Original: A multi-organ point cloud registration algorithm for abdominal CT registration
Revised: A multi-organ point cloud registration algorithm for abdominal CT registration

Original: Abstract.
Revised: Abstract.

Original: Registering CT images of the chest is a crucial step for several tasks such as disease progression tracking or surgical planning.
Revised: Registering CT images of the chest is a crucial step for several tasks such as disease progression tracking or surgical planning.

Original: It is also a challenging step because of the heterogeneous content of the human abdomen which implies complex deformations.
Revised: It is also a challenging step because of the heterogeneous content of the human abdomen which implies complex deformations.

Original: In this work, we focus on accurately registering a subset of organs of interest.
Revised: In this work, we focus on accurately registering a subset of organs of interest.

Original: We register organ surface point clouds, as may typically be extracted from an automatic segmentation pipeline, by expanding the Bayesian Coherent
Revised: We register organ surface point clouds, as may typically be extracted from an automatic segmentation pipeline, by expanding the Bayesian Coherent

Original: Point Drift algorithm (BCPD).
Revised: Point Drift algorithm (BCPD).

Original: We introduce MO-BCPD, a multi-organ version of the BCPD algorithm which explicitly models three important aspects of this task: organ individual elastic properties, inter-organ motion coherence and segmentation inaccuracy.
Revised: We introduce MO-BCPD, a multi-organ version of the BCPD algorithm which explicitly models three important aspects of this task: organ individual elastic properties, inter-organ motion coherence and segmentation inaccuracy.

Original: This model also provides an interpolation framework to estimate the deformation of the entire volume.
Revised: This model also provides an interpolation framework to estimate the deformation of the entire volume.

Original: We demonstrate the efficiency of our method by registering different patients from the LITS challenge dataset.
Revised: We demonstrate the efficiency of our method by registering different patients from the LITS challenge dataset.

Original: The target registration error on anatomical landmarks is almost twice as small for MO-BCPD compared to standard BCPD while imposing the same constraints on individual organs deformation.
Revised: The target registration error on anatomical landmarks is almost twice as small for MO-BCPD compared to standard BCPD while imposing the same constraints on individual organs deformation.

Original:  Introduction
Revised:  Introduction

Original: Registering CT images of the chest is an important step for several pipelines such as surgical planning for liver cancer resection or disease progression tracking [2, 1, 10, 15].
Revised: Registering CT images of the chest is an important step for several pipelines such as surgical planning for liver cancer resection or disease progression tracking [2, 1, 10, 15].

Original: This step is both crucial and challenging as the deformations involved are large and may contain complex patterns such as sliding motion between organs.
Revised: This step is both crucial and challenging as the deformations involved are large and may contain complex patterns such as sliding motion between organs.

Original: While traditional registration methods tend to fail on this task, learning approaches such as [12, 6, 5] obtained promising results at the Learn2Regchallenge, task 3 [7].
Revised: While traditional registration methods tend to fail on this task, learning approaches such as [12, 6, 5] obtained promising results at the Learn2Regchallenge, task 3 [7].

Original: Yet, traditional and learning approaches both aims at registering the whole image content instead of focusing on the relevant structures of interests.
Revised: Yet, traditional and learning approaches both aims at registering the whole image content instead of focusing on the relevant structures of interests.

Original: This introduces undesired noise and complexity to the registration process.
Revised: This introduces undesired noise and complexity to the registration process.

Original: To tackle this issue, we propose to exploit the recent availability of high quality automatic segmentation pipelines such as [17, 3] and register the segmented structures.
Revised: To tackle this issue, we propose to exploit the recent availability of high quality automatic segmentation pipelines such as [17, 3] and register the segmented structures.

Original: Specifically, structures are registered using their surface point cloud representation, allowing for exploiting meaningful geometric information of the different organs and finely modeling their dynamic properties.
Revised: Specifically, structures are registered using their surface point cloud representation, allowing for exploiting meaningful geometric information of the different organs and finely modeling their dynamic properties.

Original: We also stress that surface point clouds are easy to derive from segmentation masks and are 
Revised: We also stress that surface point clouds are easy to derive from segmentation masks and are a lightweight representation of the structures of interest.

