Bayesian Epipolar Geometry Estimation from

Bayesian Epipolar Geometry Estimation from Tomographic Projections Sami S. Brandt, Katrine Hommelhoff Jensen, and Fran¸cois Lauze Department of Computer Science, University of Copenhagen, Universitetsparken 1, 2100 Copenhagen, Denmark

Abstract. In this paper, we first show that the affine epipolar geometry can be estimated by identifying the common 1D projection from a pair of tomographic parallel projection images and the 1D affine transform between the common 1D projections. To our knowledge, the link between the common 1D projections and the affine epipolar geometry has been unknown previously; and in contrast to the traditional methods of estimating the epipolar geometry, no point correspondences are required. Using these properties, we then propose a Bayesian method for estimating the affine epipolar geometry, where we apply a Gaussian model for the noise and non-informative priors for the nuisance parameters. We derive an analytic form for the marginal posterior distribution, where the nuisance parameters are integrated out. The marginal posterior is sampled by a hybrid Gibbs–Metropolis–Hastings sampler and the conditional mean and the covariance over the posterior are evaluated on the homogeneous manifold of affine fundamental matrices. We obtained promising results with synthetic 3D Shepp–Logan phantom as well as with real cryo-electron microscope projections.

1

Introduction

The common line theorem states that there is a common 1D projection shared by a pair of 2D tomographic parallel projections of a 3D object [1]. The common 1D projection follows from the Fourier slice theorem or the fact that the 2D Fourier transform of a tomographic parallel projection forms a central slice of the 3D Fourier transform of the object [2]. Two central slices generally intersect on a 3D line, whose image interpretation, after using the Fourier slice theorem once again on the image plane, is a 1D projection that is identical in the two projection views. Geometrically, the common 1D projection can be identified as the projection of the rotation axis onto the calibrated views [3]. We will show that the common lines and the affine epipolar geometry [4] are related. The relation comes from the fact that the projection over the corresponding epipolar lines is identical in the normalised calibrated views since in this case it is the integral over the corresponding epipolar plane. Moreover, the common 1D projection is the corresponding line orthogonal to the epipolar lines as the rotation axis of the object is orthogonal to the epipolar planes. In the more general uncalibrated case, one has to find the 1D affine transform to link

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Sami S. Brandt, Katrine Hommelhoff Jensen, and Fran¸cois Lauze

the points on the common 1D projections, in addition to the direction of the common 1D projection, in order to recover the affine epipolar geometry. After proofing this geometric relationship, we propose a Bayesian method to estimate the affine epipolar geometry for a pair of views where no pointwise correspondences [5] nor explicit 3D reconstruction [6] are required. The marginal posterior distribution of the geometry is derived in closed-form using certain noninformative priors for nuisance parameters. To sample the posterior, we propose a hybrid Gibbs–Metropolis–Hastings sampler, where the proposal distributions are generated by spline interpolated approximations of the 1D conditional distributions. The posterior samples allow us to estimate the conditional mean and covariance over the homogeneous manifold of affine fundamental matrices. The results of this paper are central in applications that rely on the geometric relationship between parallel tomographic projections such as x-ray or transmission electron microscope images. Due to the statistical nature of the proposed method, this is especially a very promising tool to geometrically register single particle images in cryo transmission electron microscopy [7] aiming at 3D reconstruction of the particle. In cryo-EM, conventional computer vision methods, based on point correspondences [3, 8], are not applicable due to the very high level of noise. The organisation of this paper is as follows. In Section 2, we show the formal relationship between the common lines and affine epipolar geometry. In Section 3, we derive the marginal posterior distribution for the affine epipolar geometry for a pair of tomographic parallel projections. In Section 4, we describe how we sample and summarise the posterior. The experiments are in Section 5, and Section 6 concludes the paper.

2

Epipolar Geometry and Common Lines

We start with showing the geometric relationship between the common 1D projections with uncalibrated parallel projection geometry. Without a loss of generality, we may identify the common 1D projection as the line through the image origin, where the line has the direction angles θ1 , θ2 in the views one and two, respectively. The common 1D projections p1 and p2 onto the identified line are the integrals over the parallel lines orthogonal to the line (see Fig. 1). With uncalibrated projection geometry, the common 1D projections also have an unknown affine ambiguity after the directions have been identified. More formally, we state as follows. Theorem 1. Given two uncalibrated tomographic projection images I1 and I2 of a bounded object, there are corresponding 1D projections p1 (x) and p2 (x) that are related by p1 (x) = γp2 (αx + β) where α, β and γ are unknown scalar constants. Proof. Without a loss of generality, the projection of the inhomogeneous 3-vector X ∈ R3 onto the normalised, calibrated coordinates onto two 2D images can be described by the geometric projection equation u = PRX,

u0 = PR0 X

(1)

