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Published in Image Processing On Line on 2011–02–24.

Submitted on 2011–00–00, accepted on 2011–00–00.

ISSN 2105–1232

 2011 IPOL & the authors

CC–BY–NC–SA

This article is available online with supplementary materials,

software, datasets and online demo at

http://dx.doi.org/10.5201/ipol.2011.my-asift

2014/07/01 v0.5 IPOL article class

ASIFT: An Algorithm for Fully Aﬃne Invariant Comparison

Guoshen Yu

, Jean-Michel Morel

CMAP,

Ecole Polytechnique, France (yu@cmap.polytechnique.fr)

CMLA, ENS Cachan, France (moreljeanmichel@gmail.com)

Abstract

If a physical object has a smoot h or piec ewi s e smooth boundary, its images obtained by cameras

in varying positi ons undergo smoot h apparent deformations. These deformations are locally

well approximated by aﬃne transforms of the image plane. In consequence the solid object

recognition problem has often been led back to the computation of aﬃne invariant image local

features. The simil ar i ty invariance (invariance to translation, rotation, and zoom) is dealt with

rigorously by the SIFT metho d The method illustrated and demonstrated in this work, Aﬃne-

SIFT (ASIFT) , simulates a set of sample views of the initial images, obtainable by varying the

two camera axis orientation parameters, namely the latitude and the longitude angles, which

are not treated by the SIFT method. Then it applies the SIFT method itself to all images thus

generated. Thus, ASIFT covers eﬀectively all six parameters of the aﬃne transform.

Source Code

The source code (ANSI C), its documentation, and the online demo are accessible at the

IPOL

web page of this article

Keywords: SIFT; aﬃne invariant matching

1 Overview

If a physical object has a smooth or p i e cewi se smooth boundary, its images ob t a i n ed by cameras

in varying positions undergo smooth appar ent deformations. These deformations are locally well

approximated by aﬃne transforms of the im a g e plane.

In consequen ce the solid o bject recogni t ion problem has often been led back to the computation

of aﬃne invariant image local features. Such invariant features could be obtained by normalization

methods, but no fully aﬃne normalization method exists for t h e time being. Yet as shown in [

the similarity invariance (invariance t o translation, rotation, an d zoom) is dea l t with rigorously by

the SIFT m et h od [

1]. By simulating on both images zooms out and by normalizing translation and

rotation, th e SIFT method succeeds in being fully invariant to four out of the six parameters of an

aﬃne tran sf or m .

http://dx.doi.org/10.5201/ipol.2011.my-asift

Guoshen Yu, Jean-Michel Morel, ASIFT: An Algorithm for Fully Aﬃne Invariant Comparison, Image Processing On Line, 1 (2011), pp. 11–38.

http://dx.doi.org/10.5201/ipol.2011.my-asift

Guoshen Yu, Jean-Michel Morel

The method illustrated and demonstrated in this work, Aﬃne-SIFT (ASIFT), simulates a set of

sample views of the initial images , obtai n a b l e by varying the two camera axis orientation parameters,

namely the latitude and the longitude angles, wh i ch are not treated by the SIFT method. Then it

applies the SIFT method itself to all images thus generated. Thus, ASIFT covers eﬀectively a l l six

parameters of the aﬃne transform. The method is mathematically proved in [

6] to b e fully aﬃne

invaria nt. And, against any prognosis, simulating a large enough set of sample views de pending on

the two camera orientation parameters is feasible with no dr am a t i c comp u t a ti o n a l lo ad . The main

anamorphosis (deformatio n ) from an im a ge to another caused by applying aﬃne transforms can be

measured by the transition tilt, a new geometric concept introduced in [

6] and explained below.