Original: The Coherent Point Drift [13] (CPD) algorithm is one of the most popular method for deformable point cloud registration considered as state of the art [11].
Revised: The Coherent Point Drift [13] (CPD) algorithm is one of the most popular method for deformable point cloud registration considered as state of the art [11].

Original: A recent work [9] extended this framework using a Bayesian formulation and obtained more robust performances.
Revised: A recent work [9] extended this framework using a Bayesian formulation and obtained more robust performances.

Original: CDP and BCPD both assume that points move coherently as a group to preserve the structure coherence.
Revised: CDP and BCPD both assume that points move coherently as a group to preserve the structure coherence.

Original: This is mainly because these frameworks are designed to register point clouds representing a single object.
Revised: This is mainly because these frameworks are designed to register point clouds representing a single object.

Original: Consequently, [13, 9] are not adapted for registering multi-organ points clouds.
Revised: Consequently, [13, 9] are not adapted for registering multi-organ points clouds.

Original: In particular, the coherency assumption doesn’t stand for organs registration as each organ-specific point cloud may move independently to its neighbour, especially if we aim at registering inter-patient images.
Revised: In particular, the coherency assumption doesn’t stand for organs registration as each organ-specific point cloud may move independently to its neighbour, especially if we aim at registering inter-patient images.

Original: In this work, we introduce a Multi-Organ Bayesian Coherent Point Drift algorithm (MO-BCPD) that models independent coherent structures.
Revised: In this work, we introduce a Multi-Organ Bayesian Coherent Point Drift algorithm (MO-BCPD) that models independent coherent structures.

Original: The contribution of this work is four-fold.
Revised: The contribution of this work is four-fold.

Original: Firstly, we extend the Bayesian formulation of CPD to model more complex structures interactions such as organ motion independence.
Revised: Firstly, we extend the Bayesian formulation of CPD to model more complex structures interactions such as organ motion independence.

Original: Secondly, given that points clouds are obtained using automated segmentations, the proposed framework models partial segmentation errors allowing MO-BCPD to recover them.
Revised: Secondly, given that points clouds are obtained using automated segmentations, the proposed framework models partial segmentation errors allowing MO-BCPD to recover them.

Original: Thirdly, we model individual organ elasticity as part of the formulation.
Revised: Thirdly, we model individual organ elasticity as part of the formulation.

Original: Fourthly, extensive experiments on 104 patients (10,712 pairs of patients) from the LiTS public dataset [4] demonstrate the effectiveness of our approach compared to BCPD.
Revised: Fourthly, extensive experiments on 104 patients (10,712 pairs of patients) from the LiTS public dataset [4] demonstrate the effectiveness of our approach compared to BCPD.

Original: In particular, our method achieves an average target registration error on anatomical landmarks of 13mm compared to 22mm for the standard BCPD.
Revised: In particular, our method achieves an average target registration error on anatomical landmarks of 13mm compared to 22mm for the standard BCPD.

Original:  Method
Revised:  Method

Original: In this section, we present our Multi-Organ Bayesian Coherent Point Drift algorithm.
Revised: In this section, we present our Multi-Organ Bayesian Coherent Point Drift algorithm.

Original: Let y =
Revised: Let y =

Original: [ y m ] m ∈{ 1  } ∈ R M, 3 be the source point cloud and x = [ x n ] n ∈{ 1  } ∈ R N,be the target point cloud where N and M are respectively the number of source and target points.
Revised: [ y m ] m ∈{ 1 ...M } ∈ R M, 3 be the source point cloud and x = [ x n ] n ∈{ 1 ...N } ∈ R N, 3 be the target point cloud where N and M are respectively the number of source and target points.

Original: We aim at finding the transformation T that realistically aligns these point clouds.
Revised: We aim at finding the transformation T that realistically aligns these point clouds.

Original: In particular here, unlike in [13, 9], the considered point clouds both represent a set of organ surfaces.
Revised: In particular here, unlike in [13, 9], the considered point clouds both represent a set of organ surfaces.

Original: Hence, each point is associated with an organ.
Revised: Hence, each point is associated with an organ.

