Corisco: Robust edgel-basedorientation estimation forgeneric camera models

Nicolau L. Werneckhttp://nic.hpavc.net

Supervisor: Profa. Dra. Anna Helena Reali CostaIntelligent Techniques Laboratory, LTI — PCS — Poli

Universidade de São Paulo

Augsburg, Deutschland — 8/10/2015

Meta-introduction

Nicolau Werneck, Sc.D.

E.Eng. graduated from UFMG, Unicamp and USP.Specialized in signal processing and patternrecognition, especially in computer vision andparameter estimation problems.Ex-Google and ex-Geekie.Wants to help robots help us.

Presentation adapted from my doctorate defense.Results were published in Werneck and Costa [2013].

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Presentation Schedule

1 Problem introduction2 Method proposal3 Experiments and conclusion

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Anthropic EnvironmentsAn anthropic environment is composed by straight lines,parallel to the directions of a natural reference frame.

The orientation we find is a three-dimensional rotationbetween the natural and the camera reference frame.

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Edgels and straight linesEdgels are points samples over curves or (straight) lines.

Edgels have many applications.

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Proposal

We propose a monocular vision method, denominatedCorisco, that can estimate the orientation of a camerarelative to an anthropic environment.

Evolution of existing edgel based methods [Coughlan andYuille, 2003].

Aplication examples:Guiding a mobile robot.Initial estimates for multi-view reconstruction.Object orientation estimation.

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Proposal

We propose a monocular vision method, denominatedCorisco, that can estimate the orientation of a camerarelative to an anthropic environment.

Evolution of existing edgel based methods [Coughlan andYuille, 2003].

Aplication examples:Guiding a mobile robot.Initial estimates for multi-view reconstruction.Object orientation estimation.

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Proposal

We propose a monocular vision method, denominatedCorisco, that can estimate the orientation of a camerarelative to an anthropic environment.

Evolution of existing edgel based methods [Coughlan andYuille, 2003].

Aplication examples:Guiding a mobile robot.Initial estimates for multi-view reconstruction.Object orientation estimation.

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Contributions

Corisco has the following peculiarities:Supports any possible camera model.Compromise between speed and precision.Dismisses the use of very costly operations (sin,atan, exp, log). Uses the function x−1/2.

There are also no assumptions about the solution.

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High-level viewInputs: Image, intrinsic and control parameters.Output: Orientation Ψ (three-dimensional rotation).

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GeometryThe direction v of a line over the point p depends on Ψ.

(Ψ,p)→ v

http://youtu.be/PbUyvdnBIzs

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Objective

Our questionWhat could be the best way to estimate the orientationof a camera in real time in an anthropic environmentbased on a single distorted image?

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Objective

Our questionWhat could be the best way to estimate the orientationof a camera in real time in an anthropic environmentbased on a single distorted image?

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Edgels versus lines

Orientation can be found from lines (Caprile and Torre[1990], Cipolla et al. [1999], Rother [2002]).

More intuitive.Attain a good precision.

On the other hand:Restricted to perspective projection.Rely on complicated line extraction.

Edgels allow us to handle distortions and are easier toextract and sub-sample.

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Corisco in detail

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Corisco in detail

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Corisco in detail

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Corisco in detail

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Block diagram

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Block diagram

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Block diagram

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Edgel extraction

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Edgel extractionSimilar to the Canny border detection.Borders are local maxima in the gradient direction.

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Edgel extraction

Sweep the image over a set of lines and columns thatconstitute a grid mask.

Each border found produces an edgel. Its directionshould be approximately orthogonal to the line swept.

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Edgels extraídos

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Camera models

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Camera models

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Camera models

Bijective map between the image points and directionsaround the camera focal point.

q p

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Camera model examplesWide-angle, fisheye, equirectangular.

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Equações de modelos de câmera

Perspective Equiretangular (lat-lon)px = (qx/qz)f +cx

py = (qy/qz)f +cypx = f tan−1(qz ,qx )py = f sin−1 (qy/|q|)

Harris (radial distortion) Polar Equidistant (fisheye)

g(x) = 1√1−2κx2

px = p′xg(|p′|) +cx

py = p′yg(|p′|) +cy

ϕ = cos−1 (qz/|q|)px = ϕ

qx√qx 2+qy 2

f +cx

py = ϕqy√

qx 2+qy 2f +cy

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Edgel projectionA point qn from a line in the direction rk is projected onpn, producuing an edgel com with direction

vnk ∝ Jnrk (v← (Ψ,p))

The projection Jacobian matrix Jn depends on pn. Thedirection orthogonal to vn is denominated un.

