alignment visual recognition “straighten your paths” isaiah
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Alignment
Visual RecognitionVisual Recognition
“Straighten your paths” Isaiah
Main approaches to recognition:
Pattern recognitionPattern recognition InvariantsInvariants AlignmentAlignment Part decompositionPart decomposition Functional descriptionFunctional description
Alignment
An approach to recognitionAn approach to recognition
where an object is first where an object is first
aligned with an image using aligned with an image using
a small number of pairs of a small number of pairs of
model and image features, model and image features,
and then the aligned model and then the aligned model
is compared directly against is compared directly against
the image.the image.
Object Recognition Using Alignment
D. Huttenlocher and S. Ullman D. Huttenlocher and S. Ullman
11stst ICCV 1987 ICCV 1987
The Task
Matching a 2D view of a rigid object against a Matching a 2D view of a rigid object against a potential model.potential model.
The viewed object can have arbitrary 3D position, The viewed object can have arbitrary 3D position, orientation, and scale, and may be touching or orientation, and scale, and may be touching or occluded by other objectsoccluded by other objects
First the domain of flat rigid objects is considered:First the domain of flat rigid objects is considered: The problem is not 2D as the flat object positioned in 3DThe problem is not 2D as the flat object positioned in 3D It still has to handle occlusion and individuating multiple It still has to handle occlusion and individuating multiple
objects in an imageobjects in an image
Next – extension to rigid objects in general. Next – extension to rigid objects in general.
Aligning a Model With the ImageFor 2D recognition, only two pairs of corresponding model and For 2D recognition, only two pairs of corresponding model and image points are needed to align a model with an image. image points are needed to align a model with an image.
Consider two pairs, such that model point Consider two pairs, such that model point corresponds to image point and model point corresponds corresponds to image point and model point corresponds to image point . to image point . The 2D alignment of the contours has three steps:The 2D alignment of the contours has three steps: The model is translated such that is coincident with The model is translated such that is coincident with Then it is rotated about the new such that the edge Then it is rotated about the new such that the edge
is coincident with the edge is coincident with the edge Finally the scale factor is computed to make coincident Finally the scale factor is computed to make coincident
with with
, and ,m i m ia a b bma
ia mb
ib
ma iama m ma b
i ia b
m
m
b
a
mbma
These two translations, one rotation, and a scale factor These two translations, one rotation, and a scale factor make each unoccluded point of the model coincident make each unoccluded point of the model coincident with its corresponding image point, as long as the initial with its corresponding image point, as long as the initial correspondence of and is correct.correspondence of and is correct.
For 3D from 2D recognition, the alignment method is For 3D from 2D recognition, the alignment method is similar, requiring three pairs of model and image points similar, requiring three pairs of model and image points to perform a three-dimensional transformation and to perform a three-dimensional transformation and scaling of the model.scaling of the model.
,m ia a ,m ib b
The Alignment Method of Recognition
Two stage approach:Two stage approach:
First: position, orientation, and scale of an object are found First: position, orientation, and scale of an object are found using a minimal amount of information (e.g. three pairs of using a minimal amount of information (e.g. three pairs of model and image points) model and image points)
Second: alignment is used to map the object model into image Second: alignment is used to map the object model into image coordinates for comparison with the imagecoordinates for comparison with the image
Given an object Given an object OO in 3D and its 2D image in 3D and its 2D image II (perhaps along with (perhaps along withother objects). other objects).
Find Find OO in the image using the alignment computation in the image using the alignment computation..
Assume that a feature detector returns a set of potentially Assume that a feature detector returns a set of potentially
matching model and image feature pairs matching model and image feature pairs PP Since three pairs of model and image features specify a Since three pairs of model and image features specify a
potential alignment of a model with an image, any triplet in potential alignment of a model with an image, any triplet in PP may specify the position and orientation of the object may specify the position and orientation of the object
In general, some small number of triplets will specify the In general, some small number of triplets will specify the
correct position and orientation, and the rest will be due to correct position and orientation, and the rest will be due to
incorrect matching of model and image points incorrect matching of model and image points Thus the recognition problem is:Thus the recognition problem is:
Determine which alignment in Determine which alignment in PP defines the transformation defines the transformation that best maps the model into the image. that best maps the model into the image.