Original: Let l y =
Revised: Let l y =

Original: [ l ym ] m ∈{ 1 ...M } ∈ { 1 . . .
Revised: [ l ym ] m ∈{ 1 ...M } ∈ { 1 . . .

Original: L } M be the organ labels of the source point cloud and
Revised: L } M be the organ labels of the source point cloud and

Original: l x =
Revised: l x =

Original: [ l xn ] n ∈{ 1 ...N } ∈ { 1 . . .
Revised: [ l x n ] n ∈{ 1 ...N } ∈ { 1 . . .

Original: L } N be the organ labels of the target point cloud.
Revised: L } N be the organ labels of the target point cloud.

Original: L is the number of organs.
Revised: L is the number of organs.

Original: Transformation model.
Revised: Transformation model.

Original: Similarly to the BCPD, the Multi-Organ Bayesian Coherent Point Drift (MO-BCPD), decomposes the motion in two components: a similarity transform ρ : p −→ s Rp + t and a dense displacement field v .
Revised: Similarly to the BCPD, the Multi-Organ Bayesian Coherent Point Drift (MO-BCPD), decomposes the motion in two components: a similarity transform ρ : p −→ s Rp + t and a dense displacement field v .

Original: Hence the deformed source point could is [ T ( y m )] m ∈{ 1 ...M } =
Revised: Hence the deformed source point could is [ T ( y m )] m ∈{ 1 ...M } =

Original: [ ρ ( y m + v m )]
Revised: [ ρ ( y m + v m )]

Original: m ∈{ 1 ...M } .
Revised: m ∈{ 1 ...M } .

Original: While this parametrization is redundant, [9] has shown that this makes the algorithm more robust to target rotations.
Revised: While this parametrization is redundant, [9] has shown that this makes the algorithm more robust to target rotations.

Original: Moreover, it is equivalent to performing a rigid alignment followed by a non-rigid refinement which corresponds to the common practice in medical image registration.Generative model.
Revised: Moreover, it is equivalent to performing a rigid alignment followed by a non-rigid refinement which corresponds to the common practice in medical image registration.

Original: As in [9], MO-BCPD assumes that all points from the target point cloud [ x n ] n ∈{ 1 ...N } are sampled independently from a generative model.

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: while
Revised: while

Original: [Table]
Revised: [Table]

Original: more than ϵ do Update P and related terms:
Revised: more than ϵ do Update P and related terms:

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: Update displacement field and related terms:
Revised: Update displacement field and related terms:

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: end
Revised: end

Revised: Generative model.

Revised: As in [9], MO-BCPD assumes that all points from the target point cloud [ x n ] n ∈{ 1 ...N } are sampled independently from a generative model.

Original: A point x n from the target point cloud is either an outlier or an inlier which is indicated by a hidden binary variable c n .
Revised: A point x n from the target point cloud is either an outlier or an inlier which is indicated by a hidden binary variable c n .

Original: We note the probability for a point to be an outlier ω (i.e. P ( c n = 0) = ω ).
Revised: We note the probability for a point to be an outlier ω (i.e. P ( c n = 0) = ω ).

Original: If x n is an outlier, it is sampled from an outlier distribution of density p out (typically, a uniform distribution over a volume containing the target point cloud).
Revised: If x n is an outlier, it is sampled from an outlier distribution of density p out (typically, a uniform distribution over a volume containing the target point cloud).

Original: If x n is an inlier ( c n = 1), x n is associated with a point T ( y m ) in the deformed source point cloud.
Revised: If x n is an inlier ( c n = 1), x n is associated with a point T ( y m ) in the deformed source point cloud.

Original: Let e n be a multinomial variable indicating the index of the point of the deformed source point cloud with which x n is associated (i.e. e n = m in our example).
Revised: Let e n be a multinomial variable indicating the index of the point of the deformed source point cloud with which x n is associated (i.e. e n = m in our example).

Original: Let α m be the probability of selecting the point T ( y m ) to generate a point of the target point cloud (i.e. ∀ n P ( e n = m | c n = 1) =
Revised: Let α m be the probability of selecting the point T ( y m ) to generate a point of the target point cloud (i.e. ∀ n P ( e n = m | c n = 1) =

Original: α m ).
Revised: α m ).

