Last month I posted a query to CVNet that began with the following:
I'd like to understand the current concensus about the suitability of the 2D
Gabor function model of simple cell receptive fields...
Enclosed below is a collection of the replies that I received, listed in
alphabetical order by author name. Each message is separated by a line of
'%' characters. Comments enclosed in square brackets '[' and ']' are mine.
A few author's replies have been omitted by prior arrangement.
I would like to express my gratitude to the many people that replied:
Curtis L. Baker Janus Kulikowski
A. B. Bonds William Levick
David Burr Golshah Naghdy
Matteo Carandini Kevin O'Regan
Greg DeAngelis Stijn Oomes
Allan Dobbins Andrew Parker
Tom Freeman Philip Quinlan
Peter Foldiak Adam Reeves
Keith Grasse Eero Simoncelli
Lewis O. Harvey, Jr. Susan te Pas
Judd Jones Simon A. J. Winder
Anupam Joshi Florentin Woergoetter
Peter Kovesi Richard A. Young
Jun Zhang
I started out attempting to write an overview to summarize the replies, but
decided against it, because opinions were varied enough that it would be
difficult to summarize them without introducing a bias. Instead I think it
is better (and easier for me :-)) to let the messages speak for themselves.
Everyone: thank you again,
Mike Hucka E-MAIL: michael.hucka@umich.edu
University of Michigan AI Lab <URL: http://ai.eecs.umich.edu/people/hucka/>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: "Curtis L. Baker" <curtis@spot.vision.mcgill.ca>
Date: Tue, 9 May 1995 14:54:19 -0400
Hi,
In my experience of recording from visual cortex receptive fields,
I would not want to place a very great deal of faith in any one form of
mathematical model of them - they vary quite a lot one from another, in
many ways. I think any of the popular models are probably equally good,
as far as they go - the problem is that receptive fields actually are
nonlinear, in various ways, and these kinds of models are all linear
convolution operators, perhaps followed by simple intensive nonlinearities.
A simple example of how this might be wrong is seen in the currently
popular "contrast normalization" models, such as those of Heeger, in
which a given cortical neuron gets excitatory drive from a linear spatial
operator, but also gets nonlinear (divisive) inhibition from a pool of
other neurons having receptive fields at the same location but tuned
to different orientations, spatial frequencies, etc; such a structure
can give rise to various nonlinear properties. I suspect there may be
a number of such nonlinear twists to the linear model metaphor, which
altogether will make it seem a bit irrelevant to argue much about
the alternative linear models which you mention. For practical
purposes, I would choose whichever one is most convenient for
practical reasons (eg, mathematical properties, etc).
Hope this helps,
Curtis Baker
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: "A. B. Bonds" <ab@vuse.vanderbilt.edu>
Date: 08 May 1995 10:12:26 -0500
Yes, well I find it amusing that the most ardent defenders of the
Gabor function model are those that record from computers and not
critters....
I would suggest that you have a look at Sun & Bonds (1994),
Two-dimensional receptive field organization in striate cortical
neurons of the cat, Visual Neuroscience 11, 703-720. The bottom line
is that sometimes--perhaps as much as 10-15% of the time--a Gabor
function model is spot on. But most of the time it's not. The
problem has been that usually receptive fields have been mapped under
the assumption of one-dimensional uniformity (more or less), i.e.,
with bars or one-dimensional gratings, and these receptive fields are
not one-dimensionally uniform. The mapping method constrains
the result and you end up thinking that they are. (When your only
tool is a hammer, everything ends up looking like a nail....)
There are only three groups (best of my knowledge) who have tried
real two-dimensional mapping, Heggelund, Palmer and us. Both
Heggelund and Palmer used flashed spots. Problem is that simple
cells, though spatially linear (sort of) are not _amplitude_ linear,
they have a real threshold, as well as lots of inhibitory regions.
Mapping with a flashing spot give you only the tip of the iceberg,
which appears unstructured and could be imagined to resemble Gabor
functions. We used a flashing line at many orientations and
reconstructed the field via filtered back-projection (cf. CAT scans).
This improved the threshold problem and also revealed the purely
inhibitory areas. Bottom line is that even simple cells are (usually)
far more structured than we thought. You could take one of our
reconstructed fields, that looks kind of like the Rocky Mountains, and
"map" it (response-plane style) with a flashed line at one orientation
and voila! Sort of a Gabor function.
You don't say why you need the info you ask for. If you are trying to
do a cortical model, I'm afraid I can't really help, because there
seems to be no simple analytical description of even simple cell
fields. But then, is that really surprising?
If you are going to be at ARVO I would be glad to chat with you on
this issue.
