=Paper=
{{Paper
|id=None
|storemode=property
|title=Statistical Invariants of Spatial Form: From Local AND to Numerosity
|pdfUrl=https://ceur-ws.org/Vol-1007/paper14.pdf
|volume=Vol-1007
|dblpUrl=https://dblp.org/rec/conf/shapes/ZetzscheGK13
}}
==Statistical Invariants of Spatial Form: From Local AND to Numerosity==
https://ceur-ws.org/Vol-1007/paper14.pdf
Statistical Invariants of Spatial Form:
From Local AND to Numerosity
Christoph ZETZSCHE 1 , Konrad GADZICKI and Tobias KLUTH
Cognitive Neuroinformatics, University of Bremen, Germany
Abstract Theories of the processing and representation of spatial form have to take
into account recent results on the importance of holistic properties. Numerous ex-
periments showed the importance of “set properties”, “ensemble representations”
and “summary statistics”, ranging from the “gist of a scene” to something like “nu-
merosity”. These results are sometimes difficult to interpret, since we do not ex-
actly know how and on which level they can be computed by the neural machinery
of the cortex. According to the standard model of a local-to-global neural hierarchy
with a gradual increase of scale and complexity, the ensemble properties have to
be regarded as high-level features. But empirical results indicate that many of them
are primary perceptual properties and may thus be attributed to earlier processing
stages. Here we investigate the prerequisites and the neurobiological plausibility
for the computation of ensemble properties. We show that the cortex can easily
compute common statistical functions, like a probability distribution function or an
autocorrelation function, and that it can also compute abstract invariants, like the
number of items in a set. These computations can be performed on fairly early lev-
els and require only two well-accepted properties of cortical neurons, linear sum-
mation of afferent inputs and variants of nonlinear cortical gain control.
Keywords. shape invariants, peripheral vision, ensemble statistics, numerosity
Introduction
Recent evidence shows that our representation of the world is essentially determined by
holistic properties [1,2,3,4,5,6]. These properties are described as “set properties”, “en-
semble properties”, or they are characterized as “summary statistics”. They reach from
the average orientation of elements in a display [1] over the “gist of a scene”[7,8], to the
“numerosity” of objects in a scene [9]. For many of these properties we do not exactly
know by which kind of neural mechanisms and on which level of the cortex they are
computed. According to the standard view of the cortical representation of shape, these
properties have to be considered as high-level features because the cortex is organized in
form of a local-to-global processing hierarchy in which features with increasing order of
abstraction are computed in a progression of levels [10]. At the bottom, simple and lo-
cally restricted geometrical features are computed, whereas global and complex proper-
ties are represented at the top levels of the hierarchy. Across levels, invariance is system-
1 Corresponding Author: Christoph Zetzsche, Cognitive Neuroinformatics, FB3, University of Bremen, P.O.
Box 330 440, 28334 Bremen, Germany; E-mail: zetzsche@informatik.uni-bremen.de
Research supported by DFG (SFB/TR 8 Spatial Cognition, A5-[ActionSpace])
163
atically increased such that the final stages are independent of translations, rotations, size
changes, and other transformations of the input. However convincing this view seems on
first sight, it creates some conceptual difficulties.
The major difficulty concerns the question of what exactly is a low-level and a high-
level property. Gestalt theorists already claimed that features considered high-level ac-
cording to a structuralistic view are primary and basic in terms of perception. Further
doubts have been raised by global precedence effects [11]. Similar problems arise with
the recently discovered ensemble properties. The gist of a scene, a high-level feature
according to the classical view, can be recognized in 150 msec [7,12,13,14] and can be
modeled using low-level visual features [8]. In addition, categories can be shown to be
faster processed than basic objects, contrary to the established view of the latter as entry-
level representations [15]. A summary statistics approach, also based on low-level visual
features, can explain the holistic processing properties in the periphery of the visual field
[4,16,17]. What is additionally required in these models are statistical measures, like
probability distributions and autocorrelation functions, from which it is not known how
and on which level of the cortical hierarchy they can be realized.
One of the most abstract ensemble properties seems to be the number of elements
in a spatial configuration. However, the ability to recognize this number is not restricted
to humans with mature cognitive abilities but has also been found in infants and animals
[9,18], recently even in invertebrates [19]. Neural reactions to numerosity are fast (100
msecs in macaques [20]). And finally there is evidence for a “direct visual sense for
number” since number seems to be a primary visual property like color, orientation or
motion, to which the visual system can be adapted by prolonged viewing [21].