Original: x n is then sampled from a
Revised: x n is then sampled from a

Original: Gaussian distribution with covariance-matrix σ 2 Id 3 ( Id 3 is the identity matrix of R 3 ) centered on T ( y m ).
Revised: Gaussian distribution with covariance-matrix σ 2 Id 3 ( Id 3 is the identity matrix of R 3 ) centered on T ( y m ).

Original: Finally, the organ label l xn is sampled according to the label transition distribution P ( l xn | l ym )
Revised: Finally, the organ label l x n is sampled according to the label transition distribution P ( l xn | l ym )

Original: =
Revised: =

Original: u l xn ,l ym .
Revised: u l xn ,l ym .

Original: The addition of the label transition term is our contribution to the original generative model [9].
Revised: The addition of the label transition term is our contribution to the original generative model [9].

Original: This[Equation] term encourages to map corresponding organs between the different anatomies while allowing to recover from partial segmentation errors from the automatic segmentation tool.
Revised: This term encourages to map corresponding organs between the different anatomies while allowing to recover from partial segmentation errors from the automatic segmentation tool.

Original: We can now write the following conditional probability density:
Revised: We can now write the following conditional probability density:

Original: [Equation]
Revised: [Equation]

Original: where ϕ µ,Σ is the density of a multivariate Gaussian distribution N ( µ, Σ ) and δ is the Kronecker symbol.
Revised: where ϕ µ,Σ is the density of a multivariate Gaussian distribution N ( µ, Σ ) and δ is the Kronecker symbol.

Original: Prior distributions MO-BCPD also relies on prior distributions in order to regularize the registration process and obtain realistic solutions.
Revised: Prior distributions MO-BCPD also relies on prior distributions in order to regularize the registration process and obtain realistic solutions.

Original: As in [9], MO-BCPD defines two prior distributions: p v ( v | y , l y ) that regularizes the dense displacement field and p α ( α ) that regularizes the parameters α of the source point cloud selection multinomial distribution mentioned in the generative model.
Revised: As in [9], MO-BCPD defines two prior distributions: p v ( v | y , l y ) that regularizes the dense displacement field and p α ( α ) that regularizes the parameters α of the source point cloud selection multinomial distribution mentioned in the generative model.

Original: The prior on α follows a Dirichlet distribution of parameter κ 1 M .
Revised: The prior on α follows a Dirichlet distribution of parameter κ 1 M .

Original: In practice, κ is set to a very high value which forces α m ≈ 1 /M for all m .
Revised: In practice, κ is set to a very high value which forces α m ≈ 1 /M for all m .

Original: To decouple motion characteristics within and between organs, we propose a novel formulation of the displacement field prior p v .
Revised: To decouple motion characteristics within and between organs, we propose a novel formulation of the displacement field prior p v .

Original: Specifically, we introduce 3 parameters: a symmetric matrix
Revised: Specifically, we introduce 3 parameters: a symmetric matrix

Original: S =
Revised: S =

Original: [ s l,l ′ ] l,l ′ ∈{ 0 ...L } and two vectors Λ =
Revised: [ s l,l ′ ] l,l ′ ∈{ 0 ...L } and two vectors Λ =

Original: [ Λ l ] l ∈{ 0 ...L } and B =
Revised: [ Λ l ] l ∈{ 0 ...L } and B =

Original: [ B l ] l ∈{ 0 ...L } .
Revised: [ B l ] l ∈{ 0 ...L } .

Original: The matrix S parametrizes the motion coherence inter-organs.
Revised: The matrix S parametrizes the motion coherence inter-organs.

Original: The vectors Λ and B respectively characterizes the variance of the deformation magnitude and motion coherence bandwidth within each organ.
Revised: The vectors Λ and B respectively characterizes the variance of the deformation magnitude and motion coherence bandwidth within each organ.

Original: We define the displacement field prior for the MO-BCPD as:
Revised: We define the displacement field prior for the MO-BCPD as:

Original: [Equation]
Revised: [Equation]

Original: Note that G must be definite-positive, leading to strictly positive values for variance of the displacement magnitude Λ l and mild constraints on S .
Revised: Note that G must be definite-positive, leading to strictly positive values for variance of the displacement magnitude Λ l and mild constraints on S .

