Biol. Cybernetics 21, 29--35 (1976) (~)by Springer-Verlag1976
Geometry of Binocular Vision and a Model for Stereopsis J. J. Koenderink and A. J. van D o o r n Dept. Medical and Physiological Physics, Physics Laboratory, State University Utrecht, Utrecht, The Netherlands Received: March 29, 1975
Abstract
pieces (supposed to be present in the surface detail) in the retinal images of the left and right eye. This If a binocular observer looks at surfaces, the disparity is a continuous vector field defined on the manifold of cyclopeanvisual method is insensitive to differences in magnification directions. We derive this field for the general case that the observer between the eyes (uniform aniseikonia), or differences is presented with a curved surface and fixates an arbitrary point. in orientation of the eyes around the visual axes We expand the disparity field in the neighbourhood of a visual (differential cyclotorsion). direction. The first order approximation can be decomposed into Any mechanism that is proposed to explain congruences, similarities and deformations. The deformation component is described by the traceless part of the symmetric part of stereopsis should allow for the fact that it is easy to the gradient of the disparity. The deformation component carries fuse two pictures even if one is up to 15 % larger, or all information concerning the slant of a surface element that is if one of them is rotated up to 6 degrees (Julesz, 1971). contained in the disparity field itself; it is invariant for changes of Some well known models do not tolerate even the fixation, differential cyclotorsion and uniform aniseikonia. The slightest rotation or scale change [e.g. Julesz's (1971) deformation component can be found from a comparison of the orientation of surface details in the left and right retinal images. automap-1 or Dodwell and Engel's (1963) model]. The theory provides a geometric explanation of the percepts ob-. This has incited some authors to postulate zooming tained with uniform and oblique meridional aniseikonia. We utilize mechanisms and image rotators, or even mechanisms. the geometric theory to construct a mechanistic model of stereopsis with the plasticity of "rubber sheets" or "jelly" to be that obviates the need for internal zodming mechanisms, but nevertheless is insensitive to differential cyclotorsion or uniform present in the cortex (Ogle, 1962; van Heerden, 1970; aniseikonia. Julesz, 1971). Julesz (1971) remarks about this that 1. Introduction correlators based upon such mechanisms would take an unduly long time to fuse a pattern (in the case of The content of this paper divides naturally into automap-1) or would fail to explain the percepts two parts: the first part treats the geometry of binocular evoked by uniform or meridional aniseikonia (in the vision, in the second part we describe a mechanistic case of the magnetic dipole model). (but necessarily speculative) model of stereopsis. It follows from our geometric analysis that the In binocular vision the left and right eye catch deformation component of the disparity field (the sight of different aspects of objects in visual space. traceless part of the symmetric part of the gradient of We explore the difference in the natural perspective the disparity) is not at all sensitive to uniform (natural perspective describes the angle relations expansions or rotations. If we exploit this fact, the between visual directions) in the case an observer need for an internal zooming mechanism is obviated. looks at surfaces. If the object of vision is a surface, It is shown that the deformation component of the instead of a set of points or lines, we obtain for the disparity field can be computed by receptive fields difference in natural perspective a continuous vector that are sensitive to the orientation of line pieces, field defined on the manifold of visual directions. without bringing the left and right images into exact It is in this respect that our analysis differs from the register. usual treatment. The perceptual facts that relate to uniform and We obtain the first order approximation (the meridional aniseikonia are well explained by the model. gradient) of the difference field, and decompose it into elementary transformations (congruences, simi2. Geometry of Binocular Vision larities and deformations). It is shown that all information contained in the difference field, is also 2.1. Eye Movements contained in the deformation component alone. The In the sequel we assume that Donder's and deformation component can be found from a comparison of relations between the orientations of line Listing's law are valid for both eyes separately. This
30 assumption permits the derivation of a closed formula for the disparity field. For the validity of the model discussed in Section 3 these assumptions are not necessary. Even large deviations from Donder's or Listing's law are irrelevant to our final conclusions (Section 2.3). Donder's law states that the eye utilizes only two of its possible three degrees of freedom. Listing's law states explicitely that the axis of rotation that describes the result of an eye movement lies in the equatorial plane. (The plane through the center of rotation of the eye, parallel to the fronto-paralM plane.) The assumption that Listing's law is valid for both eyes separately, suffices to find the differential cyclotorsion of the two eyes (Appendix I).