Bayesian Epipolar Geometry Estimation from Tomographic Projections

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Fig. 1. An illustration of the connection between the common 1D projection and affine epipolar geometry. Two 2D parallel projection images I and I 0 of the 3D object are illustrated with the common 1D projections p1 and p2 as well as matching 2D points x and x0 along the corresponding epipolar lines l and l0 . The value on the common 1D projection represents the 3D object density integrated over the epipolar plane.

where u, u0 ∈ R2 are the corresponding projected 2D coordinates vector in the normalised views I˜1 , I˜2 , P is the 2 × 3 orthographic projection matrix with the unity diagonal, and R, R0 are 3D rotations. The inhomogeneous image coordinates relate to the normalised coordinates by the affine transform x = Ku + t,

x0 = K0 u0 + t0 ,

(2)

where K, K0 are non-singular, 2 × 2 upper triangular matrices containing the affine parameters of the cameras, and t, t0 are 2 × 1 translation vectors. From the well known common line theorem it follows that there is a common 1D projection between the normalised projection images. In other words, the integral over an epipolar plane is equal to the integral over the corresponding epipolar lines in the normalised images (Fig. 1). The common 1D projection is Z Z p˜(s) = I˜1 (u)dL = I˜2 (u0 )dL (3) ˜ T u=s n

T n˜0 u0 =s

where the normalised images I˜1 (u) = I1 (Ku + t) and I˜2 (u0 ) = I2 (K0 u0 + t0 ), ˜, n ˜ 0 are orthogonal to the epipolar lines in the normalised and the unit vectors n images, respectively. For simplicity, we define the sign of the unit vectors so that they point towards the positive direction of the 3D rotation axis defined by the 3D rotation from the view 1 to the view 2. Let us first consider the first view. We may make the coordinate transform for the normalised images I˜1 by making the substitution u = K−1 (x − t). The Jacobian of the substitution mapping is K−1 , hence, Z Z p˜(s) = I˜1 (u)dL = c I1 (x)dL = cp1 (t), (4) ˜ T u=s n

nT u=t

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Sami S. Brandt, Katrine Hommelhoff Jensen, and Fran¸cois Lauze

where c = | det(K)|−1 , n is the unit vector orthogonal to the epipolar lines in ˜ , p1 (t) is the unnormalised coordinate frame, where the sign of n set as for n the 1D projection of the image into perpendicular direction of n, and t is the signed orthogonal distance of the epipolar line from t. Making the corresponding analysis to the second view yields p˜(s) = c0 p2 (t0 ), where c0 = | det(K0 )|−1 . Since c, c0 6= 0, it follows that p1 (t) = γp2 (t0 ), where γ = c0 /c. It remains to be shown that the 1D coordinates t and t0 on the common 1D projections, respectively, are related by an 1D affine transform on the unnormalised image planes. For the unnormalised first view, the line corresponding to the common 1D projection in the normalised frame is sK˜ n +t, where s is the real parameter. On the other hand, we may equivalently parameterise the common 1D projection by the line tn which meets the origin, where t = snT K˜ n + nT t. Sim0 0T 0 ˜ 0 0T 0 ilarly for the second view, t = sn K n + n t , which implies that t0 = αt + β, 0T 0 0 ˜0 T ˜ n0T K0 n n 0T 0 where α = nnTK K˜ n and β = n t − nT K˜ n n t.  Corollary 1. The scaling parameter γ between the common uncalibrated 1D projections of two views defines the scalar constraint γ = | det(K)/ det(K0 )| for the affine camera parameters. The geometry of common 1D projections for a view pair is defined by the four parameters (θ1 , θ2 , α, β). On the other hand, the affine epipolar geometry is defined by the linear epipolar equation f13 x0 + f23 y 0 + f31 x + f32 y + f33 = 0,