Original: Learning.
Revised: Learning.

Original: Combining equations (1) and (2), the joint probability distribution of the variables y , l y , x , l x , θ , where θ = ( v , α, c , e , ρ, σ 2 ) is defined as:
Revised: Combining equations (1) and (2), the joint probability distribution of the variables y , l y , x , l x , θ , where θ = ( v , α, c , e , ρ, σ 2 ) is defined as:

Original: [Equation]
Revised: [Equation]

Original: As in [9], we use variational inference to approximate the posterior distribution p ( θ | x , y ) with a factorized distribution q ( θ ) =
Revised: As in [9], we use variational inference to approximate the posterior distribution p ( θ | x , y ) with a factorized distribution q ( θ ) =

Original: q 1 ( v , α ) q 2 ( c , e ) q 3 ( ρ, σ 2 ) so that q = arg min q 1 ,q 2 ,q 3 KL ( q | p ( .
Revised: q 1 ( v , α ) q 2 ( c , e ) q 3 ( ρ, σ 2 ) so that q = arg min q 1 ,q 2 ,q 3 KL ( q | p ( .

Original: | x , y )) where KL is the Kullback–Leibler divergence.
Revised: | x , y )) where KL is the Kullback–Leibler divergence.

Original: Similarly to [9], we derive the MO-BCPD algorithm presented in Algorithm 1.
Revised: Similarly to [9], we derive the MO-BCPD algorithm presented in algorithm 1.

Original: The steps detailed in Algorithm 1 perform coordinate ascent on the evidence lower bound L ( θ ) =
Revised: The steps detailed in Algorithm 1 perform coordinate ascent on the evidence lower bound L ( θ ) =

Original: (cid:82) θ q ( θ ) ln p ( x , y ,θ ) q ( θ ) dθ .
Revised: (cid:82) θ q ( θ ) ln p ( x , y ,θ ) q ( θ ) dθ .

Original: In algorithm 1, γ is a hyper-parameter used to scale the initial estimation of σ 2 and ϵ is used for stopping criteria.
Revised: In algorithm 1, γ is a hyper-parameter used to scale the initial estimation of σ 2 and ϵ is used for stopping criteria.

Original: We note ∆ ( ν ) the diagonal matrix with diagonal entries equal to ν .
Revised: We note ∆ ( ν ) the diagonal matrix with diagonal entries equal to ν .

Original: Hyper-parameter setting.
Revised: Hyper-parameter setting.

Original: The model has a large number of hyper-parameters which can impact the performance of the algorithm.
Revised: The model has a large number of hyper-parameters which can impact the performance of the algorithm.

Original: Regarding κ , the parameter of the prior distribution p α , and γ , the scaling applied to the initial estimation of σ 2 , we followed the guidelines in [9].
Revised: Regarding κ , the parameter of the prior distribution p α , and γ , the scaling applied to the initial estimation of σ 2 , we followed the guidelines in [9].

Original: ω is set based on an estimate of the proportion of outliers on a representative testing set.
Revised: ω is set based on an estimate of the proportion of outliers on a representative testing set.

Original: Regarding B and Λ , respectively the vector of organ-specific motion coherence bandwidth and expected deformation magnitude, they characterise organs elastic properties.
Revised: Regarding B and Λ , respectively the vector of organ-specific motion coherence bandwidth and expected deformation magnitude, they characterise organs elastic properties.

Original: Concretely, a larger motion coherence bandwidth B l increases the range of displacement correlation for organ l (points that are further away are encouraged to move in the same direction).
Revised: Concretely, a larger motion coherence bandwidth B l increases the range of displacement correlation for organ l (points that are further away are encouraged to move in the same direction).

Original: A larger expected deformation magnitude Λ l increases the probability of larger displacements for organ l .
Revised: A larger expected deformation magnitude Λ l increases the probability of larger displacements for organ l .

Original: These are physical quantities expressed in mm that could be set based on organs physical properties.
Revised: These are physical quantities expressed in mm that could be set based on organs physical properties.

Original: The inter-organ motion coherence matrix S should be a symmetric matrix containing values between 0 and 1.
Revised: The inter-organ motion coherence matrix S should be a symmetric matrix containing values between 0 and 1.