The above observations on ensemble properties raise a number of questions, from
which the following are addressed in this paper: Sect. 1: Can the cortex compute a prob-
ability distribution? Sect. 2: And also an autocorrelation function? By which kind of
neural hardware can this be achieved? Sect.3: Can the shape of individual objects also
be characterized by such mechanisms? Sect. 4: What is necessary to compute such an
abstract property like the number of elements in a spatial configuration? Can this be
achieved in early sensory stages?
1. Neural Computation of a Probability Distribution
Formally, the probability density function pe (e) of a random variable e is defined via the
cumulative distribution function: pe (e) , dPdee (e)
with Pe (e) = Pr[e e]. Their empirical
counterparts, the histogram and the cumulative histogram, are defined by use of indicator
functions. For this we divide the real line into m bins (e(i) , e(i+1) ] with bin size De =
e(i+1) e(i) . For each bin i, an indicator function is defined as
⇢
1, if e(i) < e e(i+1)
Qi (e) = 1i (e) = (1)
0, else
An illustration of such a function is shown in Fig. 1a. From N samples ek of the ran-
dom variable e we then obtain the histogram as h(i) = N1 ÂNk=1 Qi (ek ). The cumulative
histogram He (e) can be computed by changing the bins to (e(1) , e(i+1) ] (cf. Fig. 1b), and
by performing the same summation as for the normal histogram. The reverse cumulative
164
(a) (b) (c)
Figure 1. Indicator functions. Basic types are: (a) indicator function for computation of a classical histogram.
(b) indicator function for a cumulative histogram. (c) indicator function for a reverse cumulative histogram.
histogram H̄(i) is simply the reversed version of the cumulative histogram. The corre-
sponding bins are Dei = (e(i) , e(m+1) ] and the indicator functions are defined as (Fig. 1c)
⇢
1, if e e(i)
Qi (e) = 1i (e) = (2)
0, else
The corresponding system is shown in Fig. 2.
The three types of histograms have identical information content since they are re-
lated to each other as
i
h(i) = H((i + 1)) H(i) = H̄(i) H̄(i + 1) and H(i) = 1 H̄(i) = Â h( j). (3)
j=1
Measurements Indicator Functions Summation Histogram
Q1(e1)
1
Q2(e1) e1
1
e1
e1 100% H(i)
Qk(e1)
S1
1
S2
e1
Sk
Q1(eN)
1
i
Q2(eN) eN
1
eN
eN
Qk(eN)
1
eN Albrecht and Hamilton (1982)
(a) (b)
Figure 2. Computation of the reverse cumulative histogram. (a) shows the set of input variables e1 to en over
which the histogram should be computed. Each of these variables is input to a set of indicator functions Qi (ek ).
For each bin of the histogram there is a summation unit Si which sums over all indicator function outputs with
index i, i.e. over all Qi (ek ).
(b) The response functions of three neurons in the visual cortex [22]. They show a remarkable similarity to the
indicator functions for the reverse cumulative histogram. First, they come with different sensitivities. Second,
they exhibit an independence on the input strength: once the threshold and the following transition range is
exceeded the output remains constant and does no longer increase when the input level is increased.
165
Figure 3. Neurobiological computation of a reverse cumulative histogram. The upper row shows several ex-
amples of input probability distributions. The second row shows the corresponding reverse cumulative his-
tograms computed by a dense set of simulated neurons. The third row shows the estimated probability distri-
butions as derived from the neural representation by use of Eq. (3).
How does all this relate to visual cortex? Has the architecture shown in Fig. 2a any
neurobiological plausibility? The final summation stage is no problem since the most
basic capability of neurons is computation of a linear sum of their inputs. But how about
the indicator functions? They have two special properties: First, the indicator functions
come with different sensitivities. An individual function does only generate a non-zero
output if the input e exceeds a certain level, a kind of threshold, which determines the
sensitivity of the element e(i) in Eq. (2) and Fig. 1c. To cover the complete range of
values, different functions with different sensitivities are needed (Fig. 2a). Second, the
indicator functions exhibit a certain independence of the input level. Once the input is
clearly larger than the threshold, the output remains constant (Fig. 1c).