While state-of-the-art methods before ASIFT hardly exceeded transition ti l t s of 2 (SIFT), 2.5

(Harris-Aﬃne and Hessian-Aﬃ n e [

3]) and 10 (MSER [2]), ASIFT can handle transition tilts up 32 and

higher. MSER can actually deal with transition tilts as high as 10 only when both objects ar e taken

roughly at the same distance. Indeed, contrarily to SIFT, MSER is not scale invariant, because i t does

not simulate the blur due to an increasing distance to the object. The aﬃne transforms considered in

the celebrated compariso n paper [

4] do not have high transition til t s as those that can be handled by

ASIFT. As shown by the experiment archive, most scenes with negli g i b l e or moderate camera view

angle change that match with ASIFT also match with SIFT (with usually fewer ma tching points).

But, when the view angle change becomes important, SIFT and other meth ods fail while ASIFT

continues to work, as we sha l l see in t h e examples (Section

6).

2 Disclaimer

The present work publishes only the ASIFT algorit h m a s des cr i bed below. It does not publish the

SIFT and ORSA subroutin es which are called by the ASIFT code. SIFT and ORSA may be later

updated or rep l a ced by other s u b ro u t i n es.

3 Aﬃne Camera Model

3.1 Image Acquisition Mode l

As illustrated by the cam e ra model in Figu re 1, digital image a cq u i si t i o n of a ﬂat object can be

described as

u = S

AT u

where u is a digi t al image and u

is an (ideal) inﬁn i t e resolution frontal view of the ﬂat object. The

parameters T and A are respectively a plane translation and a pl a n a r projective map due to the

camera motion. G

is a Gaussian convolution modeling the optical blur, and S

is the standa rd

sampling operator on a regular grid with m esh 1. The Gaussian kernel is a ssu m ed to be broad

enough to ensure no aliasing by the 1-samplin g , therefore with a Shannon-Whittaker interpolation

I, one can recover the continuous image from its discrete version: IS

AT u

= G

AT u

. S

will

be thus omi t t ed in the following.

3.2 Aﬃne Local Approximation

We shall proceed to a further simp l i ﬁ ca ti o n of the above model, by reducing A to an aﬃne map.

Figure

2 shows one of the ﬁrst perspectively correct Renaissance paintings by Paolo Uccello. The

perspective on the ground is strongly projective: the rectangular pavement of the room becomes a

ASIFT: An Algorithm for Fully Affine Invariant Comparison

Figure 1: The projective camera model.

Figure 2: Aﬃne local approximation.

trapezoid. However, e ach tile on the pavement is almost a parallelogram. This illustrates the loca l

tangency of perspecti ve defor m a ti o n s to aﬃne maps. Indeed, by the ﬁrst order Taylor formula, any

planar smooth deformation can be approximated around each poi nt by an aﬃne map. The per spective

deformation of a plane object induced by a camera motion is a planar homographic transform, which

is smooth, and therefore locally tangent to aﬃne transforms u(x, y) 7→ u(ax + by + e, cx + dy + f) in

each im a g e region.

3.3 Aﬃne Map Decomposition, with Geometric Interpretation as a Cam-

era Motion

Any aﬃne map A with str i ct l y positive determinant which is not a similarity has a unique decom-

position

A =



a b

c d



= H

(ψ)T

(φ) = λ



cos ψ −sin ψ

sin ψ cos ψ

 

t 0

0 1

 

cos φ −sin φ

sin φ cos φ



Guoshen Yu, Jean-Michel Morel

where λ > 0, λt is the determinant of A, R

are rotation s, φ ∈ [0, Π), and T

is a tilt, n a m el y a

diagonal matrix with ﬁrst eigenvalue t > 1 and t h e second one equal to 1.

Figure 3: Geometric interpretation of aﬃne decomposition.

Figure

3 shows a camera motion interpretation of the aﬃne decomposition: φ and θ = arccos 1/t

are the viewpoint angles, ψ parameterizes the camera spin and λ corresponds to the zoom. The

camera (the small parallelogram on th e top - r i ght) is assu m ed to stay far away from the image u

and starts from a frontal view u, i.e., λ = 1, t = 1, φ = ψ = 0. The camera can ﬁrst move

parallel to th e object’s plane: this motion induces a translation T tha t is eliminated by assu m i n g

without loss of generality that t h e camera axis meets the image plane at a ﬁxed point. The plane

containing the nor m al and the optical axi s makes an angle φ with a ﬁxed vertical plane. This angle

is called “longitude”. Its optic al axis then makes a θ angle with the normal to the image plane u.