(5)

where f·· represents the elements of the affine fundamental matrix, (x, y) and (x0 , y 0 ) are the inhomogeneous coordinates of the points x and x0 in the first and second view, respectively. The following theorem states the equivalence between the common line geometry and the affine epipolar geometry. Theorem 2. Given two uncalibrated tomographic parallel projection images I and I 0 , identifying the corresponding 1D projections together with the 1D affine transform between the 1D coordinates is equivalent to identifying the affine epipolar geometry between the views. Proof. Assume first that the affine epipolar geometry is known, i.e., there is an affine fundamental matrix F [4] available. The epipolar line corresponding to a point x = (x, y, 1) in the first view is l0 = Fx = (f13 , f23 , f31 x + f32 y + f33 ) in the second view. Let the unit direction vector orthogonal to the epipolar line, defining the orientation of the 1D projection be n0 = −a0 (f13 , f23 )/s0 where s0 = k(f13 , f23 )k > 0, a0 = ±1 and correspondingly let n = a(f31 , f32 )/s where s = k(f31 , f32 )k > 0, a = ±1 in the first view.1 The cases s = 0 or s0 = 0 are not possible since these cases do not represent valid affine epipolar geometries between the views. Hence, we may write l0 =(−a0 s0 n0 , asnT (x, y) + f33 ). 1

The signs of a and a0 can be chosen arbitrarily, but the selection a0 = a corresponds to the case where the normals n and n0 point to the same side of the epipolar plane.

Bayesian Epipolar Geometry Estimation from Tomographic Projections

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Now, using the standard formula for the distance between a point and a line, the signed orthogonal distance between the origin O0 and the epipolar line l0 is d0 = ζ|asnT (x, y) + f33 |/s0 = ζ|asd + f33 |/s0 where the sign is set to ζ = a0 sign(asnT (x, y) + f33 ). This corresponds to the choice that the signed distance d = nT (x, y) in the first view has the equal sign to the corresponding distance d0 in the case where the normals n and n0 point towards the same side of the epipolar plane, and the opposite sign otherwise.2 Hence, d0 = a0 (asd + f33 )/s0 = αd + β, where α = (as)/(a0 s0 ) and β = f33 /(a0 s0 ). We have thus shown that the 1D projection directions and the parameters α and β of the affine transform between the identified 1D projections can be computed from a known affine fundamental matrix between the views. Conversely, assume that we have identified the common 1D projection parameterised by the respective unit direction vectors n, n0 ∈ R2 , defining the positive directions of the 1D coordinate systems; and the constants α 6= 0 and β defining the affine transform between the common 1D projections. Without a loss of generality, we may assume that the origins of the 1D projections have been set to the projections of the respective 2D image origins. The unit normal of epipolar line l0 is n0 , hence, (f13 , f23 ) = t0 n0 , where t0 6= 0 is a scalar. Similarly, (f31 , f32 ) = tn, t 6= 0. The corresponding points on the 1D projections x = (dn, 1) and x0 = ((αd + β)n0 , 1) must satisfy the epipolar T equation x0 Fx = 0 for all d and d0 that yields T

0 = t0 (αd + β)n0 n0 + tdnT n + f33 = (t + t0 α)d + (f33 + t0 β),

(6)

thus, t0 = −t/α and f33 = tβ/α. Collecting the constraint equations into a matrix from yields    f13 −α/t 0 0 0 0  0  0 −α/t 0 0 0  f23  n         0 0 1/t 0 0   f31  = n   0 0 0 1/t 0  f32  β 0 0 0 0 α/t f33

(7)

where there is a unique solution for all α, t 6= 0. The fundamental matrix is a homogeneous quantity so, it is defined only up to scale, thus the scale parameter t can be set to an arbitrary non-zero value. We have thus shown that identifying the corresponding 1D projections and the 1D affine transform between the 1D projections imply that the affine epipolar geometry is identified between the views.  2

This can be seen from the fact that limd→∞ sign(asd+f33 ) = a, and a = a0 when the normals point towards the same side of the epipolar plane and a = −a0 otherwise.

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Sami S. Brandt, Katrine Hommelhoff Jensen, and Fran¸cois Lauze