Regards A. B. Bonds
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: David Burr <dave@neuro.in.pi.cnr.it>
Date: Mon, 8 May 1995 12:32:08 -0700 (PDT)
I think the Gabor is probably a reasonable approximation where
receptive fields have a few numbers of cycles. However, when they get
small (compared with preferred periodicity) there are problems: odd and
even have quite different bandwidth for example. This is apparent for
motion detectors of very low preferred spatial frequency. Dogs are fine,
but we found the easiest functions to use were "log-Gabor", Gaussian
frequency distributions as a function of log-SF. Here is a list of
references -- I can't remember which are most relevant, but I think the
Morrone and Burr gives full equations of the band-pass we prefer.
Hope this may be of some use.
David Burr
Anderson, S.J & Burr, D.C. (1989) Receptive field properties of
human motion detector units inferred from spatial frequency
masking. Vision Res. 29 1343-1358.
Anderson, S.J & Burr, D.C. (1991) Receptive field length and width
of human motion detector units: spatial summation. J. Opt. Soc. Am. A
8 1330-1339.
Anderson, S.J. & Burr, D.C. (1985) Spatial and temporal
selectivity of the human motion detection system Vision Res. 25
1147-1154.
Anderson, S.J. & Burr, D.C. (1987) Receptive field sizes of human
motion detectors Vision Res. 27 621-635.
Anderson, S.J., Burr, D.C. & M.C. Morrone (1991) The
two-dimensional spatial and spatial frequency properties of motion
sensitive mechanisms in human vision. J. Opt. Soc. Am. A 8
1340-1351.
Morrone, M.C. & Burr, D.C. (1988) Feature detection in human
vision: a phase dependent energy model. Proc. R. Soc. (Lond) B235
221-245.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: matteo@cns.NYU.EDU (Matteo Carandini)
Date: Fri, 12 May 95 07:50:16 EDT
you should read
@article{Hawken87,
author = "M J Hawken and A J Parker",
title = "Spatial properties of neurons in the monkey striate cortex",
journal = "Proc. R. Soc. Lon. B",
volume = 231,
pages = "251--288",
year = 1987}
-MC
*
Matteo Carandini
Center for Neural Science
New York University
6 Washington Place, #809
New York NY 10003
vox: (212) 998 7898
fax: (212) 995 4011
matteo@cns.nyu.edu
*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: gregd@monkeybiz.stanford.edu (Greg DeAngelis)
To: michael.hucka@umich.edu
Date: Mon, 8 May 1995 12:09:38 -0700 (PDT)
Hi Michael,
My name is Greg DeAngelis. I've done a
lot of work in recent years mapping the spatio-
temporal RFs of simple cells, and also fitting
them (this was when I was in Ralph Freeman's lab
in Berkeley, which was until last month).
Your question about the Gabor model is a good
one. To me, there's two distinct issues, which are
partly confused in your question. One is whether a
Gabor function fits the simple cell RF, which it
clearly does quite well. The second is whether a
Gabor-like RF is somehow optimal for encoding spatial
information; this is more the issue that has been
argued back and forth by people like Daugman, Stork,
and Klein.
Of course, a Gabor function is not the only
function you could come up with to fit simple RFs.
So how you look at this whole issue depends on what
you want to accomplish in your modeling. I like
a Gabor function because it generally fits the data
well (probably as well as just about anything else,
I suspect), and it has "nice" parameters, in the sense
that the parameters have physiological meaning.
Hope this rambling helps. I'd be happy to talk more
to you about this. I've recently been working on
modelling the whole space-time RF of simple cells.
You have to keep in mind that, in general, a particular
Gabor function only describes the spatial structure of
the RF at a single point in time, because most simple
RFs are not space-time separable.
Cheers,
Greg DeAngelis
gregd@monkeybiz.stanford.edu
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: allan@aotus.bio.caltech.edu (Allan Dobbins)
Date: Mon, 8 May 1995 10:14:48 -0700
It is not a current research topic. Simple cell data can generally be
well-fit by Gabor functions, but they can also be well-fit by other functions.
I don't think I've read Stork and Wilson (1990), but I saw their ARVO abstract
that preceded the paper. I don't think they take issue with whether Gabor
functions fit the experimental data well, but rather with if there is any
significance (to visual processing) of the fact that Gabor functions minimize
the joint space---spatial frequency uncertainly for the 2-norm. No one knows
if this is significant. Presumably some other class of functions minimize
uncertainty products for 1-norms, infinity-norms etc. By the mid-80s wavelets
was the topic of interest to theoreticians. However, the properties generally
selected for wavelet basis functions by their creators e.g. orthogonality
is not oa property of the set of visual cortical simple cells. The message is
this: the brain may or may not be interested in subscribing to the pet theoires
(sorry, theories) of mathematicians and engineers (I am an engineer by background
by the way). At present I think it is fair so say that the field has moved on
to other issues. I can't tell you more without knowing what your goals are.
If you simply want to have a reasonable simple cell model in some simulations
Gabor functions are fine. No one will criticize you for that. On the other
hand, if you are hoping to do a research project on a new set of basis functions
to describe simple cells, that is now an area that promises very diminished
returns.