(e0, e~oare unit vectors in the directions of increasing 0, (p respectively). At the fixation point the disparity vanishes: 2(00, (Po; 0o, ~Oo)=0. Take two infinitesimal close but otherwise arbitrary points Q1, Q2 on the surface Q=O(0, q~). We transport the disparity vector at gQ~ parallel to itself (in the sense of Levi-Civita) to ~Q~ and form the difference field (ge~, ge~ are unit vectors in the cyclopean visual directions of respectively Q~, (22):
= - ( V ~ ' c~) ffang+ praa + (sin0 (/sincPoCOS~Oo(l - cos0o) cos ~o
2.2 The Disparity We use a coordinate system with the origin at the cyclopean eye (a point midway between the centers of rotation of the left and fight eye). The x-axis points from the right to the left, the y-axis vertically upwards, and the z-axis in the primary direction (straight ahead). Polar coordinates (~, 0, (p) are introduced in the usual manner (0 = 0 corresponds with the primary direction). If both eyes fixate a point P with polar coordinates (~o, 0o, (Po), the coordinates of a point (2 [-polar coordinates (0, 0, cp)] are different as seen from the position of the left or right eye. The difference, to first order in l/~ (1 is the distance between the centers of rotation of the eyes), is derived in Appendix I. Because the eyes register only angle relations, and not distances, the angular part of this difference suffices. This is called the disparity 0~). The disparity is a displacement vector defined on the manifold of cyclopean visual directions. Using the results of Appendix I we find:
(( cos0cos + - - cosq~ 00
I I-c~176
0o
sin0o
(2)
d~ = (02 - 0060+ sin01(q)2 - (p1)d~o~o
(3)
/~ang and Prad are the angular and radial part of the vector lgx; and
is the gradient of the reciprocal distance to the surface. At the fixation point we have:
A2(go; d?) = g(go + ~ ; go)
l sinOsincp (O - ~) ~ . 1 +cos0 \1 (5)
- - sinqosinCpoCOSCpo(1 - cos0o) 0o + sinZq~
1 - 10) d~
with
-= - ( V ~ ' ~ ) ffang+ prad d~
2(0, e; 0o, Co) =
)
+ - - (cos q~ocos0o + sin2q~o)sincp 0o
Because we choose to regard a surface the disparity field is a continuous vector field.
) eo
+ (~ sincp---001cos0cos~osinqoocoscpo(1-cos00)
2.3. Decomposition of the Disparity Field l Qo -
~/sin0
~o
The disparity field (1) can be written:
cos0sinqo(cosZcPoCOS00+ sin2~Oo) 1-cos00
) ) sinq) 0 ge
(1)
~2s 2 f f + s ~o)=2ff; go)+ ~
...
(6)
31
with
0~ = (aj_ 1 al2 ] ~;~ \ a 2 1 a22)'
(7)
a11=-
(8)
000 cos0cos(p+ -/sin0cos(p,0
A
1 ( )cos0cos(p
a12-
sin0
l _ __ sin(poCOS(po(1- cos00)sin0cos(p 00 l _ __ (cos2(poCOS00+ sina(po)sin0sin(p 00 l 1-cos00 . + sm(poCOS0, 00 sin00
L
(9)
C
R
Fig. 1. A binocular observer is confronted with a plane surface p. The plane p is perpendicular to the x-z plane (horizontal plane), x-axis: direction from the right to the left eye, z-axis: primary direction, L: left eye, R: right eye, C: cyclopean eye, U: is the point on p that is nearest to C
a21 = ~
sin(p + - - sin(poCOS(po(1-cos00)sin0cos(p 00 l 2 + - - (cos (poCOS00+ sin2(po)sin0sin(p 00 1 1-cos00 - sin(pocos0 (10) 00 sin00
a22= s~n0 ~ 0
sin(p+ 0 sin0cos(p.
We write ~~)~ =
(~0 2 )++ ( ~ ) w h e_,r e
symmetric, ( ~ ) Since ~ - -
(11) ( ~ ) i +s
the
is the antisymmetric part of ~)~
is symmetric the characteristic values, /
,\
and consequently the trace of ( ~ / \GT]
are real. We write +
(0)~) ~ + = B + C' sO that C has vanishing trace- It fOll~ that B = 89 ( ~ )
"I and C = (0~) _ l t r ( ~ +
) "I.