Posterior Distribution for Epipolar Geometry

After showing the relationship between the common-lines and the affine epipolar geometry, we now derive the posterior distribution for the affine epipolar geometry. Let D denote the measured data or the pixel intensities in the images, and p1 and p2 the vectors representing the discretised 1D projections of the 2D planes, respectively. Assuming i.i.d. zero mean Gaussian noise on the 1D projections, we should consider the likelihood, associated with the reprojection error,   1  2 2 −2N ˆ 2 k + kp2 − p ˆ 2k (8) L(D|θ) = σ exp − 2 kp1 − γ p 2σ where θ = (ˆ p2 , θ1 , θ2 , α, β, γ, σ), N is the number of discrete points on the proˆ 1 and p ˆ 2 are related jection pi = pi (θi ), i = 1, 2; and the noise free projections p ˆ1 = γp ˆ 2 according to Theorem 1. by p The noise-free projections, the scaling parameter γ, and the noise variance σ 2 are nuisance parameters and should be integrated out. First, using a uniform ˆ 2 , after some algebra the marginalisation yields prior for p   Z ∞ σ −N 1 2 L(D|θ)dˆ p2 ∝ kp1 − γp2 k exp − 2 2σ (1 + γ 2 ) (1 + γ 2 )N/2 −∞   1 (9) 2 ∝ σr−N exp − 2 kp1 − γp2 k 2σr = L(D|θ0 ), which can still be interpreted as another likelihood function, since σr2 = σ 2 (1+γ 2 ) is the variance of the residual r = p1 − γp2 , and θ0 = (θ1 , θ2 , α, β, γ, σr ). We assume a uniform prior for γ and the non-informative, Jeffrey’s prior p(σr ) ∝ σ1r for the residual deviation [9, 10]. The marginal posterior finally is ZZ p(θ1 , θ2 , α, β|D) ∝ L(D|θ0 )p(σr )dγdσr   Z ∞Z ∞ 1 −N −1 2 ∝ σr exp − 2 kp1 − γp2 k dγdσr 2σr 0 −∞    Z ∞ T 2 (10) 1 1 N ¯2 − (u2 p1 ) ∝ exp − p dσr 1 N 2 kp2 k 0 σr 2σr N  1−N  2 2 p¯21 − N1 (uT 2 p1 ) , ∝ kp2 k where u2 is the vector p2 normalised to the unit length.

4 4.1

Estimation Algorithm Sampling the Posterior

We propose to sample the posterior (10) by a hybrid method that combines Gibbs sampling with the Metropolis–Hastings method [11]. We have an outer

Bayesian Epipolar Geometry Estimation from Tomographic Projections

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Algorithm 1 Gibbs–Metropolis–Hastings Sampler (0)

(0)

Input: The projection images I and I 0 , initial guess ϑ(0) = (θ1 , θ2 , α(0) , β (0) ) for the parameters. Output: Effectively independent samples from the posterior p(θ1 , θ2 , α, β|D). 1. Set n = 1, d = 1. (n) (n) (n−1) (n−1) 2. Draw a sample ϑ0d from papprox (ϑd |ϑ1 , . . . , ϑd−1 , ϑd+1 , . . . , ϑ4 ). (n)

(n)

(n)

(n)

3. Accept or reject ϑ0d to be a sample from p(ϑd |ϑ1 , . . . , ϑd−1 , ϑd+1 , . . . , ϑ4 ) with (n)

the Metropolis–Hastings criterion (12). If accepted, set ϑd = ϑ0d , otherwise go to Step 2. 4. Increment d by one. Repeat from Step 2 until all the four dimensions have been processed. 5. Increment n by one, set d = 1. Repeat from Step 2 until enough samples have been obtained.

Gibbs sampler that sequentially samples the one-dimensional conditionals while the conditional densities are sampled the Metropolis–Hastings method. Assuming bounded parameter space, the proposal distribution can be generated by discretising one-dimensional conditional density and generating a continuous approximation by spline interpolation. Let ϑ = (θ1 , θ2 , α, β). Looking at the dth variable ϑd , the conditional is approximated as papprox (ϑd |ϑ1 , . . . , ϑd−1 , ϑd+1 , . . . , ϑ4 ) P m g(ϑd − ϑm d )p(ϑd |ϑ1 , . . . , ϑd−1 , ϑd+1 , . . . , ϑ4 ) = m P , m m p(ϑd |ϑ1 , . . . , ϑd−1 , ϑd+1 , . . . , ϑ4 )

(11)

1 th where ϑm d = ϑd + (m − 1)∆ϑd , m = 1, . . . , M are equidistant samples of the d variable, g is the interpolation kernel, where we use the zeroth order spline. Sampling from the conditional distribution is now straightforward. First a sample is generated from the proposal density by drawing a sample s ∼ Uniform(0, 1), and finding the inverse of the conditional probability distribu(−1) tion vd0 = Papprox (s|ϑ1 , . . . , ϑd−1 , ϑd+1 , . . . , ϑ4 ), which can be implemented by linearly interpolating the cumulative sum of the discretised conditional density. The proposed sample vd0 is accepted by the Metropolis–Hastings criterion or it is accepted with the probability  p(ϑ0d |ϑ1 , . . . , ϑd−1 , ϑd+1 , . . . , ϑ4 ) min 1, × p(ϑd |ϑ1 , . . . , ϑd−1 , ϑd+1 , . . . , ϑ4 )  (12) papprox (ϑd |ϑ1 , . . . , ϑd−1 , ϑd+1 , . . . , ϑ4 ) × , papprox (ϑ0d |ϑ1 , . . . , ϑd−1 , ϑd+1 , . . . , ϑ4 )

otherwise it is rejected. The method is summarised in Algorithm 1.