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Edgel projectionA point qn from a line in the direction rk is projected onpn, producuing an edgel com with direction

vnk ∝ Jnrk (v← (Ψ,p))

The projection Jacobian matrix Jn depends on pn. Thedirection orthogonal to vn is denominated un.

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Edgel normal vector

Direction of a plane defined by the focal point and by anedgel, or a corresponding environment line.

nk = uxkJx

k +uykJ

yk

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Objective function

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Objective function

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Objective function

The objective function is the heart of the method.Does not depend on the extraction technique.Determines the result.Conducts the choice of the optimization algorithm.

The expression has the shape of a summation of errorsobtained from each observation.

yn = xn− x̂n(Ψ)

F (Ψ) = ∑n

(yn)2

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Rotation matrixAn edgel direction prediction starts by finding thedirections of the natural frame rk as a function of Ψ.

Corisco works with quaternions.

Ψ = (Ψa,Ψb,Ψc ,Ψd) |Ψ|= 1

For a general orientation Ψ we have

R(Ψ) =[rx ry rz

]T =[Ψa2 + Ψb2−Ψc2−Ψd 2 2ΨbΨc +2ΨaΨd 2ΨbΨd −2ΨaΨc

2ΨbΨc −2ΨaΨd Ψa2−Ψb2 + Ψc2−Ψd 2 2Ψc Ψd +2ΨaΨb

2ΨbΨd +2ΨaΨc 2Ψc Ψd −2ΨaΨb Ψa2−Ψb2−Ψc2 + Ψd 2

]

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Edgel residueGiven the rk , we calculate the predicted vnk using Jk .

Ψ→ rk → vnk

vnk ∝ Jnrk

The residue compares:vn (ou un) taken from the imagem.vnk predicted from Ψ e pn.

Previous methods: Corisco:∠v = arctan(vx ,vy ) unvnk∠vn−∠vnk

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Error functionComparison with existing edgel based techiques:

MAP EM M-estimationModels Probabilistic Probabilistic ρ(x)Classes 4 4 3Classification × Iterative Direct

0.2 0.1 0.0 0.1 0.20.0

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Funções de Tukey, e quadrática

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Objective function expressionMAP estimation (Coughlan and Yuille [2003]):

F (Ψ) =−∑nlog(

∑k

p(cn = k)p(∠vn|Ψ,pn,cn = k))

EM estimation (Schindler and Dellaert [2004]):

F (Ψ) =−∑n

∑k

p(cn = k) log (p(∠vn|Ψ,pn,cn = k))

F (Ψ) = ∑n

∑k

p(cn = k)(∠vn−∠vnk)2

Corisco:F (Ψ) = ∑

nmin

kρ(unvnk)

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Objective function expressionMAP estimation (Coughlan and Yuille [2003]):

F (Ψ) =−∑nlog(

∑k

p(cn = k)p(∠vn|Ψ,pn,cn = k))

EM estimation (Schindler and Dellaert [2004]):

F (Ψ) =−∑n

∑k

p(cn = k) log (p(∠vn|Ψ,pn,cn = k))

F (Ψ) = ∑n

∑k

p(cn = k)(∠vn−∠vnk)2

Corisco:F (Ψ) = ∑

nmin

kρ(unvnk)

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F (Ψ) =−∑n

∑k

p(cn = k) log (p(∠vn|Ψ,pn,cn = k))

Objective function expressionMAP estimation (Coughlan and Yuille [2003]):

F (Ψ) =−∑nlog(

∑k

p(cn = k)p(∠vn|Ψ,pn,cn = k))

EM estimation (Schindler and Dellaert [2004]):

F (Ψ) =−∑n

∑k

p(cn = k) log (p(∠vn|Ψ,pn,cn = k))

F (Ψ) = ∑n

∑k

p(cn = k)(∠vn−∠vnk)2

Corisco:F (Ψ) = ∑

nmin

kρ(unvnk)

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F (Ψ) = ∑n

∑k

p(cn = k)(∠vn−∠vnk)2

Objective function expressionMAP estimation (Coughlan and Yuille [2003]):

F (Ψ) =−∑nlog(

∑k

p(cn = k)p(∠vn|Ψ,pn,cn = k))

EM estimation (Schindler and Dellaert [2004]):

F (Ψ) =−∑n

∑k

p(cn = k) log (p(∠vn|Ψ,pn,cn = k))

F (Ψ) = ∑n

∑k

p(cn = k)(∠vn−∠vnk)2

Corisco:F (Ψ) = ∑

nmin

kρ(unvnk)

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Corisco:F (Ψ) = ∑

nmin

kρ(unvnk)

Optimization

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Optimization

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Optimization

First step: RANSAC, stochastic search guided by data.Inherently inefficient and imprecise.

