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Fake uniformity in a shape inversion formula

We revisit a shape inversion formula derived by Panaretos in the context of a particle density estimation problem with unknown rotation of the particle. A distribution is presented which imitates, or 'fakes', the uniformity or Haar distribution that is part of that formula.


Introduction
Random rotations in Euclidean space R d (d ≥ 2) arise in a variety of contexts in stochastic geometry, especially in the construction of other stochastic models, for example random hyperplane processes; see the monograph [13]. In these contexts, the rotation distribution is usually taken as the Haar probability measure µ = µ d on the d-dimensional rotation group SO(d ). Recall that µ is characterised by the property that µ(QA) = µ(AQ) = µ(A) for any Q ∈ SO(d ) and any measurable A ⊆ SO(d ), where SO(d ) is endowed with its Borel σ-field, QA = {QP : P ∈ SO(d )}, and AQ is defined analogously. See [13,Ch. 13] for more information and one of several constructions of Haar measure on SO(d ). Most applications concern the case d = 3, not only in stochastic geometry but also statistics. Two important instances are a statistical model for random orientations of limbs of experimental subjects performing drilling tasks, studied in [4], and a model for tomographic imaging introduced in [7] and further studied in [8]. In [4], a class of rotation distributions in R d was introduced that was shown to be amenable to exact computations of several key quantities including first and second moments. In [7], the goal was to estimate the shape (as defined in that paper) of a particle imaged with cryo-electron microscopy, where the orientation of the particle is not only random but also unknown, its orientation being governed by µ. As was shown in that paper, this last assumption is necessary in order to produce a well-defined estimator of the shape of the particle.
The basis of the present paper is a synthesis of [4] and [7]. Our focus is on probabilistic rather than statistical aspects of the model. Specifically, we study a probability model that arises when the rotation distribution in the stochastic Radon transform introduced in [7] is not Haar measure, but only possesses the more general symmetry property of conjugation-invariance, whose definition we recall in Section 2. Our results will be more general than what we need for the tomography application, and in particular, we will obtain several new results relating to three-dimensional rotation distributions. We also derive the Bayes classification rule with two classes which are shifts of a common conjugation-invariant distribution, and examine the behaviour for small shifts. For the case of two multivariate Normal distributions on R d with equal covariance matrices, a corresponding result is well known; see [10, pp. 21-22]. As far as we are aware, our approach is essentially new, although there is some similarity in some instances with the geometric reasoning in [3], which studied the distribution of the projected shape under Haar measure.
Our two main results, namely Theorems 1 and 2 in Sections 4 and 5 respectively, yield two different ways of quantifying the effect of moving away from the Haar measure situation. In relation to the first of these results, a finding of particular interest is what we call a fake uniformity property, 'uniform' being taken here and below as a synonym for Haar measure µ (see the discussion around Definition 1 in Section 4). The second result quantifies how well one may distinguish two distributions for a shift of 'modal' angle, as explained later; as expected, we find better accuracy for stronger deviation of the common 'centred' distribution from µ (see the discussion around (12) in Section 5). We shall not attempt to develop a variational approach using distances between rotation distributions as was done in [9], as this seems a secondary matter to the phenomena we study. Our results also provide a new interpretation of the concentration parameter in statistical models for random rotations. Finally, our results may provide a foundation for statistical tests that are based on the quantities we consider. It seems clear that the fake uniformity property is a problem for the most basic of such tests, namely that for uniformity of the rotation. Section 2 contains background on rotations, as well as notation. Section 3 investigates the connection between two univariate distributions which describe conjugation-invariant rotations. Section 4 contains the main result on the stochastic Radon transform for what we call shiftsymmetric rotations, along with an example that reveals the fake uniformity property and motivates us to define it separately. Finally, in Section 5 the Bayes classifier is derived.