+
can be written:
We now write:
~(F+~;Fo)=~(f;fo)+ ~
Fig. 2. The lines of expansion (closed curves) and contraction (open curves) of the deformation component of the disparity field [Lines of expansion (contraction) are everywhere tangent to the axis of expansion (contraction)]. A central projection on p (Fig. 1) from the cyclopean eye C was used. The point at the center of the figure is U (Fig. 1); the other point corresponds to V. We took 2 = 60 ~ (Fig. 1)
_dr+Bd~§
(12)
The first and second term describe a translation and a rigid rotation (congruences). The third term describes a uniform expansion or contraction (a simil'arity). The fourth term describes an expansion in one and a contraction in an orthogonal direction with a conservation of solid angle (a deformation). The second term is proportional to the curl, the third to the divergence of the disparity field. The fourth term
C = 89
R-I(~_
01)R.
(13)
R is a rigid rotation that specifies the axes of expansion and contraction, def2 is a (by definition positive) real quantity. A shift of the fixation point affects only the antisymmetric part of the gradient of the disparity field. The symmetric part is invariant under shifts of fixation. It can be shown that the gradient of the reciprocal distance can be found from the traceless part C alone.
32
with ( R=
cos# sin#] - s i n # cos/4"
(16)
# specifies the axes of the deformation field. To a direction 2 in the field of the right eye [a unit vector in the direction 2 is (cos2E0+ sin2E~)] corresponds a direction 2+A2 in the field of the left eye: A2(2) = - 89rot~ + 89def)~sin2(2-#)
Fig. 3. Lines of equal magnitude of the deformation component of the disparity field. A central projection on p (Fig. 1) from the cyclopean eye C was used. The magnitude of the deformation vanishes at the points U (center of the figure, see Fig. 1) and V (Fig. 1). We took 2 = 60 ~ (Fig. 1)
The magnitude of the deformation is def2= lY~ "ll/~angll
(14)
and the axis of contraction bisects the direction of the gradient of the reciprocal distance and the direction from the center of rotation of the right to that of the left eye. Hence, the deformation component of the disparity field taken alone specifies the gradient of the reciprocal distance. Shifts of fixation have no influence. It can be shown that the deformation component possesses even more desirable invariant properties: if we subject the image of one of the eyes to a uniform expansion or a rigid rotation, then only congruences or similarities change, the deformation component is unaffected. By way of an example we show in Figs. 1--3 the deformation component of the disparity field for the case that an observer looks at a slanted surface.
(17)
(to the first order in dive, rot2, def)~). This is an immediate consequence of Eq. (15). If we take two directions 2~, )~2 in the field of the right eye, then we have A(21- 22)= def)~sin(2~- 22)cos2 (2-~2 22
#).
(18)
This difference depends only on the deformation component of the field. It follows that the information contained in the disparity field, can be obtained from the mere comparison of angles in the images of left and fight eyes. This comparison cannot be influenced by magnification differences or differential rotations of the left and right images.
3. A Model for Stereopsis
3.1. Correlation of Left and Right Images Assume that the surface is covered with random textural detail. In that case a certain number of cortical line detectors (Hubel and Wiesel, 1965) will be excited for any small patch of the visual field. For any patch of the visual field we have an excitation as function of the orientation 2 of the line detectors of the following type: E(2)= ~ = 1 6()~-21),
(19)
where the 2i are randomly distributed. If we correlate the excitations of two patches, one 2.4. The Deformation Component Can be Found from from the right, one from the left image, then the the Orientation of Line Pieces resulting crosscorrelation function will almost vanish, The disparity field in the neighbourhood of a except when the left and right images correspond visual direction can be written: to the same patch on the surface. If we crosscorrelate the excitation of a small patch from the right image with the excitation of a large patch from the left ~(~'+ d~; 7o) = ~(r'; ~'o)+ (ldiv~ (~ 01) + l r o t ~ (~ - ~) image (or vice versa) then the resulting crosscorrelation depends principally on the excitation of a certain small patch, contained in the large patch of the left image, + ~ def)~R- 1 (~ _ 01)R) dr (15) that corresponds to the small patch of the right image.
33 In this way left and right images can be compared without bringing them into exact register. It is only necessary that the cortical projection from one of the eyes is more diffuse than that of the other eye. Evidence for such an organization has been presented by Blakemore and Pettigrew (1970). It is possible to show (Appendix II) that the width of the crosscorrelation function equals the magnitude of the deformation component of the disparity field.