→Ψ inicial

Second step: FilterSQP, continuous optimization, moreefficient and precise than RANSAC.

→Ψ final

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Optimization

First step: RANSAC, stochastic search guided by data.Inherently inefficient and imprecise.

→Ψ inicial

Second step: FilterSQP, continuous optimization, moreefficient and precise than RANSAC.

→Ψ final

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Optimization

First step: RANSAC, stochastic search guided by data.Inherently inefficient and imprecise.

→Ψ inicial

Second step: FilterSQP, continuous optimization, moreefficient and precise than RANSAC.

→Ψ final

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RANSACObservation triplets are randomly selected. From eachone we calculate a hypothetical Ψ, using nk .

The Ψ with the smallest F (Ψ) is the initial estimate.

http://i.imgur.com/O9jP8tz.gifv

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FilterSQPMinimize F (Ψ) over the 4D and constrained to |Ψ|= 1.

Non-linear program → SQP: Sequential QuadraticProgramming

FilterSQP (Fletcher and Leyffer [2002]) dismisses penaltyfunctions.

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Derivative

Corisco calculates the Ψ derivatives by closed formulas.

∂F∂ Ψa (Ψ) = ∑

nkKnk ρ

′(unvnk)(

uxn

∂vxnk

∂ Ψa + uyn

∂vynk

∂ Ψa

)

The derivatives of the directions rk are trivial:

∂ rx∂ Ψa = 2(Ψa,Ψd ,−Ψc)

∂ rx∂ Ψb = 2(Ψb,Ψc ,Ψd)

· · ·

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Experiments

We performed 3 experiments to assess the performanceof Corisco.

Each experiment used a different image set, and methodto obtain the reference orientations.

The observed error is the displacement in degrees of the“rotação residual” entre cada estimativa e referência.

Corisco was executed changing the settings:grid size Cg ,number of RANSAC iterations Cr .

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YorkUrbanDB

Images: 101 imagens from anthropic environments.

Model: Perspective.

Reference: Semi-automatic method based on lines.

Comparison: Methods studied by Denis et al. [2008].

Camera parameters, reference orientations andperformance statistics provided by the authors.

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YorkUrbanDB

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YorkUrbanDB

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Method Time Error[s] Mean σ 1/4 Median 3/4

EM Newton 27+? 4.00◦ 1.00◦ 1.15◦ 2.61◦ 4.10◦MAP Quasi-Newton 6+? 4.00◦ 1.00◦ 1.32◦ 2.39◦ 4.07◦EM Quasi-Newton 1+? 9.00◦ 1.00◦ 4.04◦ 6.21◦ 10.33◦J-linkage 1.13 8.23◦ 13.76◦ 1.14◦ 2.36◦ 4.44◦Corisco Cr = 104 Cg = 1 47.20 1.51◦ 3.26◦ 0.69◦ 1.09◦ 1.51◦Corisco Cr = 104 Cg = 4 16.68 1.71◦ 3.35◦ 0.72◦ 1.14◦ 1.64◦Corisco Cr = 104 Cg = 32 7.57 2.43◦ 4.03◦ 0.97◦ 1.54◦ 2.42◦Corisco Cr = 103 Cg = 1 8.12 1.70◦ 3.22◦ 0.70◦ 1.11◦ 1.77◦Corisco Cr = 103 Cg = 4 2.50 2.02◦ 3.86◦ 0.81◦ 1.24◦ 1.80◦Corisco Cr = 103 Cg = 32 0.99 2.44◦ 3.54◦ 1.00◦ 1.68◦ 2.64◦Corisco Cr = 200 Cg = 1 5.34 2.08◦ 3.38◦ 0.72◦ 1.22◦ 1.78◦Corisco Cr = 200 Cg = 4 1.89 3.27◦ 6.38◦ 0.85◦ 1.34◦ 2.35◦Corisco Cr = 200 Cg = 32 0.45 3.29◦ 4.99◦ 0.99◦ 1.72◦ 3.46◦

YorkUrbanDB

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Method Time Error[s] Mean σ 1/4 Median 3/4