Notation and Background
All random variables and random elements on other state spaces are taken to reside on a common probability space (Ω, F , P), with E(·) denoting the corresponding expectation operator.
We shall be concerned not only with random variables on R, but also on the sphere in three dimensions S 2 := {x ∈ R 3 : x = 1}, and on SO(3). If T denotes a random element in either of these three spaces, we shall denote its density by f T , it being understood that the dominating measure for this density is ordinary Lebesgue measure, normalised Lebesgue surface measure σ on S 2 , or Haar probability measure µ on SO(3) respectively. As done above, we use boldfacing to distinguish deterministic and random rotations, but will use uppercase and lowercase letters for vectors and scalars as is customary. All vectors are regarded as column vectors. We write v i = v · e (i) for v ∈ R 3 , where the dot denotes the usual inner product and e (i) is the ith column of the 3 × 3 identity matrix, for i = 1, 2, 3. The matrix diag(a 1 , a 2 , a 3 ) is the 3 × 3 diagonal matrix with entries a 1 , a 2 , a 3 along its main diagonal. The Beta integral is and Beta(p, q) is the Beta distribution with parameters p, q > 0. Recall that if W ∼ Beta(p, q) then E(W r ) = (p) r (p + q) r for any integer r ≥ 0, where (x) n = Γ(x + 1)/Γ(x − n + 1) is the Pochhammer symbol and Γ(·) is the Gamma function.
We use the notation E(X; B) := E(X1 B ) for a scalar random variable X and a measurable set Recall that any rotation P ∈ SO(3) (except if it belongs to the null set comprising the identity and rotations by an angle π) may be expressed uniquely in terms of the rotation axis u ∈ S 2 and rotation angle θ ∈ [0, π), as follows (see [11, pp. 188-189] or [12, p. 99]): We assume throughout that the letter R denotes a random three-dimensional rotation which The terminology here follows [12]. We say that R (or its distribution) belongs to I s . This definition for I s is somewhat less restrictive than the one given in [11, p. 190]. In the context of random P at (2), one has f P ∈ I s if and only if U ∼ σ 2 (cf. [11, p. 191]). In the sequel, when referring to random elements U and Θ, we always assume them to be defined through P via the bijection (up to a null set) P ↔ (U, Θ).
To date, only few rotation distributions have been studied in the literature. We shall consider the distribution introduced in [4], referred to below as the Cayley-LMR distribution: and the symmetric matrix Fisher-von Mises distribution, whose density is proportional to exp{κ(P M T )}; see [5, pp. 289ff] for more information on the latter distribution. In both cases, κ ≥ 0 is a dispersion parameter, with κ = 0 corresponding to the case f P = f R = µ, and M may be regarded as a modal matrix. We shall make the minimal assumption that our rotation distributions are parametrised by modal matrices in the sense of a group model, see [9, p. 653], and a dispersion parameter κ, with the case κ = 0 corresponding to Haar measure µ. They are allowed to have other parameters as well.
We shall on several occasions consider the random variable and its (central) moments , p. 425) and thus by (1), It will become clear later why we regard these as functions of κ.
Finally, we introduce some convenient terminology. For a random rotation P which is such that there is a conjugation-invariant rotation R ∈ SO(3) and a deterministic rotation M ∈ SO (3) such that P M T d = R, we write P ∈ I M s . We call Q shift-symmetric, and write Q ∈ I + s , if there exists an M such that Q ∈ I M s . The same terminology applies to the underlying distributions of the random variables.

Relation between two distributions and moments
In this section we investigate the connection between two key distributions derived from a three-dimensional rotation. The first proposition shows that if f R ∈ I s then the density f Re (3) is simply that of In the proof and in several later instances we shall need the following simple result on integration on the sphere, which is a special case of [1, Proposition 9.1.2, p. 189]: .
where ∠(v, w) = arccos(v · w) ∈ [0, π) is the angle between the vectors v, w ∈ S 2 . Hence 3 ) almost surely. The equality of the densities follows from Lemma 1, cf. also the remark in [1, p. 189]. Now (2) shows that the distribution of R, and thus also that of Re (3) , is determined uniquely.
and differentiation with respect to s completes the proof.
The relationship between the densities in Proposition 2 will now be used to relate the corresponding moments up to some finite order. Consider the vector τ = (τ j ) m j=0 , m ≥ 1, where τ j = E(Z j ). Lemma 1 and Proposition 2 yield for k ≥ 1, by partial integration, Another partial integration gives, for any k ≥ 1 and −1 ≤ x ≤ 1, where G 0 ≡ 0 and

By induction,
It seems impossible to reduce this further. We have which combined with (8) gives The coefficient of ρ k in the expression for τ k is readily seen to be equal to k!2 k−1 √ π/Γ(k+3/2) → 2 as k → ∞, by Stirling's formula.