Original: S l,l = 1 for all organs l ∈ { 1 . . .
Revised: S l,l = 1 for all organs l ∈ { 1 . . .

Original: L } and S l,l ′ is closer to 0 if organs l and l ′ can move independently.
Revised: L } and S l,l ′ is closer to 0 if organs l and l ′ can move independently.

Original: u l,l ′ is the probability that a point with label l ′ generates a point with label l .
Revised: u l,l ′ is the probability that a point with label l ′ generates a point with label l .

Original: As points labels are in practice obtained from an automatic segmentation tool, we note [ g ym ] m ∈{ 1 ...M } and [ g xn ] n ∈{ 1 ...n } respectively the unknown true organ labels of the source and target point clouds (as opposed to the estimated ones [ l ym ] m ∈{ 1 ...M } and [ l ym ] m ∈{ 1 ...M } ).
Revised: As points labels are in practice obtained from an automatic segmentation tool, we note [ g ym ] m ∈{ 1 ...M } and [ g xn ] n ∈{ 1 ...n } respectively the unknown true organ labels of the source and target point clouds (as opposed to the estimated ones [ l ym ] m ∈{ 1 ...M } and [ l ym ] m ∈{ 1 ...M } ).

Original: We assume that points from the deformed source point cloud generate points with the same true labels (i.e. P ( g xn = g ym | e n = m ) = 1).
Revised: We assume that points from the deformed source point cloud generate points with the same true labels (i.e. P ( g xn = g ym | e n = m ) = 1).

Original: Hence, the probability for y m , with estimated organ label l ym to generate a point with label l xn is given by: u l xn ,l ym = (cid:80) k p ( g ym = [Equation]
Revised: Hence, the probability for y m , with estimated organ label l ym to generate a point with label l xn is given by: u l xn ,l ym = (cid:80) k p ( g ym = k

Original: | [Equation] ) p ( l 
Revised: | l ym ) p ( l xn |

Original: [Equation] =
Revised: g xn = k ) where p ( g ym =

Original: [Equation] ) where p ( g ym = k | l ym ) is the probability that a point labelled l ym by the automatic segmentation tool has true label k and p ( l xn | g xn = k ) is the probability that the automatic segmentation tool predicts the organ label l xn for a point with true label k .
Revised: k | l ym ) is the probability that a point labelled l ym by the automatic segmentation tool has true label k and p ( l xn | g xn = k ) is the probability that the automatic segmentation tool predicts the organ label l xn for a point with true label k .

Original: These probabilities need to be estimated on a representative testing set.
Revised: These probabilities need to be estimated on a representative testing set.

Original: We note that if the segmentation is error-free, the formula above gives U = Id L .
Revised: We note that if the segmentation is error-free, the formula above gives U = Id L .

Original: Indeed, in that case the points organ labels correspond exactly to the organ true labels so a point belonging to an certain organ can only generate a point from the same organ in the target anatomy.
Revised: Indeed, in that case the points organ labels correspond exactly to the organ true labels so a point belonging to a certain organ can only generate a point from the same organ in the target anatomy.

Original: Properly setting the organ label transition probability matrix U is crucial to recover from potential partial segmentation errors.
Revised: Properly setting the organ label transition probability matrix U is crucial to recover from potential partial segmentation errors.

Original: Fig.
Revised: Fig.

Original: 1 illustrates with a toy example a situation where the algorithm converges to an undesired state if the segmentation error is not modeled properly.
Revised: 1 illustrates with a toy example a situation where the algorithm converges to an undesired state if the segmentation error is not modeled properly.

Original: Interpolation.
Revised: Interpolation.

Original: Once the deformation on the organ point clouds is known, one might want to interpolate the deformation back to image space in order to resample the whole volume.
Revised: Once the deformation on the organ point clouds is known, one might want to interpolate the deformation back to image space in order to resample the whole volume.

Original: As in [8] we propose to use Gaussian process regression to interpolate the deformation obtained by the MO-BCPD algorithm.
Revised: As in [8] we propose to use Gaussian process regression to interpolate the deformation obtained by the MO-BCPD algorithm.