Do we know of neurons which have such properties, a range of different sensitivi-
ties, and a certain independence of the input strength? Indeed, cortical gain control (or
normalization), as first described in early visual cortex (e.g. [22]) but now believed to
exist throughout the brain [23], yields exactly these properties. Gain-controlled neurons
(Fig. 2b) exhibit a remarkable similarity to the indicator functions used to compute the
reverse cumulative histogram, since they (i) come with different sensitivities, and (ii) pro-
vide an independence of the input strength in certain response ranges.
The computation of a reverse cumulative histogram thus is well in reach of the cor-
tex. We only have to modify the architecture of Fig. 2a by the smoother response func-
tions of cortical neurons. The information about a probability distribution available to the
visual cortex is illustrated in Fig. 3. The reconstructed distributions, as estimated from the
neural reverse cumulative histograms, are a kind of Parzen-windowed (lowpass-filtered)
versions of the original distributions.
2. Neural Implementation of Auto- and Cross-Correlation Functions
A key feature of the recent statistical summary approach to peripheral vision [4,6,24,16]
is the usage of auto- and cross-correlation functions. These functions are defined as
N/2
1
N k= Â
h(i) = e(k) g(i + k), (4)
N/2+1
166
Figure 4. Different types of AND-like functions. Each function is of the type gk = g(si , s j ), i.e. assigns
an output value to each combination of the two input values. The upper row shows the functions as surface
plots, the lower row as iso-response curves. Left: Mathematical multiplication of two inputs. Center: AND-like
combinations that can be obtained by use of cortical gain control (normalization). The upper left figure shows
the classical gain control without additional threshold. The upper right figure shows the same mechanism with
an additional threshold. This results in a full-fledged AND with a definite zero response in case that only one
of the two inputs is active. Right: The linear sum of the two input values for comparison purposes.
where autocorrelation results if e(k) = g(k) and where indicates multiplication. With
respect to their neural computation, the outer summation is no problem, but the cru-
cial function is the nonlinear multiplicative interaction between two variables. A neu-
ral implementation could make use of the Babylonian trick ab = 14 [(a + b)2 (a b)2 ]
[25,26,27], but this requires two or more neurons for the computation and thus far there
is neither evidence for such a systematic pairing of neurons nor for actual multiplicative
interactions in the visual cortex. However, exact multiplication is not the key factor: a
reasonable statistical measure merely requires provision of a matching function such that
e(k) and g(i + k) generate a large contribution to the autocorrelation function if they are
similar, and a small contribution if they are dissimilar. For this, it is sufficient to provide
a neural operation which is AND-like [27,28]. Surprisingly, such an AND-like operation
can be achieved by the very same neural hardware as used before, the cortical gain con-
trol mechanism, as shown in [28]. Cortical gain control [22,29] applied to two different
features si (x, y) and s j (x, y) can be written as
0 1
si + s j
gk (x, y) = g(si (x, y), s j (x, y)) := max @0, q p QA (5)
( s2i + s2j + e) 2
where k = k(i, j), e is a constant which controls the steepness of the response and Q is a
threshold. The resulting nonlinear combination is comparable with an AND-like opera-
tion of two features and causes a substantial nonlinear increase of the neural selectivity,
as illustrated in Fig. 4.
Of course there will be differences between a formal autocorrelation function and
the neurobiological version, but the essential feature, the signaling of good matches in
dependence of the relative shifts will be preserved (Fig. 5).
167
(a) (b)
Figure 5. Mathematical and neurobiological autocorrelation functions. (a) shows a test input and (b) the cor-
responding mathematical (red dotted) and neurobiological (blue) autocorrelation function.
Figure 6. Different shapes and the corresponding integral features. We used parameter combinations of six
different orientations qi = (i 1)p/6, i = 1, . . . , 6, and four different scales ri = 2 i , i = 1, . . . , 4. The radial
half-bandwidth was set to fr,h = 13 r and the angular half-bandwidth was constant with fq ,h = p/12. Each
parameter combination creates pairs of variables for each x,y-position which are AND-combined by the gain
control mechanism described in Eq. (5) as gk (x, y) = g(si (x, y), s j (x, y)).