Allan Dobbins
216-76 Division of Biology
Caltech
Pasadena, CA 91125
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: tom@john.Berkeley.EDU (Tom Freeman)
Date: Tue, 9 May 1995 09:19:44 -0800
Whilst I can't help you resolve your problem, you may be interested in
adding to your reference list:
1. Young, R.A. (1985). "The Gaussian derivative theory of spatial vision:
analysis of cortical cell receptive field line-weighting profiles", General
Motors Reserach Publication, GMR-4920 and 1.
2.Young R.A.(1987) "The Gaussian derivative model for spatial vision: I.
Retinal mechanisms", Spatial Vision, 2(4):273-93.
for yet another possible model...
------------------------------------
Tom C.A. Freeman
U.C. Berkeley School of Optometry
360 Minor Hall, Berkeley, CA 94720-2020
Tel: (510) 642-0229
Fax: (510) 643-5109
E-mail: tom@john.berkeley.edu
------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: Peter Foldiak <pf2@st-andrews.ac.uk>
Date: Wed, 10 May 95 09:24:06 BST
Hi, Sorry.
I meant to say that I think there was a paper by
Andrews Parker, Oxford (Hawken & Parker ???)
on Gabor functions. Thay said that you can fit
simple cells with a difference of difference of
Gaussians better than with Gabors. The problem
there is that the DDG has more parameters, so it is
not really a surprise that it can be made to fit
better. I wouldn't think there is a big
difference anyway.
I asked someone to try to find the precise reference.
I will email it to you.
Peter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: Peter Foldiak <pf2@st-andrews.ac.uk>
Date: Wed, 10 May 95 09:24:19 BST
TI: TWO-DIMENSIONAL SPATIAL STRUCTURE OF RECEPTIVE-FIELDS IN MONKEY STRIATE
CORTEX
AU: PARKER_AJ, HAWKEN_MJ
JN: JOURNAL OF THE OPTICAL SOCIETY OF AMERICA A-OPTICS AND IMAGE SCIENCE
1988 Vol.5 No.4 pp.598-605
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: Lew Harvey <lharvey@psych.colorado.edu> (Lewis O. Harvey, Jr.)
Date: Mon, 8 May 1995 08:48:18 -0600
Check out:
Harvey, L. O., Jr., & Doan, V. V. (1990). Visual masking at different polar
angles in the two-dimensional Fourier plane. Journal of the Optical Society
of America A, 7(1), 116-127.
It offers strong psychophysical evidence that Gabor functions fit human
data and that the parameters of these functions are the same as those
describing single cells in monkey.
Lew Harvey
------------------------------------------------------------------------
Lewis O. Harvey, Jr. Department of Psychology
tel: +1 (303) 492-8882 Campus Box 345
fax: +1 (303) 492-2967 University of Colorado
lharvey@clipr.colorado.edu Boulder, Colorado 80309-0345 USA
------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: jov@picoflop.EPM.ORNL.GOV (Judd Jones)
Date: Tue, 9 May 95 13:44:52 EDT
Mike:
> I'd like to understand the current concensus about the suitability of the 2D
> Gabor function model of simple cell receptive fields
As an interested party, I'm not sure I should participate in the debate.
But I'd be most interested in the results!
Regards,
Judd Jones
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: "Anupam Joshi" <joshi@cs.purdue.edu>
Date: Sun, 7 May 1995 19:57:06 -0500
Hi,
I don't believe there is a consensus, although as you rightly point out, most
folks are quite comfortable with Gabor functions. If you look at retinal
ganglion cells, they too are generally regarded as having an RF described by
Gabor like funcions. Some computational modelling work I did with backprop
NNs howeverseemed to fit the old LOG model of Marr better (Joshi & Lee, Biol.
Cyb, V70, no.1). An earlier BCyb (in 93) also had a paper by Gaudino of BU on
a coupled PDE model (derived from Grossbergs work) describing retinal
functinality.
Anupam
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: Peter Kovesi <Peter.Kovesi@sophia.inria.fr>
Date: Thu, 11 May 1995 10:45:02 +0200
Mike,
Someone passed your posting regarding receptive fields to CVnet onto
me. I have had some interest in receptive field models for my work on
detecting image features on the basis of phase congruency. (If you
are interested you can ftp a recent tech report I did from
cs.uwa.edu.au in pub/techreports/95/4.ps.gz) Until recently I have
been using quadrature pairs of oriented 2D Gabor functions. However I
now believe Log Gabor functions may be more appropriate.
For info on Log Gabor functions see the (very interesting) paper by
D. J. Field "Relations between the statistics of natural images and
the response properties of cortical cells" Journal of the Optical
Society of America A. December 1987 pp 2379-2394.