This parameter is called “latitude”. Both parameters are classical coord i n a t es on the “ observation

hemisphere”. The camera can rotate around its optical axis (rotatio n parameter ψ). Last but not

least, the camera can move forward or backward, as measured by the zoom parameter λ. We have

seen that the aﬃne model is enough t o give an account o f loc al projective deform a t i on s. If the came ra

were not assumed to be far away, the plan e i m a g e def or m a ti o n u n d e r a camera motion would be a

homography. Yet, as explain ed above, a homography is locally tangent to an aﬃ n e map.

3.4 Transition Tilts, why can they be so Hig h?

The parameter t deﬁned above is called absolute tilt, since it measures the tilt between the frontal

view and a slanted view. In real appl i ca t ion s, both compared images are usually sla nted views. The

transition tilt is designed to quantify the amount of tilt between two such im a ge s. Assume that

v(x, y) = u(A(x, y)) and w(x, y) = u(B(x, y)) are two slanted views of a ﬂat scene whose image is

u(x, y), where A and B are two aﬃ n e m ap s . Then v(x, y) = w(AB

−1

(x, y)). The transit i o n tilt

between v and w is deﬁn ed as the absolute tilt associated with the aﬃne map AB

−1

. Let t and t

′

the

absolute tilts of two images u and u

′

, and let φ and φ

′

be their longitude angles. The transition tilt

τ(u, u

′

) between th e two images depends on the absolute tilts and the longitude angles, and satisﬁes

t/t

′

≤ τ(u, u

′

) = τ(u

′

, u) ≤ t t

′

where we assume t ≥ t

′

. The transition tilt can theref or e be much h i g h er than an absol u t e tilt.

Hence, it is important for an image matching algorithm to be invariant to high transition tilts.

ASIFT: An Algorithm for Fully Affine Invariant Comparison

Figure 4: High transition tilt.

Figure

4 illustrates an example of high transition tilt. The frontal image (above) is squeezed

in one di r ec tio n on the left image by a slanted view, and squeezed in an orthogonal direction by

another slanted view. The absolute tilt (compression factor) is about 6 in each view. The resulting

compression factor, or transition tilt from left to right is actual l y 6 × 6 = 36.

4 Description of the ASIFT Algorithm

A fully aﬃne invariant image matching algorit h m needs to cover the 6 aﬃne parameters. The SIFT

method covers 4 parameters by normalizing rotations and translations, and simulating all zooms out

of the query and of the search images.

As illustrated in Figure

5, ASIFT compl em ents SIFT by si mulating the two parameters that model

the camera optical axis dir ect i on (the original and simulated images are represented respectively by

squares and pa ra l l el o gr a m s) , and then appli e s the SIFT method to compa re the simulated images,

so that all the 6 parameters are covered. In other words, ASIFT simulates three paramete rs :

the scale, the camera longitude angle a n d the latitude angle (which is equivalent to the tilt) and

normalizes the other three (translation and rotation). ASIFT can thus be mathematicall y shown

to b e fully aﬃne invaria nt [6]. Against any prognos i s, simulating the whole aﬃn e space is not

prohibitive at all with the pro posed aﬃn e space sampling [

6, 8].

4.1 Algorithm Description

ASIFT proceeds by the following steps.

1. Each image is transformed by simulating all possible aﬃne distortions caused by the change

of camera optical axis orientation from a frontal position. These distortions depend upon two

parameters: the longitude φ and the latitude θ. The images undergo rotations wi t h angle φ

followed by ti l ts with parameter t = 1/|cos θ| (a tilt by t in t h e direction of x is the operation

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