3.2. The Directions of the Deformation Axes Because of the crosscorrelation technique, phase information is lost. Because phase information is necessary in order to find the directions of the deformation axes, the crosscorrelation scheme outlined in Section 3.1. has to be altered. Phase information can be retained if we do not crosscorrelate over the interval 0 =
Geometry of Binocular Vision and a Model for Stereopsis J. J. Koenderink and A. J. van D o o r n Dept. Medical and Physiological Physics, Physics Laboratory, State University Utrecht, Utrecht, The Netherlands Received: March 29, 1975
Abstract
pieces (supposed to be present in the surface detail) in the retinal images of the left and right eye. This If a binocular observer looks at surfaces, the disparity is a continuous vector field defined on the manifold of cyclopeanvisual method is insensitive to differences in magnification directions. We derive this field for the general case that the observer between the eyes (uniform aniseikonia), or differences is presented with a curved surface and fixates an arbitrary point. in orientation of the eyes around the visual axes We expand the disparity field in the neighbourhood of a visual (differential cyclotorsion). direction. The first order approximation can be decomposed into Any mechanism that is proposed to explain congruences, similarities and deformations. The deformation component is described by the traceless part of the symmetric part of stereopsis should allow for the fact that it is easy to the gradient of the disparity. The deformation component carries fuse two pictures even if one is up to 15 % larger, or all information concerning the slant of a surface element that is if one of them is rotated up to 6 degrees (Julesz, 1971). contained in the disparity field itself; it is invariant for changes of Some well known models do not tolerate even the fixation, differential cyclotorsion and uniform aniseikonia. The slightest rotation or scale change [e.g. Julesz's (1971) deformation component can be found from a comparison of the orientation of surface details in the left and right retinal images. automap-1 or Dodwell and Engel's (1963) model]. The theory provides a geometric explanation of the percepts ob-. This has incited some authors to postulate zooming tained with uniform and oblique meridional aniseikonia. We utilize mechanisms and image rotators, or even mechanisms. the geometric theory to construct a mechanistic model of stereopsis with the plasticity of "rubber sheets" or "jelly" to be that obviates the need for internal zodming mechanisms, but nevertheless is insensitive to differential cyclotorsion or uniform present in the cortex (Ogle, 1962; van Heerden, 1970; aniseikonia. Julesz, 1971). Julesz (1971) remarks about this that 1. Introduction correlators based upon such mechanisms would take an unduly long time to fuse a pattern (in the case of The content of this paper divides naturally into automap-1) or would fail to explain the percepts two parts: the first part treats the geometry of binocular evoked by uniform or meridional aniseikonia (in the vision, in the second part we describe a mechanistic case of the magnetic dipole model). (but necessarily speculative) model of stereopsis. It follows from our geometric analysis that the In binocular vision the left and right eye catch deformation component of the disparity field (the sight of different aspects of objects in visual space. traceless part of the symmetric part of the gradient of We explore the difference in the natural perspective the disparity) is not at all sensitive to uniform (natural perspective describes the angle relations expansions or rotations. If we exploit this fact, the between visual directions) in the case an observer need for an internal zooming mechanism is obviated. looks at surfaces. If the object of vision is a surface, It is shown that the deformation component of the instead of a set of points or lines, we obtain for the disparity field can be computed by receptive fields difference in natural perspective a continuous vector that are sensitive to the orientation of line pieces, field defined on the manifold of visual directions. without bringing the left and right images into exact It is in this respect that our analysis differs from the register. usual treatment. The perceptual facts that relate to uniform and We obtain the first order approximation (the meridional aniseikonia are well explained by the model. gradient) of the difference field, and decompose it into elementary transformations (congruences, simi2. Geometry of Binocular Vision larities and deformations). It is shown that all information contained in the difference field, is also 2.1. Eye Movements contained in the deformation component alone. The In the sequel we assume that Donder's and deformation component can be found from a comparison of relations between the orientations of line Listing's law are valid for both eyes separately. This
30 assumption permits the derivation of a closed formula for the disparity field. For the validity of the model discussed in Section 3 these assumptions are not necessary. Even large deviations from Donder's or Listing's law are irrelevant to our final conclusions (Section 2.3). Donder's law states that the eye utilizes only two of its possible three degrees of freedom. Listing's law states explicitely that the axis of rotation that describes the result of an eye movement lies in the equatorial plane. (The plane through the center of rotation of the eye, parallel to the fronto-paralM plane.) The assumption that Listing's law is valid for both eyes separately, suffices to find the differential cyclotorsion of the two eyes (Appendix I).