Log Gabor functions have a transfer function that is Gaussian when
viewed on the *log frequency* scale. Unfortunately because of the
singularity with log at 0 I don't know of anyone who has worked out an
analytic expression for the function shape in the spatial domain - one
is reduced to contructing the filter in the frequency domain and then
doing an inverse FFT to see what it looks like. In general they are
similar to Gabor functions - only 'sharper'.
Their most important property as far as I am concerned is that one can
construct arbitrarily broad bandwidth filters and still always
maintain a 0 DC value. With standard Gabor filters the maximum
bandwidth one can have before the DC component becomes excessive is
about 1 octave. The question is, what is the appropriate bandwidth to
use with a log Gabor function? Recently I looked at the variation of
spatial width of the filters with bandwidth. I looked at two measures
of 'width'
1) The width required to represent 99% of the absolute area of the filter
2) The 2nd Moment about the middle of the filter with respect to absolute
value of the filter.
The fascinating thing is that both measures are minimised when the
filter bandwidth is about 2 octaves. This matches well with
psychophysical measurements of the bandwidth of cells. The spatial
width at this bandwidth corresponds roughly to the width of a standard
Gabor function having a bandwidth of 1 octave. Thus it seems log Gabor
functions are good at obtaining broad frequency information over a small
spatial window.
Hope this is of some interest to you. Perhaps you can let me know
what your research interests are regarding receptive fields.
Regards,
Peter Kovesi
Department of Computer Science. The University of Western Australia
(Currently visiting INRIA, France)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: Janus Kulikowski <YMUM21@mh1.mcc.ac.uk>
Date: Thu, 11 May 1995 05:03:24 BST
Dear Mike,
I am just about to leave my lab but I wanted to draw your attention
to the following points:
1/ The Gabor system is linear and at best can be a good approximation
of the simple RF; incidentally 2d model is presented in the following
paper
*Kulikowski jJ, Marcelja S and Bishop PO 1982, Theory of spatial
position and spatial frequency relations in the receptive fields of
simple cells in the visual cortex. Biological Cybernetics 43:187-198
My data considering the fit are published in several papers, among
which three are worth considering:
* Kulikowski&Bishop 1981, Fourier analysis and spatial representation
in the visual cortex, Experientia 37: 160-163 (first data)
* Kulikowski & Vidyasagar T, 1986, Space and spatial frequency ..
Experimental brain Research 64: 5-18. (shows discrepancies ofthe fit)
In brief: depending how enthousiatic you are the Gabor functions can
be considered as good or fair approximation of the simple RFs.
Even if it is not accurate the Gabor theory may provide a conceptual
guidance, and help to formulate testeble predictions (see below).
2/ Irrespective of how good the fit is to a particular RF of a cell
tested in isolation (with optimized stimuli), the cells interact with
one another in a nonlinear manner, as shown by processes of
suppression and facilitation -both nonlinear.
I have desribed these interaction psychophysically, and in cells. Eg:
* Kulikowski & King-Smith PE, 1973, Spatial arrangements of the line,
edge and grating detectors revealed by subthreshold summation. Vision
Res. 13: 1455-1475 gabor equations are shown there to approximate
only narrow band processes.
* A book SEEING CONTOUR AND COLOUR, Eds: Kulikowski Dickinson & Murray
pergamon Press 1989 Oxford, ISBN 0-08-036136-6, contains two chapters
of mine (Liner/nonlinear analysis...) and several others relevant to
the theme plus discussions.
In brief: A purely linear system (even if Gabor approximation was
accurate) could not work since the nervous system uses a different
logic. Recently i lectured on these prob;ems to AI people who put in
doubt even the rationale of calling the simple cells essentially
linear. I argued that the simple=cell system is responsible for
distinct processing of half-tone pictures (which depend also on the
polarity of contrast - we cannot recognise negatives of faces), as
opposed to contour outlining and silhouettes (which do not depend on
polarity or contrast).
Thus all depends which aspects you are interested in (fidelity of
pseudo-linear processing, or simple contour processing).
I would not like to burden you with further details, unless you are
interested specifically with some aspects of thi s problem. Please,
let me know your views on the matter.
Best wishes, Janus Kulikowski.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[This is a second reply from Dr. Kulikowski, in response to a followup
message I sent him about non-linear interactions in early vision.]
From: Janus Kulikowski <YMUM21@mh1.mcc.ac.uk>
Date: Tue, 16 May 1995 08:31:39 BST
This is just a brief reply. I am glad that you are looking into these
problems since I intended to return to it myself, but introducing two
different parameters - time and chroma. Although hasty, my reply is
not hopefully too ambigous, so you can publish summary, if you wish.
I should add that the summary of the psychophysical part (and of
many references which are relevant to the debate) is contained in:
JJ Kulikowski (Chapter 22), What really limits vision? in "Limits of
Vision", eds JJ Kulikowski, V Walsh & IJ Murray, Macmillan Press (UK)
1991, pp.286-329.
