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