Given a set pairs of model and image features, Given a set pairs of model and image features, PP ,we solve ,we solve for the alignment specified by each triplet in for the alignment specified by each triplet in PP
For some triplets, there will be no way to position and For some triplets, there will be no way to position and orient the three model points such that they project onto orient the three model points such that they project onto their corresponding image points their corresponding image points
Such triplets do not specify a possible alignment of the Such triplets do not specify a possible alignment of the model and the image model and the image
The remaining triplets each specify a transformation The remaining triplets each specify a transformation mapping model points to image pointsmapping model points to image points
An alignment is scored by using the transformation to map An alignment is scored by using the transformation to map the model edges into the image, and comparing the the model edges into the image, and comparing the transformed model edges with the image edgestransformed model edges with the image edges
The best alignment is the one that maps the most model The best alignment is the one that maps the most model edges onto image edgesedges onto image edges
For For mm model features and model features and ii image features, the number of pairs image features, the number of pairs of model and image features, of model and image features, p, p, is at most is at most
A good labeling scheme will bring A good labeling scheme will bring p p close to close to m (m (then each then each model point has one corresponding image point)model point has one corresponding image point)
Given Given pp pairs of features,there are , or an upper bound of pairs of features,there are , or an upper bound of
triplets of pairs. Each specifies a possible alignmenttriplets of pairs. Each specifies a possible alignment An alignment is scored by mapping the model edges into the An alignment is scored by mapping the model edges into the
imageimage If the model edges are of length If the model edges are of length l,l,then the worst case running then the worst case running
time of the algorithm is time of the algorithm is Alignment transforms recognition from exponential problem of Alignment transforms recognition from exponential problem of
finding the largest consistent set of model and image points, to finding the largest consistent set of model and image points, to polynomial problem of finding the best triplet of model and polynomial problem of finding the best triplet of model and image points.image points.
.i m
3p
3O p
3O lp
Complexity
Alignment points
It is important to label features distinctively in It is important to label features distinctively in order to limit the number of pairsorder to limit the number of pairs
The labels must be relatively insensitive to partial The labels must be relatively insensitive to partial occlusion, juxtaposition,and projective occlusion, juxtaposition,and projective
distortion, while being as distinctive as possibledistortion, while being as distinctive as possible If the number of pairs, If the number of pairs, p, p, is kept small then little is kept small then little
or no search is necessary to find the correct or no search is necessary to find the correct alignment. alignment.
Multi-Scale Description
Using significant inflection points and low-Using significant inflection points and low-curvature regions to segment edge contourscurvature regions to segment edge contours
Edge segments are labeled to produce distinctive Edge segments are labeled to produce distinctive labels for use in pairing together potentially labels for use in pairing together potentially matching image and model pointsmatching image and model points
Context – edge contour is smoothed at various Context – edge contour is smoothed at various scales and the finer scale descriptions are used to scales and the finer scale descriptions are used to label the coarser scale segmentslabel the coarser scale segments
The coarser scale segments are used to group finer The coarser scale segments are used to group finer scale segments togetherscale segments together
The tree corresponding to the curvature scale space segmentation The contours are segmented at inflections in the smoothed curvature
Alignment of a widget with an image that does not match themodel edge contour with image edges
Left - Matching a widget against an image of two widgets in the planeRight – Matching a widget against an image of a foreshortened widget
3D from 2D Alignment
It is shown that the position,orientation,and scale of It is shown that the position,orientation,and scale of an object in 3D can be determined from a 2D image an object in 3D can be determined from a 2D image using three pairs of corresponding model and image using three pairs of corresponding model and image points under weak perspective modelpoints under weak perspective model
Under full perspective – up to four distinct solutionsUnder full perspective – up to four distinct solutions Next:Next:
The use of orthographic projection and a linear The use of orthographic projection and a linear scale factor (weak perspective) as approximation scale factor (weak perspective) as approximation for perspective viewingfor perspective viewing
The alignment method using explicit 3D rotationThe alignment method using explicit 3D rotation Alignment method can be simulated using only Alignment method can be simulated using only planar operations. planar operations.