EM Newton 27+? 4.00◦ 1.00◦ 1.15◦ 2.61◦ 4.10◦MAP Quasi-Newton 6+? 4.00◦ 1.00◦ 1.32◦ 2.39◦ 4.07◦EM Quasi-Newton 1+? 9.00◦ 1.00◦ 4.04◦ 6.21◦ 10.33◦J-linkage 1.13 8.23◦ 13.76◦ 1.14◦ 2.36◦ 4.44◦Corisco Cr = 104 Cg = 1 47.20 1.51◦ 3.26◦ 0.69◦ 1.09◦ 1.51◦Corisco Cr = 104 Cg = 4 16.68 1.71◦ 3.35◦ 0.72◦ 1.14◦ 1.64◦Corisco Cr = 104 Cg = 32 7.57 2.43◦ 4.03◦ 0.97◦ 1.54◦ 2.42◦Corisco Cr = 103 Cg = 1 8.12 1.70◦ 3.22◦ 0.70◦ 1.11◦ 1.77◦Corisco Cr = 103 Cg = 4 2.50 2.02◦ 3.86◦ 0.81◦ 1.24◦ 1.80◦Corisco Cr = 103 Cg = 32 0.99 2.44◦ 3.54◦ 1.00◦ 1.68◦ 2.64◦Corisco Cr = 200 Cg = 1 5.34 2.08◦ 3.38◦ 0.72◦ 1.22◦ 1.78◦Corisco Cr = 200 Cg = 4 1.89 3.27◦ 6.38◦ 0.85◦ 1.34◦ 2.35◦Corisco Cr = 200 Cg = 32 0.45 3.29◦ 4.99◦ 0.99◦ 1.72◦ 3.46◦

YorkUrbanDB

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Method Time Error[s] Mean σ 1/4 Median 3/4

EM Newton 27+? 4.00◦ 1.00◦ 1.15◦ 2.61◦ 4.10◦MAP Quasi-Newton 6+? 4.00◦ 1.00◦ 1.32◦ 2.39◦ 4.07◦EM Quasi-Newton 1+? 9.00◦ 1.00◦ 4.04◦ 6.21◦ 10.33◦J-linkage 1.13 8.23◦ 13.76◦ 1.14◦ 2.36◦ 4.44◦Corisco Cr = 104 Cg = 1 47.20 1.51◦ 3.26◦ 0.69◦ 1.09◦ 1.51◦Corisco Cr = 104 Cg = 4 16.68 1.71◦ 3.35◦ 0.72◦ 1.14◦ 1.64◦Corisco Cr = 104 Cg = 32 7.57 2.43◦ 4.03◦ 0.97◦ 1.54◦ 2.42◦Corisco Cr = 103 Cg = 1 8.12 1.70◦ 3.22◦ 0.70◦ 1.11◦ 1.77◦Corisco Cr = 103 Cg = 4 2.50 2.02◦ 3.86◦ 0.81◦ 1.24◦ 1.80◦Corisco Cr = 103 Cg = 32 0.99 2.44◦ 3.54◦ 1.00◦ 1.68◦ 2.64◦Corisco Cr = 200 Cg = 1 5.34 2.08◦ 3.38◦ 0.72◦ 1.22◦ 1.78◦Corisco Cr = 200 Cg = 4 1.89 3.27◦ 6.38◦ 0.85◦ 1.34◦ 2.35◦Corisco Cr = 200 Cg = 32 0.45 3.29◦ 4.99◦ 0.99◦ 1.72◦ 3.46◦

YorkUrbanDB

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Method Time Error[s] Mean σ 1/4 Median 3/4

EM Newton 27+? 4.00◦ 1.00◦ 1.15◦ 2.61◦ 4.10◦MAP Quasi-Newton 6+? 4.00◦ 1.00◦ 1.32◦ 2.39◦ 4.07◦EM Quasi-Newton 1+? 9.00◦ 1.00◦ 4.04◦ 6.21◦ 10.33◦J-linkage 1.13 8.23◦ 13.76◦ 1.14◦ 2.36◦ 4.44◦Corisco Cr = 104 Cg = 1 47.20 1.51◦ 3.26◦ 0.69◦ 1.09◦ 1.51◦Corisco Cr = 104 Cg = 4 16.68 1.71◦ 3.35◦ 0.72◦ 1.14◦ 1.64◦Corisco Cr = 104 Cg = 32 7.57 2.43◦ 4.03◦ 0.97◦ 1.54◦ 2.42◦Corisco Cr = 103 Cg = 1 8.12 1.70◦ 3.22◦ 0.70◦ 1.11◦ 1.77◦Corisco Cr = 103 Cg = 4 2.50 2.02◦ 3.86◦ 0.81◦ 1.24◦ 1.80◦Corisco Cr = 103 Cg = 32 0.99 2.44◦ 3.54◦ 1.00◦ 1.68◦ 2.64◦Corisco Cr = 200 Cg = 1 5.34 2.08◦ 3.38◦ 0.72◦ 1.22◦ 1.78◦Corisco Cr = 200 Cg = 4 1.89 3.27◦ 6.38◦ 0.85◦ 1.34◦ 2.35◦Corisco Cr = 200 Cg = 32 0.45 3.29◦ 4.99◦ 0.99◦ 1.72◦ 3.46◦

ApaSt

Images: 24+24 building images.