3. Figural Properties from Integrals
We extracted different features sr,q from the image luminance function l = l(x, y) by
applying a Gabor-like filter operation sr,q (x, y) = (l ⇤ F 1 (Hr,q ))(x, y) where F 1 de-
notes the inverse Fourier transformation and the filter kernel Hr,q is defined in the spec-
tral space. We distinguish two cases (even and odd) which can be seen in the following
definition in polar coordinates:
( ⇣ ⌘ ⇣ ⌘
even cos2 p2 2frf r cos2 p2 2fqf q , ( fr , fq ) 2 Wr,q
Hr,q ( fr , fq ) := r,h q ,h
0 , else,
with Wr,q := {( fr , fq )| fr 2 [r 2 fr,h , r + 2 fr,h ] ^ fq 2 [q 2 fq ,h , q + 2 fq ,h ] \ [q + p
2 fq ,h , q + p + 2 fq ,h ]}, where fr,h denotes the half-bandwidth in radial direction and fq ,h
denotes the half-bandwidth in angular direction. Hr,q odd is defined as the Hilbert trans-
formed even symmetric filter kernel.
Various AND combinations of these oriented features (see caption Fig. 6) are ob-
tained by the gain-control mechanism described R in Eq. (5). The integration over the
whole domain results in global features Fk := R2 gk (x, y) d(x, y) which capture basic
shape properties (Fig. 6).
4. Numerosity and Topology
One of the most fundamental and abstract ensemble properties is the number of elements
of a set. Recent evidence (see Introduction) raised the question at which cortical level
168
the underlying computations are performed. In this processing, a high degree of invari-
ance has to be achieved, since numerosity can be recognized largely independent of other
properties like size, shape and positioning of elements. Models which address this ques-
tion in a neurobiologically plausible fashion, starting from individual pixels or neural re-
ceptors instead of an abstract type of input, are rare. To our knowledge, the first approach
in this direction has been made in [30]. A widely known model [31] has a shape-invariant
mapping to number which is based on linear DOG filters of different sizes, which sub-
stantially limits the invariance properties. A more recent model is based on unsupervised
learning but has only employed moderate shape variations [32]. In [30] we suggested
that the necessary invariance properties may be obtained by use of a theorem which con-
nects local measurements of the differential geometry of the image surface with global
topological properties [30,33]. In the following we will build upon this concept.
The key factor of our approach is a relation between surface properties and a topo-
logical invariant as described by the famous Gauss-Bonnet theorem. In order to apply
this to the image luminance function l = l(x, y) we interpret this function as a surface
S := {(x, y, z) 2 R3 |(x, y) 2 W, z = l(x, y)} in three-dimensional real space. We then apply
the formula for the Gaussian curvature
lxx (x, y)lyy (x, y) lxy (x, y)2
K(x, y) = , (6)
(1 + lx (x, y)2 + ly (x, y)2 )2
2
where subscript denotes the differentiation in the respective direction (e.g. lxy = ∂∂x∂ly ).
The numerator of (6) can also be written as D = l1 l2 where l1,2 are the eigenvalues of
the Hessian matrix of the luminance function l(x, y) which represent the partial second
derivatives in the principal directions. The values and signs of the eigenvalues give us
the information about the shape of the luminance surface S in each point, whether it
is elliptic, hyperbolic, parabolic, or planar. Since Gaussian curvature results from the
multiplication of the second derivatives l1,2 it is zero for the latter two cases. It has been
shown that this measure can be generalized in various ways, in particular towards the use
of neurophysiologically realistic Gabor-like filters instead of the derivatives [27,30]. The
crucial point, however, is the need for AND combinations of oriented features [27,30]
which can be obtained as before by the neural mechanism of cortical gain control [28].
The following corollary from the Gauss-Bonnet theorem is the basis for the invari-
ance properties in the context of numerosity.
Corollary 4.1 Let S ⇢ R3 be a closed two-dimensional Riemannian manifold. Then
Z
K dA = 4p(1 g) (7)
S
where K is the Gaussian curvature and g is the genus of the surface S.