Weak Perspective Projection
Given a set of points Given a set of points In the new image: In the new image:
, ,T
i i i iP X Y Z
(1,2)' ( ' , ' )Ti i i ip x y sRP t
Projection model: W.P. is good enough
A point (X,Y,Z) is projected:A point (X,Y,Z) is projected:
under perspective:under perspective:
under weak perspective:under weak perspective:
The error is expressed by: The error is expressed by:
oror
0 0
1 1 1 1x x fX xZ
Z Z Z Z
0
1Z
E xZ
( , ) ( / , / )x y fX Z fY Z
0ˆ ˆ( , ) ( , ), /x y sX sY s f Z
Allowed depth ratios as a function of x
Error is small when:
The measured feature is close to the optical axis The measured feature is close to the optical axis
oror The estimate for the depth is close to the real depth The estimate for the depth is close to the real depth
(average depth of the observed environment)(average depth of the observed environment)
Supports the intuition that for images with low Supports the intuition that for images with low
depth variance and for fixed regions near the depth variance and for fixed regions near the
center - perspective distortions are relatively small center - perspective distortions are relatively small
Alignment Consider three model points and and three Consider three model points and and three corresponding image points and ,where the model corresponding image points and ,where the model points specify 3D positions points specify 3D positions (x,y,z) (x,y,z) and the image points specifyand the image points specify positions in the image plane,positions in the image plane,(x,y,0)(x,y,0) The alignment task is to find a transformation that maps the The alignment task is to find a transformation that maps the
plane defined by the three model points onto the image plane, plane defined by the three model points onto the image plane, such that each model point coincides with its corresponding such that each model point coincides with its corresponding image point. If no such transformation exists,then the image point. If no such transformation exists,then the alignment process must determine this factalignment process must determine this fact
Since the viewing direction is along the Since the viewing direction is along the z-z-axis,an alignment is axis,an alignment is a transformation that positions the model such that projects a transformation that positions the model such that projects
along the along the z-z-axis onto , and similarly for onto ,and axis onto , and similarly for onto ,and onto onto
,m ma b mc,i ia b ic
ma
ia ibmb mc
ic
The transformation consists of translations in The transformation consists of translations in x x and and yy,and,and rotations about three orthogonal axes. There is no translation rotations about three orthogonal axes. There is no translation
in in zz (all points along the viewing axis are equivalent under (all points along the viewing axis are equivalent under orthographic projection)orthographic projection) First we show how to solve for the alignment assuming no First we show how to solve for the alignment assuming no
change in scale,and then modify the computation to allow for change in scale,and then modify the computation to allow for a linear scale factora linear scale factor
First translate the model points so that one point projects First translate the model points so that one point projects along the along the zz-axis onto corresponding image point-axis onto corresponding image point
Using for this purpose, the model points are translated by Using for this purpose, the model points are translated by
yielding the model points and yielding the model points and This brings ,the projection of into the image plane This brings ,the projection of into the image plane into correspondence with into correspondence with
ma , ,0ai am ai amx x y y ' ',m ma b
'mc
'mia '
maia
Now it is necessary to rotate the model about three orthogonal Now it is necessary to rotate the model about three orthogonal
axes to align and with their corresponding image points.axes to align and with their corresponding image points.mb mc
First we align one of the model edges with its corresponding First we align one of the model edges with its corresponding image edge by rotating the model about the image edge by rotating the model about the zz-axis-axis
Using the edge we rotate the model by the angel Using the edge we rotate the model by the angel between the image edge , and the projected model edge between the image edge , and the projected model edge
,yielding the model points and (stage b),yielding the model points and (stage b) To simplify the presentation, the coordinate axes are now To simplify the presentation, the coordinate axes are now
shiftedshifted Because ,the projection of into the image plane, lies Because ,the projection of into the image plane, lies
along the along the xx-axis,it can be brought into correspondence with -axis,it can be brought into correspondence with by simply rotating the model about the by simply rotating the model about the yy-axis-axis