(e0, e~oare unit vectors in the directions of increasing 0, (p respectively). At the fixation point the disparity vanishes: 2(00, (Po; 0o, ~Oo)=0. Take two infinitesimal close but otherwise arbitrary points Q1, Q2 on the surface Q=O(0, q~). We transport the disparity vector at gQ~ parallel to itself (in the sense of Levi-Civita) to ~Q~ and form the difference field (ge~, ge~ are unit vectors in the cyclopean visual directions of respectively Q~, (22):
= - ( V ~ ' c~) ffang+ praa + (sin0 (/sincPoCOS~Oo(l - cos0o) cos ~o
2.2 The Disparity We use a coordinate system with the origin at the cyclopean eye (a point midway between the centers of rotation of the left and fight eye). The x-axis points from the right to the left, the y-axis vertically upwards, and the z-axis in the primary direction (straight ahead). Polar coordinates (~, 0, (p) are introduced in the usual manner (0 = 0 corresponds with the primary direction). If both eyes fixate a point P with polar coordinates (~o, 0o, (Po), the coordinates of a point (2 [-polar coordinates (0, 0, cp)] are different as seen from the position of the left or right eye. The difference, to first order in l/~ (1 is the distance between the centers of rotation of the eyes), is derived in Appendix I. Because the eyes register only angle relations, and not distances, the angular part of this difference suffices. This is called the disparity 0~). The disparity is a displacement vector defined on the manifold of cyclopean visual directions. Using the results of Appendix I we find:
(( cos0cos + - - cosq~ 00
I I-c~176
0o
sin0o
(2)
d~ = (02 - 0060+ sin01(q)2 - (p1)d~o~o
(3)
/~ang and Prad are the angular and radial part of the vector lgx; and
is the gradient of the reciprocal distance to the surface. At the fixation point we have:
A2(go; d?) = g(go + ~ ; go)
l sinOsincp (O - ~) ~ . 1 +cos0 \1 (5)
- - sinqosinCpoCOSCpo(1 - cos0o) 0o + sinZq~
1 - 10) d~
with
-= - ( V ~ ' ~ ) ffang+ prad d~
2(0, e; 0o, Co) =
)
+ - - (cos q~ocos0o + sin2q~o)sincp 0o
Because we choose to regard a surface the disparity field is a continuous vector field.
) eo
+ (~ sincp---001cos0cos~osinqoocoscpo(1-cos00)
2.3. Decomposition of the Disparity Field l Qo -
~/sin0
~o
The disparity field (1) can be written:
cos0sinqo(cosZcPoCOS00+ sin2~Oo) 1-cos00
) ) sinq) 0 ge
(1)
~2s 2 f f + s ~o)=2ff; go)+ ~
...
(6)
31
with
0~ = (aj_ 1 al2 ] ~;~ \ a 2 1 a22)'
(7)
a11=-
(8)
000 cos0cos(p+ -/sin0cos(p,0
A
1 ( )cos0cos(p
a12-
sin0
l _ __ sin(poCOS(po(1- cos00)sin0cos(p 00 l _ __ (cos2(poCOS00+ sina(po)sin0sin(p 00 l 1-cos00 . + sm(poCOS0, 00 sin00
L
(9)
C
R
Fig. 1. A binocular observer is confronted with a plane surface p. The plane p is perpendicular to the x-z plane (horizontal plane), x-axis: direction from the right to the left eye, z-axis: primary direction, L: left eye, R: right eye, C: cyclopean eye, U: is the point on p that is nearest to C
a21 = ~
sin(p + - - sin(poCOS(po(1-cos00)sin0cos(p 00 l 2 + - - (cos (poCOS00+ sin2(po)sin0sin(p 00 1 1-cos00 - sin(pocos0 (10) 00 sin00
a22= s~n0 ~ 0
sin(p+ 0 sin0cos(p.
We write ~~)~ =
(~0 2 )++ ( ~ ) w h e_,r e
symmetric, ( ~ ) Since ~ - -
(11) ( ~ ) i +s
the
is the antisymmetric part of ~)~
is symmetric the characteristic values, /
,\
and consequently the trace of ( ~ / \GT]
are real. We write +
(0)~) ~ + = B + C' sO that C has vanishing trace- It fOll~ that B = 89 ( ~ )
"I and C = (0~) _ l t r ( ~ +
) "I.