Interactions revealed psychophysically & electrophysiologically.
The most relevant to your problem is discussion of the methods, which
distinguish between characteristics of units and interaction between
units. The above chapter attempts to elucidate these problems by
comparing the methods of adaptation (introduced after A Gilinsky by
Blakemore & Campbell 1969) and of masking (introduced by Campbell &
myself 1966). Both are still (mis)interpreted as revealing channels,
whereas they reveal actually interactions between channels.
At first it may seem irrelevant, since the numerical differences may
not be great if the method of subthreshold summation (corrected for
probability) id compared with the methods of masking/adaptaion (after
all if they are many channels their spatial interaction may not be
MUCH different from their individual characteristics), but
conceptually it makes difference when large sets of units are
considered.
All Gabor-like, Difference-of-Gausian, etc. approximations treat
the receptive field in isolation (no account of spatial interactions
with other units).
If other background stimuli are introduced the receptive (or now
response) fields of a cell are substantially modified as numerous
contributors noticed.
References:
Blakemore C & Campbell FW, 1969, On the existence of neurones in the
human visual system selectively sensitive ro the orientation and size
of retinal images, J Physiol. 203, 237-260.
Campbell FW & Kulikowski JJ, 1966, Orientation selectivity of the
human visual system, J Physiol. 187, 437-445.
Kulikowski JJ ,1966, On spatial filtering mechanisms in the visual
system (English translation) Prace Inst. Automat. PAN (Warsaw), 43,
33-43.
Kulikowski JJ 1969, Limiting conditions of visual perception
(English translation), Prace Inst. Automat. PAN (warsaw), 77, 1-133.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: William.Levick@anu.edu.au (William Levick)
Date: Mon, 08 May 1995 18:28:44 +1000
Michael:
Here are a couple more references on receptive field profiles derived
from Silverplatter-Medline. I would be very interested in the results of
your enquiry.
Bill Levick
TI: Initial processing of visual information within the retina and the LGN.
AU: Marcelja-S
SO: Biol-Cybern. 1979 May 2; 32(4): 217-26
ISSN: 0340-7200
PY: 1979
LA: ENGLISH
CP: GERMANY,-WEST
AB: The initial stage of information processing by the visual system reduces
the information contained in the continuous image on the retina into a
discrete set of responses which are carried from the lateral geniculate
nucleus (LGN) to the visual cortex. -- 1. The optimal sampling of the light
intensity distribution in the visual environment is achieved only if each
channel in the visual pathways carries undistorted information corresponding
to an image element. The visual system approaches as closely as possible the
scheme of optimal spatial sampling, retaining the full information on the
low spatial frequency content of the object light intensity. The ideal
receptive field of a sustained LGN cell is then of the form J1(Kr)/Kr. -- 2.
The experimentally determined receptive fields of sustained LGN cells (and
to some extent retinal ganglion cells as well) in cat closely resemble the
functional form J1(Kr)/Kr. The centre-surround organization of the receptive
fields is therefore understood as a scheme which leads to a maximal
information flow through the visual pathways. -- 3. The optimal sampling
scheme cannot be realized by the retina alone, because of restrictions on
the size of neural networks. It is therefore constructed in two stages,
ending at the LGN level. A recombination of ganglion cell signals into
optimal receptive fields is a major role of the LGN.
MESH: Fourier-Analysis; Ganglia-physiology; Photoreceptors-physiology;
Visual-Acuity; Visual-Cortex-physiology; Visual-Pathways-physiology
MESH: *Geniculate-Bodies-physiology; *Models,-Neurological;
*Models,-Psychological; *Retina-physiology; *Visual-Perception
TG: Animal; Human
PT: JOURNAL-ARTICLE
AN: 79209767
UD: 7911
TI: Theory of spatial position and spatial frequency relations in the
receptive fields of simple cells in the visual cortex.
AU: Kulikowski-JJ; Marcelja-S; Bishop-PO
SO: Biol-Cybern. 1982; 43(3): 187-98
ISSN: 0340-7200
PY: 1982
LA: ENGLISH
CP: GERMANY,-WEST
AB: Striate cells showing linear spatial summation obey very general
mathematical inequalities relating the size of their receptive fields to the
corresponding spatial frequency and orientation tuning characteristics. The
experimental data show that, in the preferred direction of stimulus motion,
the spatial response profiles of cells in the simple family are well
described by the mathematical form of Gabor elementary signals. The product
of the uncertainties in signaling spatial position (delta x) and spatial
frequency (delta f) has, therefore, a theoretical minimum value of delta x
delta f = 1/2. We examine the implications that these conclusions have for
the relationship between the spatial response profiles of simple cells and
the characteristics of their spatial frequency tuning curves. Examples of
the spatial frequency tuning curves and their associated spatial response
profiles are discussed and illustrated. The advantages for the operation of
the visual system of different relationships between the spatial response
profiles and the characteristics of the spatial frequency tuning curves are
examined. Two examples are discussed in detail, one system having a constant
receptive field size and the other a constant bandwidth.