The amount of rotation is determined by the relative lengths The amount of rotation is determined by the relative lengths of andof and
If the model edge is shorter than the image edge - there is If the model edge is shorter than the image edge - there is no such rotation, and hence the model cannot be aligned no such rotation, and hence the model cannot be aligned with the imagewith the image
' 'm ma b
i ia b' 'mi mia b
''mb ''
mc
i ia b
''mib ''
mbib
m ma b i ia b
The model points and are rotated about the The model points and are rotated about the yy-axis by -axis by to obtain and , whereto obtain and , where
(1)(1)
for (stage c)for (stage c) is brought into correspondence with by rotation about is brought into correspondence with by rotation about
the the xx-axis-axis The degree of rotation is again determined by the relative The degree of rotation is again determined by the relative
lengths of model and image edgeslengths of model and image edges In the previous case the edges were parallel to the In the previous case the edges were parallel to the xx-axis, and -axis, and
therefore the length was the same as the therefore the length was the same as the xx component of the component of the lengthlength
In this case, the edges need not be parallel to the In this case, the edges need not be parallel to the yy axis, and axis, and therefore the therefore the yy component of the lengths must be used component of the lengths must be used
''mb ''
mc '''mb '''
mc
1,0,0
cos1,0,0
i
m
b
b
0 cos 1 '''mc ic
Thus, the rotation about the Thus, the rotation about the xx-axis is ,where-axis is ,where
(2)(2) for (stage d)for (stage d)
If the model distance is shorter than the image distance,there If the model distance is shorter than the image distance,there is no transformation that aligns the model and the image is no transformation that aligns the model and the image
If the rotation does not actually bring into If the rotation does not actually bring into correspondence with ,then there is also no alignmentcorrespondence with ,then there is also no alignment Verification: The combination of translations and rotations Verification: The combination of translations and rotations
can now be used to map the model into the imagecan now be used to map the model into the image
0,1,0
cos0,1,0
i
m
c
c
0 cos 1
'''mic
ic
Scale
Linear scale factor - a sixth unknownLinear scale factor - a sixth unknown The final two rotations which align with The final two rotations which align with
, and are the only computations , and are the only computations affected by a change in scale. affected by a change in scale.
The alignment of involves movement of The alignment of involves movement of along the along the xx-axis, whereas the alignment -axis, whereas the alignment of involves movement of in both the of involves movement of in both the xx and and yy directions. directions.
mbib
ic
mb mib
mc
mic
Because the movement of is a sliding along the x-axis,only Because the movement of is a sliding along the x-axis,only the x-component, ,changes. The change is given by the the x-component, ,changes. The change is given by the rotation about the yrotation about the y-axis, as in (1).With a scale factor -axis, as in (1).With a scale factor s s this this becomes becomes (3)(3)
Similarly the movement of in the Similarly the movement of in the yy direction is given by direction is given by the rotation about the the rotation about the xx-axis, as in (2).With a scale factor this -axis, as in (2).With a scale factor this becomes becomes
(4)(4)
mib
bx
' cos .b bx sx
mic
' (cos ).c cy sy
The movement of in the The movement of in the xx direction is given by the direction is given by the
rotations about both the rotations about both the xx- and - and yy-axis we obtain -axis we obtain
Thus with the scale factor,the Thus with the scale factor,the x x component of is component of is
(5)(5)
Now we have three equations in the three unknowns, Now we have three equations in the three unknowns, ss, ,
and One method to solve for and One method to solve for ss is to substitute for , is to substitute for ,
,and in (5). From (3) we know that, ,and in (5). From (3) we know that,
(6)(6)
mic
' cos sin sin .x x y
mc
' cos sin sin .c c cx s x y
,. cos
sin sin2 2 '21
sin .b bb
s x xsx
And similarly from (4), (7)And similarly from (4), (7)
Substituting (6) and (7) into (5) and simplifying yields Substituting (6) and (7) into (5) and simplifying yields
Expanding out the terms we obtainExpanding out the terms we obtain
a quadratic in While there are generally two possible a quadratic in While there are generally two possible
solutions, it can be shown that only one of the solutions will solutions, it can be shown that only one of the solutions will
specify possible values of and .specify possible values of and .
Having solved for the scale of an object, the final two Having solved for the scale of an object, the final two
rotations and can be computed using (1)and (2) rotations and can be computed using (1)and (2)
modified to account for the scale factor modified to account for the scale factor ss. .