+
can be written:
We now write:
~(F+~;Fo)=~(f;fo)+ ~
Fig. 2. The lines of expansion (closed curves) and contraction (open curves) of the deformation component of the disparity field [Lines of expansion (contraction) are everywhere tangent to the axis of expansion (contraction)]. A central projection on p (Fig. 1) from the cyclopean eye C was used. The point at the center of the figure is U (Fig. 1); the other point corresponds to V. We took 2 = 60 ~ (Fig. 1)
_dr+Bd~§
(12)
The first and second term describe a translation and a rigid rotation (congruences). The third term describes a uniform expansion or contraction (a simil'arity). The fourth term describes an expansion in one and a contraction in an orthogonal direction with a conservation of solid angle (a deformation). The second term is proportional to the curl, the third to the divergence of the disparity field. The fourth term
C = 89
R-I(~_
01)R.
(13)
R is a rigid rotation that specifies the axes of expansion and contraction, def2 is a (by definition positive) real quantity. A shift of the fixation point affects only the antisymmetric part of the gradient of the disparity field. The symmetric part is invariant under shifts of fixation. It can be shown that the gradient of the reciprocal distance can be found from the traceless part C alone.
32
with ( R=
cos# sin#] - s i n # cos/4"
(16)
# specifies the axes of the deformation field. To a direction 2 in the field of the right eye [a unit vector in the direction 2 is (cos2E0+ sin2E~)] corresponds a direction 2+A2 in the field of the left eye: A2(2) = - 89rot~ + 89def)~sin2(2-#)
Fig. 3. Lines of equal magnitude of the deformation component of the disparity field. A central projection on p (Fig. 1) from the cyclopean eye C was used. The magnitude of the deformation vanishes at the points U (center of the figure, see Fig. 1) and V (Fig. 1). We took 2 = 60 ~ (Fig. 1)
The magnitude of the deformation is def2= lY~ "ll/~angll
(14)
and the axis of contraction bisects the direction of the gradient of the reciprocal distance and the direction from the center of rotation of the right to that of the left eye. Hence, the deformation component of the disparity field taken alone specifies the gradient of the reciprocal distance. Shifts of fixation have no influence. It can be shown that the deformation component possesses even more desirable invariant properties: if we subject the image of one of the eyes to a uniform expansion or a rigid rotation, then only congruences or similarities change, the deformation component is unaffected. By way of an example we show in Figs. 1--3 the deformation component of the disparity field for the case that an observer looks at a slanted surface.
(17)
(to the first order in dive, rot2, def)~). This is an immediate consequence of Eq. (15). If we take two directions 2~, )~2 in the field of the right eye, then we have A(21- 22)= def)~sin(2~- 22)cos2 (2-~2 22
#).
(18)
This difference depends only on the deformation component of the field. It follows that the information contained in the disparity field, can be obtained from the mere comparison of angles in the images of left and fight eyes. This comparison cannot be influenced by magnification differences or differential rotations of the left and right images.
3. A Model for Stereopsis
3.1. Correlation of Left and Right Images Assume that the surface is covered with random textural detail. In that case a certain number of cortical line detectors (Hubel and Wiesel, 1965) will be excited for any small patch of the visual field. For any patch of the visual field we have an excitation as function of the orientation 2 of the line detectors of the following type: E(2)= ~ = 1 6()~-21),
(19)
where the 2i are randomly distributed. If we correlate the excitations of two patches, one 2.4. The Deformation Component Can be Found from from the right, one from the left image, then the the Orientation of Line Pieces resulting crosscorrelation function will almost vanish, The disparity field in the neighbourhood of a except when the left and right images correspond visual direction can be written: to the same patch on the surface. If we crosscorrelate the excitation of a small patch from the right image with the excitation of a large patch from the left ~(~'+ d~; 7o) = ~(r'; ~'o)+ (ldiv~ (~ 01) + l r o t ~ (~ - ~) image (or vice versa) then the resulting crosscorrelation depends principally on the excitation of a certain small patch, contained in the large patch of the left image, + ~ def)~R- 1 (~ _ 01)R) dr (15) that corresponds to the small patch of the right image.
33 In this way left and right images can be compared without bringing them into exact register. It is only necessary that the cortical projection from one of the eyes is more diffuse than that of the other eye. Evidence for such an organization has been presented by Blakemore and Pettigrew (1970). It is possible to show (Appendix II) that the width of the crosscorrelation function equals the magnitude of the deformation component of the disparity field.
3.2. The Directions of the Deformation Axes Because of the crosscorrelation technique, phase information is lost. Because phase information is necessary in order to find the directions of the deformation axes, the crosscorrelation scheme outlined in Section 3.1. has to be altered. Phase information can be retained if we do not crosscorrelate over the interval 0 =