MESH: Cats-; Mathematics-; Models,-Neurological
MESH: *Visual-Cortex-physiology; *Visual-Perception
TG: Animal; Human
PT: JOURNAL-ARTICLE
AN: 82232251
UD: 8211
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: Golshah NAGHDY <golshah@elec.uow.edu.au>
Date: Tue, 9 May 1995 10:08:03 +1000 (EST)
Dear Michael,
I read with interest your posting to CVnet regarding Gabor filters and RF of
simple cells. I have been working on machine vision using biological vision
as a model for a while now. I have used Gabor filters as feature detectors
for natural and artificial texture classification. I have also used Gabor
filters for pre-attentive vision. I have, however, relied on Jones and Palmer's
work on similarity of Gabor filters and RF of simple cells.
I would be very interested in the outcome of your request. I will be grateful
if you could share it with me.
Regards
Golshah
Dept. of Elec and Computer Engineering
University of Wollongong
Australia
g.naghdy@uow.edu.au
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: oregan@pathfinder.cbr.com (Kevin O'Regan)
Date: Sun, 7 May 95 18:53:19 -0400
I believe there's an article by Young in Spatial vision about 2 or 3 years
ago saying that receptive fields are also accurately modelled by
derivatives of gaussians...
Kevin O'Regan
currently at: Nissan Cambridge Basic Research, 4 Cambridge Center,
Cambridge, Mass 02142. Tel. (617) 374 9671. Fax: (617) 374 9697
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: oomes@NICI.KUN.NL (Stijn Oomes)
Date: Mon, 08 May 1995 09:40:49 +0100 (MET)
Hi Michael, I think the best sources these days are:
Receptive field families
Koenderink & van Doorn
Biolological Cybernetics, 63, 291-297 (1990)
Generic neighborhood operators
Koenderink & van Doorn
IEEE Transactions on Pattern Analysis and Machine Intelligence
14(6), 597-605 (1992)
They find that the Gaussian and its derivatives model the sensitivity
function of receptive fields and deduce their results from first principles.
These ideas started a whole new branch of science called 'scale-space
theory' that has wide applications, but is particularly usefull in the
fields of human- & machine-vision.
Regards, Stijn Oomes
@@@@@@@@@@@@@@@@@@@@@@@@@
Stijn Oomes
University of Nijmegen
Nijmegen Institute for Cognition and Information
Division of Perception
P.O. Box 9104, 6500 HE Nijmegen
tel. + 31 80 615698, fax + 31 80 616066
@@@@@@@@@@@@@@@@@@@@@@@@@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From: Andrew Parker <ajp@physiol.ox.ac.uk>
Date: Mon, 8 May 1995 13:48:36 +0100
Dear Mike,
I saw your enquiry on CVnet. You may like to look
up the paper that Mike Hawken (now at NYU) and I
wrote in which we compareed contrast sensitivity
functions for spatial frequency from neurons
in monkey V1 against various models for their
receptive field structures, including Gabor and DOG
and other more esoteric variants. We came in for
some flack for discussing models with large numbers of
free parameters after publishing that paper, but this
slightly misrepresents our intentions. The data in
Fig 14 of our paper may be most relevant for your
question.
The reference is:
Proceeding of the Royal Society of London, Series B,
231, 251-288 (1987)
"Spatial properties of neurons in the monkey striate cortex."
Andrew
-- Andrew Parker (ajp@uk.ac.oxford.physiology) University Laboratory of Physiology Parks Road, Oxford, OX1 3PT +44 865 272504 +44 865 272469 (FAX)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [This is a second reply from Dr. Parker, in response to a followup message that I sent him. I asked his opinion regarding the lack of polar separability in spatial frequency + orientation tuning of simple cells and its significance for being able to characterize the selectivity of a cell.]
From: Andrew Parker <andrew.parker@physiol.ox.ac.uk> Date: Fri, 12 May 1995 10:09:13 +0100
Dear Mike,
Thanks for looking into our work and taking account of it.
You asked about separability. This is important for a full 2-D evaluation of a model for cortical cells and I accept the point that 1-D tuning can only partly characterize the selectivity of the cell. However, this does not mean that all 1-D experiments can be discarded. To reject a model only requires an inconsistency with experimental results, whereas to accept a model as final would require support from several kinds of experiment (i.e. 1-D and 2-D).
