ksj ^ 2000
and AA'^j - 3, which yields 2000 prototypes with 3 x 3 x 4 = 36 elements; while other imple­
mentations (Serre et al. 2007b) use values of ^^52 - 1000 and ANs2 in the range (4,8, 12,16[.
In summary, at each S2 location there iu^e Ksi S2 units coding each of the learned proloiypes.
During the recognition phase, the response of an S2 unit at a particular kx;ation and coding a
specific learned prototype or RBI- centre is calculated as the distance between the input patch
of ANs2 X AA's2 CI units, and the k"' stored protoiype F^. such that,
^2i,,„,AM,„,,* - exp(-/i • j|c:i{6,^„j,,) -/*(|pj (4.6)
where 0 is the square of the inverse width of the RBF and therefore defines the sharpness of the
tuning curve,
bs2TXs2^ys2 represents the band and location of the S2 unit,
{h,.x,,yi\ represents the band and location of the afferent CI units, and Is given, as a function
of the S2 unit's band and kxalion and the network parameters, by the following expressions:
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