Original: This interpolation process can also be used to register sub-sampled point clouds to decrease computation time as in [8].
Revised: This interpolation process can also be used to register sub-sampled point clouds to decrease computation time as in [8].

Original: Given a set of points ˜ y
Revised: Given a set of points ˜ y

Original: =
Revised: =

Original: [˜ y
Revised: [˜ y

Original: i ] i ∈{ 1 ... ˜ M } with labels l ˜ y
Revised: i ] i ∈{ 1 ... ˜ M } with labels l ˜ y

Original: =
Revised: =

Original: [ l ˜ yi ] i ∈{ 1 ... ˜ M } .
Revised: [ l ˜ yi ] i ∈{ 1 ... ˜ M } .

Original: We compute the displacement for the set of points ˜
Revised: We compute the displacement for the set of points ˜

Original: y as:
Revised: y as:

Original: [Equation]
Revised: [Equation]

Original: Acceleration.
Revised: Acceleration.

Original: The speed ups mentioned in [9] section 4.6 are fully transferable to the MO-BCPD pipeline.
Revised: The speed ups mentioned in [9] section 4.6 are fully transferable to the MO-BCPD pipeline.

Original: In our experiments though, the main improvement, by far, came from performing a low rank decomposition of G at the initialization of the algorithm.
Revised: In our experiments though, the main improvement, by far, came from performing a low rank decomposition of G at the initialization of the algorithm.

Original: Indeed, this yielded consistent reliable × 10 speed-ups with negligible error when using ≥ 20 eigen values.
Revised: Indeed, this yielded consistent reliable × 10 speed-ups with negligible error when using ≥ 20 eigen values.

Original: The Nystrom methods to approximate P sometime implied large error due to the stochasticity of the method while yielding up to × 2 speed-ups which is why we did not useof it.
Revised: The Nystrom methods to approximate P sometimes implied large error due to the stochasticity of the method while yielding up to × 2 speed-ups which is why we did not use it.

Original: This allows
Revised: This allows

Original: MO-BCPD to be run in a few seconds with M, N ≈ 5000.
Revised: MO-BCPD to be run in a few seconds with M, N ≈ 5000.

Original:  Experiments
Revised:  Experiments

Original: We evaluate the MO-BCPD algorithm by performing inter-patient registration from the LITS challenge training dataset [4] which contains 131 chest-CT patient images.
Revised: We evaluate the MO-BCPD algorithm by performing inter-patient registration from the LITS challenge training dataset [4] which contains 131 chest-CT patient images.

Original: 27 patients were removed due to different field of view, issues with the segmentation or landmark detection.
Revised: 27 patients were removed due to different field of view, issues with the segmentation or landmark detection.

Original: In total, 10,712 registrations were performed on all the pairs of remaining patients.
Revised: In total, 10,712 registrations were performed on all the pairs of remaining patients.

Original: The segmentation was automatically performed using an in-house tool derived from [17] which also provides
Revised: The segmentation was automatically performed using an in-house tool derived from [17] which also provides

Original: [Table]
Revised: [Table]

Original: a set of anatomical landmarks for each image which were used for evaluation
Revised: a set of anatomical landmarks for each image which were used for evaluation

Original: [Table]
Revised: [Table]

Original: We considered five organs of interest: the liver, the spleen, the left and right kidneys and the bladder.
Revised: We considered five organs of interest: the liver, the spleen, the left and right kidneys and the bladder.

Original: We compared 4 different algorithms: registration of the point clouds with a similarity transform (Sim), BCPD, GMC-MO-BCPD which is MO-BCPD with global motion coherence ( S = 1 L,L ) and OMC-MO-BCPD which is MO-BCPD with intra-organ motion coherence only ( S = Id L ).
Revised: We compared 4 different algorithms: registration of the point clouds with a similarity transform (Sim), BCPD, GMC-MO-BCPD which is MO-BCPD with global motion coherence ( S = 1 L,L ) and OMC-MO-BCPD which is MO-BCPD with intra-organ motion coherence only ( S = Id L ).

Original: As the segmentation tool performed very well on the considered organs, we set U
Revised: As the segmentation tool performed very well on the considered organs, we set U

Original: = Id L and ω = 0 for both MO-BCPD versions ( ω = 0 for BCPD as well).
Revised: = Id L and ω = 0 for both MO-BCPD versions ( ω = 0 for BCPD as well).