2 2 '21sin .c c
c
s y ysy
2 ' ' 2 2 2 '2 2 2 '2( ) ( )( ).b c c b b b c cs x x x x s x x s y y
4 2 2 2 2 '2 '2 2 ' ' 2 '2 '2( ) ( ( ) ) ,b c b c b c b c c b b cs x y s x y x y x x x x x y
2.s
cos cos
Issues
3D objects: 3D objects: Maintain a single 3-D model, use the Maintain a single 3-D model, use the
recovered T and align – occlusion ** 2recovered T and align – occlusion ** 2Store number of models and alignment Store number of models and alignment
keys representing different viewing keys representing different viewing positionspositions
Object centered vs viewer centered Object centered vs viewer centered Handling DB with multiple objectsHandling DB with multiple objects
Examples
Alignment
Recognizing objects by compensating for variationsRecognizing objects by compensating for variations
Method:Method: The stored library of objects contains their shape The stored library of objects contains their shape
and allowed transformations. and allowed transformations. Given an image and an object model, a Given an image and an object model, a
transformation is sought that brings the object to transformation is sought that brings the object to appear identical to the image.appear identical to the image.
Alignment (cont.)
Domain:Domain: Suitable mainly for recognition of specific object.Suitable mainly for recognition of specific object.
Problems:Problems: Complexity: recovering the transformation is Complexity: recovering the transformation is
time-consuming.time-consuming. Indexing: library is searched serially.Indexing: library is searched serially. Non rigidities are difficult to model.Non rigidities are difficult to model.
Linear Combinations Scheme
Relates familiar views and novel views of objects Relates familiar views and novel views of objects in a simple wayin a simple way
Novel views are expressed by linear combinations Novel views are expressed by linear combinations of the familiar viewsof the familiar views
This is used to develop a recognition system that This is used to develop a recognition system that uses viewer-centered representations uses viewer-centered representations
An object is modeled by a small set of its familiar An object is modeled by a small set of its familiar viewsviews
Recognition involves comparing the novel views Recognition involves comparing the novel views to linear combinations of the model viewsto linear combinations of the model views
Weak Perspective Projection
Given a set of points Given a set of points In the new image: In the new image:
, ,T
i i i iP X Y Z
(1,2)' ( ' , ' )Ti i i ip x y sRP t
x’ y’ belong to 4D linear subspace !
For For under weak perspective: under weak perspective:
In vector equation form:In vector equation form:
Consequently,Consequently,
, , ,i i i ip x y z 1 i n '
11 12 13
'21 22 23
i i i i x
i i i i y
x sr x sr y sr z t
y sr x sr y sr z t
' ', span , , ,1x y x y z
11 12 13
21 22 23
'
'x
y
x sr x sr y sr z t
y sr x sr y sr z t
Theorem:Theorem: The coefficients satisfy two quadratic constraints, The coefficients satisfy two quadratic constraints, which can be derived from three imageswhich can be derived from three imagesProof:Proof: Consider the coefficients Consider the coefficients SinceSince R R is a rotation matrix, its row vectors are orthonormal, is a rotation matrix, its row vectors are orthonormal,
and therefore the following equations hold for the coefficients: and therefore the following equations hold for the coefficients:
Choosing a different base to represent the object will change Choosing a different base to represent the object will change the constraintsthe constraints
The constraints depend on the transformation that separates The constraints depend on the transformation that separates the model viewsthe model views
1 4 1 4,..., , ,...,a a b b
2 2 2 2 2 21 2 3 1 2 3a a a b b b
1 1 2 2 3 3 0a b a b a b
The CoefficientsThe Coefficients
Denote the coefficients that represent a Denote the coefficients that represent a
novel view, namely novel view, namely
and denote and denote U U the rotation matrix that separates the two the rotation matrix that separates the two
model viewsmodel views
By substituting the new coefficients we obtain newBy substituting the new coefficients we obtain new
constraints:constraints:
'1 1 2 1 3 2 4
'1 1 2 1 3 2 4
1
1
x x y x
y x y x
2 2 2 2 2 21 2 3 1 2 3 1 3 1 3 11 2 3 2 3 12
1 1 2 2 3 3 1 3 3 1 11 2 3 3 2 12
2 2
0
a a a u u
u u
1 4 1 4,..., , ,...,
To derive the constraints the values of and should be To derive the constraints the values of and should be
recovered. A third view can be used for this purposerecovered. A third view can be used for this purpose When a third view of the object is given, the constraints When a third view of the object is given, the constraints
supply two linear equations in and , and, therefore, supply two linear equations in and , and, therefore, in general, their values can be recovered from the two in general, their values can be recovered from the two constraintsconstraints
This proof suggest a simple, essentially linear structure from This proof suggest a simple, essentially linear structure from
motion algorithm that resembles the method used in motion algorithm that resembles the method used in
[Ullman79, Huang and Lee89] [Ullman79, Huang and Lee89]
11u
12u
12u
11u
Linear Combination
For two views there exist coefficients andFor two views there exist coefficients and
such thatsuch that
The coefficients satisfy the following two quadratic constraints:The coefficients satisfy the following two quadratic constraints:
To derive these the transformation should be recovered – a thirdTo derive these the transformation should be recovered – a third
image is needed. image is needed.