Gram inversion for the stochastic Radon transform and fake uniformity
In this section we consider the stochastic Radon transform, originally introduced in [7] in the case where the rotation distribution is Haar measure µ, for conjugation-invariant rotation distributions, and will utilise results from the previous section. We first briefly repeat the main points of the stochastic Radon transform that are relevant for our own purposes; more information can be found in [7] and the references cited therein. The definition of the stochastic Radon transform in [7] is based on the projection operator Π of an essentially bounded integrable function ρ on R 3 , taken to have total mass unity and integral mean zero, with support contained in a ball of radius π centred at the origin: where the rotation group acts on functions on R 3 via (Aρ)(w) = ρ(A −1 w). For a random rotation, the map Π gives rise to a distribution P ρ on an L 2 space of planar functions where B ranges over the Borel sets in L 2 space. Note that here we consider what we may call the 'population version' of the stochastic Radon transform.
In [7] it was shown that P Qρ = P ρ for all Q ∈ SO(3), which implies that only the shape of ρ, that is the orbit of ρ under the above group action, may be recovered in the estimation from sample data. This motivates the consideration of the Gram matrix. The Gram matrix of V ∈ R d×k is  1, 0). Then for f P ∈ I M s , Proof. As in [7, pp. 3285-3286], one obtains from Proposition 1 and the fact that This completes the proof.
Remark 2. The first equation of the above proof is easily adaptable to rotations in R d , d ≥ 2.
The following example reveals important consequences of the results developed so far.
Definition 1. Let P be a family of rotation distributions as described in Section 2. We say that P has the fake uniformity property if there exists κ > 0 such that the Gram matrix in the context of Theorem 1 has the same distribution as for µ.
The fake uniformity property sets the Cayley-LMR distribution distinctly apart from the Fishervon Mises distribution in a novel aspect, besides the matter of the singular behaviour of the former at Θ = π already noted in [4, p. 425]. It considerably complicates, if not frustrates, any attempt at the development of a statistical test for the rotation distribution to be Haar measure, if only the random projections are known. For it to be present, a sufficient condition is that the derivative of τ 2 (κ) at κ = 0+ is negative. By (9), the latter condition is satisfied if and only if 4 3 ρ 2 (0+) < ρ 1 (0+) .

Bayes classifier for shift-symmetric rotations
We continue to use the notation and framework from Section 4. Consider now two shift- The angle α in (11) is the Riemannian distance between the rotations P i when coupled as in our definitions; see [6]. Our main result in this section is as follows. For simplicity we assume equal prior probabilities π 1 = π 2 = 0.5 for selecting an object distributed as P 1 or P 2 .
Theorem 2. With the foregoing notation and conditions, let T denote the event of correctly classifying a Class 1 [respectively, arbitrary] observation to Class 1 [respectively, its true class].
Then Proof of Theorem 2 is at the end of this section. We first discuss classification accuracy for small α. Define Note that for the Cayley-LMR distribution with M = I, we have h(x) = x κ /B(κ + 1/2, 3/2).
Regarding the left-hand side in Theorem 2 as a function ψ R (α) and writing w α = cos(α/2), we It is readily checked that, with the ± signs taken respectively, , from which we obtain after some manipulations the simple expression a weighted difference between two integral means. This expression is analogous to the wellknown one for the case of two multivariate Normal distributions with equal covariance matrix given in [10, p. 22], when α = 0. One may also regard it as a measure for deviation from uniformity, but unlike the one presented in the previous section it is a non-local one and hence probably less useful in practice.
Proof of Theorem 2. We only need to prove the assertion in the case where T is the event of correct classification into Class 1. We have P(Observation correctly assigned to Class 1) = P tr P M T Moreover, with the notation at (11), Using the representation at (2) The zeros of this quadratic in U 3 are U hence (noting that cot β − csc β = − tan(β/2) and cot β + csc β = cot(β/2)) U (1) It is easily seen that U 3 ≤ −1 if and only if Θ ≤ α/2, and U 3 ≥ 1 if and only if Θ ≤ π − α/2. Thus the integration with respect to the distribution F U 3 of U 3 is over the domains The density of arcsin U 3 is {γ → (cos γ)/2} for γ ∈ [−π/2, π/2]. Changing variables and integrating over F U 3 gives for the three expressions in (14), respectively, P(Θ ≤ α/2), π−α/2 α/2 F Θ (dθ) where we used the fact that This concludes the proof of Theorem 2.