We consider the special case where the luminance function consists of multiple objects
(polyhedra with orthogonal corners) with constant luminance level. We compare the sur-
face of this luminance function to the surface of a cuboid with holes that are shaped like
the polyhedra. The trick is that the latter surface has a genus which is determined by the
number of holes in the cuboid and which can be determined by the integration of the
local curvature according to Eq. (7). If we can find the corresponding contributions of
169
the integral on the image surface, we can use this integral to count the number of ob-
jects. We assume the corners to be locally sufficiently smooth such that the surfaces are
Riemannian manifolds. The Gaussian curvature K then is zero almost everywhere except
on the corners. We hence have to consider only the contributions of the corners. It turns
out that these contributions can be computed from the elliptic regions only if we use dif-
ferent signs for upwards and downwards oriented elliptic regions. We thus introduce the
following operator which distinguishes the different types of ellipticity in the luminance
function. Let l1 l2 , then the operator N(x, y) := | min(0, l1 (x, y))| | max(0, l2 (x, y))|
is always zero if the surface is hyperbolic and has a positive sign for positive elliptic-
ity and a negative one for negative ellipticity. We thus can calculate the numerosity fea-
ture which has the ability of counting objects in an image by counting the holes in an
imaginary cuboid as follows:
Z
N(x, y)
F= 3 d(x, y). (8)
W (1 + lx (x, y)2 + ly (x, y)2 ) 2
The crucial feature of this measure are contributions of fixed size and with appropriate
signs from the corners. The denominator can thus be replaced by a neural gain control
mechanism and an appropriate renormalization. For the implementation here we use a
shortcut which gives us straight access to the eigenvalues. The numerator D(x, y) of (6)
can be rewritten as
1 1 1
D(x, y) = lxx lyy (luu lvv )2 = [(lxx +lyy )2 ((lxx lyy )2 + (luu lvv )2 )] = (Dl 2 e 2 )
4 4 | {z } 4
=:e 2
(9)
with u := x cos(p/4) + y sin(p/4) and v := x sin(p/4) + y cos(p/4). The eigenvalues
then are l1,2 = 12 (Dl ± |e|) and we can directly use them to compute N(x, y). Application
of this computation to a number of test images is shown in Fig. 7.
1 1
0.9 0.9
50 0.8 50 0.8
0.7 0.7
100 0.6 100 0.6
0.5 0.5
150 150
0.4 0.4
0.3 0.3
200 200
0.2 0.2
0.1 0.1
250 250
0 0
50 100 150 200 250 50 100 150 200 250 quadrat c04a
image1s−1 image3s−4 v rechteckstruktur01c
20 20 20
40
15 15 15
30
50 50 50 50
10 10 10
20
100 5 100 5 5
100 100 10
0 0 0 0
150 150 150 150
−5 −5 −5 −10
−20
−10 −10 −10
200 200 200 200
−30
−15 −15 −15
−40
250 250 250 250
−20 −20 −20
50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250
1.0 2.99 4.0 1.01
Figure 7. Based on a close relation to topological invariants the spatial integration of local curvature fea-
tures can yield highly invariant numerosity estimates. The numerical values in the last row are the normalized
integrals of the filter outputs (middle row).
170
5. Conclusion
Recent evidence shows that ensemble properties play an important role in perception and
cognition. In this paper, we have investigated by which neural operations and on which
processing level statistical ensemble properties can be computed by the cortex. Compu-
tation of a probability distribution requires indicator functions with different sensitivi-
ties, and our reinterpretation of cortical gain control suggests that this could be a basic
function of this neural mechanism. The second potential of cortical gain control is the
computation of AND-like feature combinations. Together with the linear summation ca-
pabilities of neurons this enables the computation of powerful invariants and summary
features. We have repeatedly argued that AND-like feature combinations are essential
for our understanding of the visual system [27,30,34,35,36,28]. The increased selectivity
of nonlinear AND operators, as compared to their linear counterparts, is a prerequisite
for the usefulness of integrals over the respective responses [30,28]. We have shown that
such integrals of AND features are relevant for the understanding of texture perception
[37], of numerosity estimation [30], and of invariance in general [28]. Recently, integrals
over AND-like feature combinations in form of auto- and cross-correlation functions
have been suggested for the understanding of peripheral vision [4,16,17].
A somewhat surprising point is that linear summation and cortical gain control, two
widely accepted properties of cortical neurons, are the only requirements for the com-
putation of ensemble properties. These functions are already available at early stages of
the cortex, but also in other cortical areas [23]. The computation of ensemble properties
may thus be an ubiquitous phenomenon in the cortex.
Acknowledgement
This work was supported by DFG, SFB/TR8 Spatial Cognition, project A5-[ActionSpace].
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