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Sami S. Brandt, Katrine Hommelhoff Jensen, and Fran¸cois Lauze

(a) φ = 22.5◦

(b) φ = 45◦

(c) φ = 67.5◦

(d) φ = 90◦

(e) σ/µ = 0.1

(f) σ/µ = 0.4

(g) σ/µ = 1.6

(h) σ/µ = 6.5

Fig. 2. (top) Shepp–Logan phantom rotated over the y-axis with 3D reference points (blue) superimposed; (bottom) Different levels of noise illustrated (φ = 0).

4.2

Fundamental Matrix and Its Covariance (n)

(n)

Having the samples from the posterior (θ1 , θ2 , α(n) , β (n) ), n = 1, . . . , Nsamples we also have the affine fundamental matrix estimates, according to Theorem 2. We summarise the samples by estimating the conditional mean of the affine fundamental matrices over the posterior and the posterior covariance of the fundamental matrices. However, due to the homogeneous scaling ambiguity, we need to define in which sense the mean is computed. Without a loss of generality, the affine fundamental matrix estimate can be parameterised to be a point on the four dimensional unit sphere S4 . Due to the construction, we can moreover assume that all samples are in the same halfsphere. Assuming uniform weighting, we use the standard Riemannian distance on S4 . The mean affine fundamental matrix is defined as the estimate that minimises the intrinsic variance for the distance d on the sphere [12] or, ˆf = argmin f ∈S4

X

d(f (n) , f )2 .

(13)

n

The posterior covariance is naturally estimated in the tangent space of the mean, so that  T X 1 ˆf = C logˆf f (n) logˆf f (n) , (14) Nsamples n where logˆf is the Riemannian Log map [12] computed at the mean ˆf .

Bayesian Epipolar Geometry Estimation from Tomographic Projections

9

8 Mean reprojection error in pixels

Mean reprojection error in pixels

8 7 6 5 4 3 2 1

7 6 5 4 3 2 1 0

0

0

0

10 Inverse of the Signal−to−Noise−Ratio ◦

10 Inverse of the Signal−to−Noise−Ratio ◦

(a) φ = 22.5

(b) φ = 45

15

7 Mean reprojection error in pixels

Mean reprojection error in pixels

8

6 5 4 3 2 1 0 0

10 Inverse of the Signal−to−Noise−Ratio ◦

(c) φ = 67.5

10

5

0

0

10 Inverse of the Signal−to−Noise−Ratio

(d) φ = 90◦

Fig. 3. Reprojection error on the Shepp–Logan phantom in the function of the inverse SNR (σ/µ).

5

Experiments

We first experimented our method with the classic 3D Shepp–Logan (128×128× 128), see Fig. 2. The left view was taken at the rotation φ = 0 over the y-axis and the right view at φ = 22.5◦ , 45◦ , 67.5◦ , 90◦ . For all the projections, i.i.d. Gaussian noise were added with inverse signal-to-noise-ratio (σ/µ, where µ is the mean intensity ) from 0.1 to 6.5. The parameter bounds were set so that θ1 ∈ [0, π], θ2 ∈ [0, 2π), α ∈ [−1.5, −0.5], β ∈ [−N/5, N/5]. For each image pair and noise level, our method was initialised to the MAP estimate for θ1 and θ2 (no burnin), conditioned on α = α(0) and β = β (0) , whereas α and β were initialised to minus one and zero, respectively. We drew 100 samples and computed the mean affine fundamental matrix from (13) and evaluated the standard reprojection error [4] of the reference points superimposed in Fig. 2. The computations were repeated 7 times on each noise and angle combination, and the mean of the mean estimates with the standard deviations of the means are shown in Fig. 3.