In the particular case that we studied, we looked at neurons after we had determined the peak orientation tuning. Thus we always chose a consistent point of the orientation tuning curve at which to collect our data for contrast sensitivity as a function of spatial frequency. The Gabor model makes specific predictions about the shape of this 1-D spatial frequency tuning and, although we did not develop the algebra in 2-D form in the paper, a 1-D slice through the 2-D model at the peak orientation give essentially a (complex) pair of Gaussian spectra in the frequency domain. After that is established, I think the rest of our 1-D treatment is sufficient to ask whether the Gabor model can be rejected.
There are two senses in which a Gabor model could be rejected. The first is that it is statistically inadequate to describe the experimentally-determined responses. This is the criterion that Jones and Palmer used and that we applied in Fig 11 of our paper (comparison of models against variance of data). The other way in which one might wish to reject any model, including a Gabor, is to say that there is an equally powerful model that can better account for the data. This is what we explored in Fig 14 by comparing the difference-of-Gaussian fits against the Gabor fits.
Although Jones and Palmer were able to accept the Gabor model for their spectral data by the first criterion, there is one indication in their paper that the Gabor model is stretched to account for all the spectral data. This is its tendency to predict an unusually high proportion of fits in which the phase parameter is at 90 degrees (odd symmetry). We noted this tendency too on pp 269-270 of our paper and its relates to the poor ability of the Gabor to account for the low frequency portion of the contrast sensitivity function of the cells, as you mentioned in your last reply.
Of course, nothing that Mike Hawken and I did can address the question of experimental measures of the space-domain properties of the cells in V1 and this is the great advantage of the Jones and Palmer study. But it would be reasonable to ask the following question of their space domain measurements. Granted that the Gabor model may give a statistically adequate description of the data, are there any other models with similar numbers of parameters that can give at least as good (or even a better) account of the space domain data?
There are of course a number of other differences between Jones and Palmer's study and ours that may be responsible for the different conclusions. We measured threshold sensitivity curves (less affected by suprathreshold nonlinearities), rather than using response measures. We worked in primate cortex, not cat.
Best of luck in trying to sort this out. I look forward to a summary. Please use any of these comments as you deem suitable.
Andrew
-- Andrew Parker (ajp@uk.ac.oxford.physiology) University Laboratory of Physiology Parks Road, Oxford, OX1 3PT +44 865 272504 +44 865 272469 (FAX)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% From: Philip Quinlan <ptq1@mailer.york.ac.uk> Date: Mon, 15 May 1995 14:43:49 +0100
> I'd like to understand the current concensus about the suitability of the 2D > Gabor function model of simple cell receptive fields, and I hope that > someone on this list can steer me in the right direction.
You might be interested in Hawken, M. J. & Parker, A. J. (1987). Spatial properties of neurons in the monkey striate cortex. Proc. R. Soc. Lond. B231, 251-288.
Sincerely, Philip.
----------------------------------------------------------------- Dr. Philip Quinlan E-Mail: ptq1@york.ac.uk Department of Psychology FAX: (0904) 433181 University of York Tel: (0904) 430000 Ext. 3135 Heslington Direct line: (0904) 433135 York Telex: 57933 YORKUL YO1 5DD U.K. -----------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% From: eero@tarpon.cis.upenn.edu (Eero Simoncelli) Date: Wed, 10 May 95 00:59:03 EDT
[On problems with Gabor function models:]
2) They don't tile the frequency domain evenly. Even worse, they cannot be used to give an unbiased estimate of velocity (except in the limit of infinite numbers of cells). For this, one wants to use filters that are DERIVATIVES of some common function (eg, a Gaussian).
The latter issue is part of my PhD thesis, and will be available shortly as a journal paper preprint. Send me your address if you'd like a copy.
Eero Simoncelli Assistant Professor Computer and Information Science Department University of Pennsylvania
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% From: S.F.tePas@fys.ruu.nl (Susan te Pas) Date: Mon, 8 May 1995 09:35:35 +0100
>I'd like to understand the current concensus about the suitability of the 2D >Gabor function model of simple cell receptive fields, and I hope that >someone on this list can steer me in the right direction.
I feel that you should also look at the Gaussian model that was presented by Koenderink and van Doorn. I've included some references, but they have published much more about this topic. I think that Jan Koenderink is on the CV-net mailing list, so if you have any questions you might ask him yourself.
Yours Sincerely,
Susan te Pas
REFERENCES J.J. Koenderink and A.J. van Doorn. Representation of Local Geometry in the Visual System. Biological Cybernetics 55, 367-375 (1987)
J.J. Koenderink and A.J. van Doorn. Receptive Field Families. Biological Cybernetics 63, 291-297 (1990)
J.J. Koenderink. The Brain a geometry engine? Psychological Research 52, 122-127 (1990)
J.J. Koenderink and A.J. van Doorn. Receptive Field Assembly Pattern Specificity. Journal of visual communication and image representation vol. 3 no.1 pp 1-12 (1992)
*********************************************************** // Susan F. te Pas \\ \\ // // Helmholtz Instituut, Universiteit Utrecht \\ \\ Princetonplein 5, 3584 CC Utrecht, The Netherlands // // \\ \\ phone: +31 30532809;fax: +31 30522884 // // E-mail: S.F.tePas@fys.ruu.nl \\ ***********************************************************
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% From: "S. A. J. Winder" <sajw@maths.bath.ac.uk> Date: Mon, 8 May 95 13:07:31 BST
> I'd like to understand the current concensus about the suitability of the 2 > Gabor function model of simple cell receptive fields, and I hope that > someone on this list can steer me in the right direction.