1 2 3 4, , ,a a a a1 2 3 4, , ,b b b b
'1 1 2 1 3 2 4
'1 1 2 1 3 2 4
1
1
x a x a y a x a
y b x b y b x b
2 2 2 2 2 21 2 3 12 2 3 1 3 1 3 11 2 3 2 3 122 2a a a b b b b b a a r b b a a r
1 1 2 2 3 3 1 3 3 1 11 2 3 3 2 12 0a b a b a b a b a b r a b a b r
LC - Formally
Given: P Given: P and a set of stored models and a set of stored models MM
Find:Find: such that P matches such that P matches
i.e such thati.e such that
Object – is modeled by a set images with correspondence Object – is modeled by a set images with correspondence (quadratic constraints may also be stored)(quadratic constraints may also be stored)
Recognition – recovering the LC that aligns model and imageRecognition – recovering the LC that aligns model and image 3-4 points are sufficient to determine the coefficients3-4 points are sufficient to determine the coefficients Predict the appearance and verifyPredict the appearance and verify Worse case complexity - no. of models, no. of model Worse case complexity - no. of models, no. of model
points, no. of image points, no. of points used for verification points, no. of image points, no. of points used for verification
ii
1
k jj jj
Mi
j
4 4( ) 'k m n m
Results
The bottom 2 lines were created by linear combinations of the top 2 lines:
Results (cont.)
a) The model pictures
b) A linear combination
c) True images
d) The error between b) and c)
e) The error between b) and another car
Recognition Operator
Operators that are invariants for a given Operators that are invariants for a given space of viewsspace of views
Return constant value for all views of the Return constant value for all views of the object and different value for views of other object and different value for views of other objectsobjects
Correspondence needed, but no need for Correspondence needed, but no need for explicit recovery of the alignment explicit recovery of the alignment coefficient coefficient
Idea
LC => a view is a point in R LC => a view is a point in R n n
Object’s view => belongs to the space of Object’s view => belongs to the space of views spanned by the object modelviews spanned by the object model
Recognition = how far the in view from the Recognition = how far the in view from the views space of the objectviews space of the object
Project the given view on the views space Project the given view on the views space of the object and compute distance of the object and compute distance
Recognition Operator (cont)
Represent the model picture by the vector
Construct a matrix , such that
We get:
So if an image is a view of the object, it is mapped to the same vector up to a scale.
(for every )
L measures the distance of the new view L measures the distance of the new view from the linear space spanned by the object from the linear space spanned by the object model viewsmodel views
Note that L does not verify any of the Note that L does not verify any of the quadratic constraints quadratic constraints
To verify these a quadratic invariant can be To verify these a quadratic invariant can be constructedconstructed
How to find ?
Such that all the vectors are independent
We get:
if (noise is mapped to itself)
For quantitative results, we can chose
and test the ratio
For a view of the object (recognition)
and for pure noise (no recognition)
Results
Left column:
A view of the object
Middle column:
Transformed to the canonical view by
Right column:
Another object transformed
Recognize!
Projection model: W.P. is good enough
A point (X,Y,Z) is projected:A point (X,Y,Z) is projected:
under perspective:under perspective:
under weak perspective:under weak perspective:
The error is expressed by: The error is expressed by:
oror
0 0
1 1 1 1x x fX xZ
Z Z Z Z
0
1Z
E xZ
( , ) ( / , / )x y fX Z fY Z
0ˆ ˆ( , ) ( , ), /x y sX sY s f Z
Error is small when:
The measured feature is close to the optical axis The measured feature is close to the optical axis oror The estimate for the depth is close to the real The estimate for the depth is close to the real
depth (average depth of the observed depth (average depth of the observed environment)environment)
Supports the intuition that for images with low depthSupports the intuition that for images with low depth variance and for fixed regions near the center - variance and for fixed regions near the center - perspective distortions are relatively small perspective distortions are relatively small
Allowed depth ratios as a function of x
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