Model: Perspective with radial distortion (Harris).

Reference: Bundler.

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ApaStBundler

(Snavely et al. [2006]) foi used to obtain the referenceorientations and intrinsic parameters.

Point-based multi-view method.

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ApaSt

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Corisco performance on ApaSt, Cr=10000 RANSAC iterations

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ApaSt

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Corisco performance on ApaSt, Cr=1000 RANSAC iterations

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StreetViewImages: 250 images from an urban environment.

Model: Equiretangular projection.

Reference: Physical sensors (IMU).

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StreetViewErro ' 1◦, time ' 17s.

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Estimativa Referência

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Conclusion

Corisco is a method with great potential for immediateapplication.

Relatively simple implementation.Great robustness (distortions, RANSAC).Good performance.

We demonstrated the advantages of applying the gridmask and M-estimation to the problem.

We also demonstrated how to use FilterSQP in order towork with quaternions in a convenient way.

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Future Work

Automatic parameter control. Replace RANSAC.Apply to multi-view reconstruction, monocularSLAM, object tracking...Use directional filters to measure the angular error.Use the grid mask in other problems.Estimate orientation, camera parameters, andextract curves all in a single unified process. (MRF?)

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Fim

Obrigado!

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Referências BibliográficasB. Caprile and V. Torre. Using vanishing points for camera calibration.International Journal of Computer Vision, 4(2):127–139, March 1990.ISSN 0920-5691. doi: 10.1007/BF00127813. URLhttp://www.springerlink.com/content/k75077108473tm15/.

R Cipolla, T Drummond, and D Robertson. Camera calibration fromvanishing points in images of architectural scenes. In British MachineVision Conference, volume 2, pages 382–391, Nottingham, England,1999. BMVA.

James M. Coughlan and A. L. Yuille. Manhattan World: Orientation andoutlier detection by Bayesian inference. Neural Computation, 15(5):1063—-1088, March 2003. URL http://www.mitpressjournals.org/doi/abs/10.1162/089976603765202668.

Patrick Denis, James H Elder, and Francisco J Estrada. Efficientedge-based methods for estimating Manhattan frames in urban imagery.In European Conference on Computer Vision, pages 197–210, Marselha,França, 2008. Springer. doi: 10.1007/978-3-540-88688-4\_15.

Roger Fletcher and Sven Leyffer. Nonlinear programming without a penaltyfunction. Mathematical Programming, 91(2):239–269, January 2002.ISSN 0025-5610. doi: 10.1007/s101070100244. URLhttp://www.springerlink.com/content/qqj37x00y79ygdl8/.

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Carsten Rother. A new approach to vanishing point detection inarchitectural environments. Image and Vision Computing, 20(9-10):647–655, 2002. doi: 10.1016/S0262-8856(02)00054-9. URLhttp://dx.doi.org/10.1016/S0262-8856(02)00054-9.

G. Schindler and F. Dellaert. Atlanta World: An expectation maximizationframework for simultaneous low-level edge grouping and cameracalibration in complex man-made environments. In Conference onComputer Vision and Pattern Recognition, pages 203–209, Washington,DC, USA, 2004. IEEE. ISBN 0-7695-2158-4. doi:10.1109/CVPR.2004.1315033. URL http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1315033.

Noah Snavely, Steven M. Seitz, and Richard Szeliski. Photo Tourism:Exploring photo collections in 3D. In SIGGRAPH, pages 835–846,Boston, MA, USA, 2006. ACM. URLhttp://dl.acm.org/citation.cfm?id=1141964.

Nicolau Leal Werneck and Anna Helena Reali Costa. Corisco: Robustedgel-based orientation estimation for generic camera models. Imageand Vision Computing, 12(31):969—-981, 2013. URLhttp://nic.hpavc.net/almoxarifado/imavis2013-final.pdf.

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Calibration test

Estimating focal distance with the same F (Ψ).

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Calibração baseada no Corisco - Função objetivo variando com a distância focal

Referência (Bundler) Estimado (Corisco)Referência (Bundler) Estimado (Corisco)Referência (Bundler) Estimado (Corisco)Referência (Bundler) Estimado (Corisco)

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