The Gabor function is a useful first approximation to simple cell receptive fields. However, there are two main problems, as far as I can see:
Firstly, the bandwidth of most cortical filters, judged by electrophysiology, is of the order of 1.0 -- 2.5 octaves. Gabor filters with even phase and bandwidth greater than about 1 octave show a very large response to diffuse illumination (DC brightness) and this is not true for cortical responses. In other words, Gabor filters have a low frequency roll-off which is too slow. This lead some people to modify the function when simulating this stage using image processing techniques, e.g. Heitger et al. (1992), Winder (1995).
Secondly, the high frequency roll-off of real cortical filters often has a Gaussian low pass shape because the neurons receive drive from retinal units with centres that have a Gaussian profile. The Gabor function falls off too fast at high frequencies because its high frequency limb is made from a Gaussian that it centred around a frequency greater than zero. (It should also be remembered, however, that cortical inhibition acts to sharpen spatial frequency selectivities.)
A useful comparison between the contrast sensitivity functions of cortical neurons and those predicted by a number of theoretical models, e.g. Gabor, difference-of-Gaussians, can be found in Hawken and Parker (1987).
I hope this helps,
Simon
Refs ====
F. Heitger, L. Rosenthaler, von der Heydt, R., E. Peterhans and O. K\"{u}bler. Simulation of Neural Contour Mechanisms: From Simple to End-stopped Cells. Vision Research. 1992, Vol. 32, pp 963--981.
S. A. J. Winder. From Cones to Contours: A Parallel Simulation of Neural Mechanisms in the Primate Vision System. PhD Thesis. School of Mathematical Sciences, University of Bath, UK. 1995.
M. J. Hawken and A. J. Parker. Spatial Properties of Neurons in the Monkey Striate Cortex. Proceedings of the Royal Society of London, Series B. 1987, Vol. 231, pp 251--288.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Simon A.J. Winder *********** Vision Research *********** sajw@maths.bath.ac.uk ** University of Bath Computing Group ** Tel: +44 (0)1225 826183 http://www.bath.ac.uk/~mapsajw/home.html
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% From: worgott@smart.neurop.ruhr-uni-bochum.de (Florentin Woergoetter) Date: Wed, 10 May 1995 10:00:06 +0000
Check also out Hawken and Parker Proc Roy Soc. Lond. B 231, 251-288, 1987
F. Woergoetter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% From: young@predator.cs.gmr.com (Richard A. Young) Date: Fri, 26 May 95 14:07:51 -0400
I have published 4-6 papers on this topic and argue that the Gaussian derivative function better describes simple cell receptive fields than Gabor functions. I have talked several times at UM but not in last several years. Perhaps you would consider inviting me out again? Also, I can send you papers. - Dick
Richard A. Young Phone: 810-986-1471 (GM:8-226-1471) Computer Science (AP50) FAX: 810-986-9356 GM Research & Devt. Center Internet: young@gmr.com Warren, Mich. 48090-9055 GM_PROFS: RYOUNG--GMRCMSA
From: young@predator.cs.gmr.com Date: Tue, 6 Jun 95 09:01:14 -0400
>Hi Dr. Young, > >I'd like to post a summary to CVNet of the replies I received about the Gabor >question. Would you be willing to send me a short list of your recent papers >on the subject? I think a number of people would be interested in seeing it. > >Thanks, >Mike
R. A. Young, "The Gaussian derivative theory of spatial vision: Analysis of cortical cell receptive field line-weighting profiles," GM Research Laboratories Technical Pub. #4920, May 28 (1985).
R. A. Young, "The Gaussian derivative model for machine and biological image processing," General Motors Research Laboratories Technical Publication #5128, August (1985).
R. A. Young, "The Gaussian derivative model for machine vision: visual cortex simulation," General Motors Research Laboratories Technical Publication #5323, July 7 (1986).
R. A. Young, "The Gaussian derivative model for spatial vision: I. Retinal mechanisms," Spatial Vision 2, No. 4, 273-293 (1987).
R. A. Young, "Oh say, can you see? The physiology of vision," In Human Vision, Visual Processing, and Digital Display II," Proceedings SPIE, 1453 (1991).
R. A. Young, "A physiological model of motion analysis for machine vision," In Human Vision, Visual Processing, and Digital Display IX, Proceedings SPIE, 1913 (1993). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%