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Sami S. Brandt, Katrine Hommelhoff Jensen, and Fran¸cois Lauze

It can be seen that the method is very robust to noise, as the reprojection error was mostly below 2 pixels, and it does not break down until the level of noise is about six times higher than the level of the signal. In the the real data experiment, we experimented the CroEl, Mm-Cpn, and Ribosome particles taken by a cryo transmission electron microscope. The first experiments were made with class average images, i.e., the mean images over noisy sets of 53 to 418 projections that have been classified to indicate a similar projection view. The second set of real experiments were performed on the raw particle images. In the experiments, the parameter bounds and the initial guess were set similarly as above. Fig. 5 illustrates the log posterior of the parameters conditioned on the initial guess for α and β in all the six cases experimented. We drew samples for the parameters, converted them to affine fundamental matrices, and computed their conditional mean as the result. Fig. 4 shows the results on class averages, where we have shown three points on one image and show the corresponding, estimated mean epipolar lines in the other image. In all the three cases the correct epipolar geometry has been found, which can be concluded from the known 3D structure and symmetry of the particles. Fig. 6 displays the results on the raw particle images. It can be seen that a reasonable estimate was found in all these cases, but the high noise level slightly deviates the direction of the epipolar lines with the GroEl particle from the ground truth.

6

Conclusions

In this paper we have shown the equivalence between uncalibrated common line geometry and the affine epipolar geometry. In contrast to the traditional methods, no point correspondences are required to estimate the epipolar geometry. Using the common-line geometry and Gaussian noise model, we then derived the marginal posterior distribution for the affine epipolar geometry using certain non-informative priors for the nuisance parameters. We additionally proposed a hybrid Gibbs-Metropolis-Hastings sampler to sample the posterior distribution. We summarised the samples by the conditional mean on the manifold of the affine fundamental matrices. Our experiments on both synthetic and real data show that our approach is successful in recovering the affine geometry of tomographic parallel projections with a very low signal to noise ratio.

References 1. Crowther, R., Amos, L., Finch, J., De Rosier, D., Klug, A.: Three dimensional reconstructions of spherical viruses by fourier synthesis from electron micrographs. Nature 226 (1970) 421–425 2. Kak, A.C., Slaney, M.: Principles of Computerized Tomographic Imaging. IEEE Press (1988) 3. Brandt, S.S.: Markerless alignment in electron tomography. In Frank, J., ed.: Electron Tomography. Springer (2006) 4. Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge (2000)

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Fig. 4. Estimated Epipolar Geometry between the real class average images of cryo TEM projections: (a,d) GroEL, (b,e) Mm-Cpn particle, (c,f) Ribosome. The features pointed by the three points meet the corresponding points on the epipolar lines. Original images are available from http://blake.bcm.edu/emanwiki/Ws2011/Eman2.

5. Brandt, S., Heikkonen, J.: Optimal method for the affine F-matrix and its uncertainty estimation in the sense of both noise and outliers. In: Proc. ICCV. Volume 2. (2001) 166–173 6. Brandt, S.S., Kolehmainen, V.: Structure-from-motion without correspondence from tomographic projections by bayesian inversion theory. IEEE Trans. Med. Imaging 26 (2007) 238–248 7. Frank, J.: Three-Dimensional Electron Microscopy of Macromolecular Assemblies. Oxford (2006) 8. Brandt, S.S., Ziese, U.: Automatic TEM image alignment by trifocal geometry. Journal of Microscopy 222 (2006) 1–14 9. Bretthorst, G.L.: Bayesian Spectrum Analysis and Parameter Estimation. Springer (1988) 10. Brandt, S.S., Palander, K.: A Bayesian approach for affine auto-calibration. In: Proc. SCIA 2005). (2005) 577–587 11. MacKay, D.: Information Theory, Inference and Learning Algorithms. Cambridge (2003) 12. Pennec, X.: Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements. JMIV 25 (2006) 127–154

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(a) GroEL Class

(b) Mm-Cpn Class

(c) Ribosome Class

(d) GroEL Raw

(e) Mm-Cpn Raw

(f) Ribosome Raw

Fig. 5. Log posterior maps of (θ1 , θ2 ) conditioned on the α = α(0) , β = β (0) , using (a-c) the class average image pairs, and (d-e) individual particle image pairs. The red colour indicates the highest probability, blue the lowest.

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Fig. 6. Real particle projections by a cryo TEM with three points and the estimated epipolar lines; (a,d) GroEL; (b,e) Mm-Cpn particle; (c,f) Ribosome; original images available from http://blake.bcm.edu/emanwiki/Ws2011/Eman2.