Original: We used for all organs the same values for Λ l and B l respectively 10mm and 30mm as a trade off between shape matching and preservation of individual organs appearance ( β = 30 and λ = 0 . 1 for BCPD which is the equivalent configuration).
Revised: We used for all organs the same values for Λ l and B l respectively 10mm and 30mm as a trade off between shape matching and preservation of individual organs appearance ( β = 30 and λ = 0 . 1 for BCPD which is the equivalent configuration).

Original: We also set, γ = 1 and ϵ = 0 .
Revised: We also set, γ = 1 and ϵ = 0 .

Original: We compared those algorithms by computing the registration error on the anatomical landmarks belonging to those organs.
Revised: We compared those algorithms by computing the registration error on the anatomical landmarks belonging to those organs.

Original: We chose this generic, relatively simple setting (same rigidity values for all organs, no outlier modeling, only two extreme configurations for S ) in order to perform large scale inter-patient registration experiments but we would like to stress that further fine tuning of these parameters for a specific application or even for a specific patient would further improve the modeling and hence the registration outcome.
Revised: We chose this generic, relatively simple setting (same rigidity values for all organs, no outlier modeling, only two extreme configurations for S ) in order to perform large scale inter-patient registration experiments but we would like to stress that further fine tuning of these parameters for a specific application or even for a specific patient would further improve the modeling and hence the registration outcome.

Original: Results are presented in Table 1.
Revised: Results are presented in Table 1.

Original: We observe that while GMC-MO-BCPD induces some marginal improvements with respect to BCPD, OMC-MO-BCPD allows a much more precise registration.
Revised: We observe that while GMC-MO-BCPD induces some marginal improvements with respect to BCPD, OMC-MO-BCPD allows a much more precise registration.

Original: As illustrated in Fig.
Revised: As illustrated in Fig.

Original: 2, the main improvement from
Revised: 2, the main improvement from

Original: BCPD to GMC-MO-BCPD is that different organs no longer overlap.
Revised: BCPD to GMC-MO-BCPD is that different organs no longer overlap.

Original: Indeed, as highlighted by the green and blue ellipses, the spleen and the right kidney from the deformed source patient overlap the liver of the target patient when using
Revised: Indeed, as highlighted by the green and blue ellipses, the spleen and the right kidney from the deformed source patient overlap the liver of the target patient when using

Original: BCPD.
Revised: BCPD.

Original: OMC-MO-BCPD properly aligns the different organs while preserving their shape (see orange and purple ellipses for instance).
Revised: OMC-MO-BCPD properly aligns the different organs while preserving their shape (see orange and purple ellipses for instance).

Original:  Conclusion
Revised:  Conclusion

Original: We introduced MO-BCPD, an extension of the BCPD algorithm specifically adapted to abdominal organ registration.
Revised: We introduced MO-BCPD, an extension of the BCPD algorithm specifically adapted to abdominal organ registration.

Original: We identified three limitations of the original work [9] on this task and proposed solutions to model: the segmentation error between neighboring organs of interest, the heterogeneous elastic properties of the abdominal organs and the complex interaction between various organs in terms of motion coherence.
Revised: We identified three limitations of the original work [9] on this task and proposed solutions to model: the segmentation error between neighboring organs of interest, the heterogeneous elastic properties of the abdominal organs and the complex interaction between various organs in terms of motion coherence.

Original: We demonstrated significant improvements over
Revised: We demonstrated significant improvements over

Original: BCPD on a large validation set (N=10,712).
Revised: BCPD on a large validation set (N=10,712).

Original: Moreover, we would like to highlight that segmentation error could also be taken into account by tuning the outlier probability distribution p out and the probability of being an outlier ω .
Revised: Moreover, we would like to highlight that segmentation error could also be taken into account by tuning the outlier probability distribution p out and the probability of being an outlier ω .

Original: When the point is estimated as a potential outlier by the algorithm, its contribution to the estimation of the transformation
Revised: When the point is estimated as a potential outlier by the algorithm, its contribution to the estimation of the transformation

Original: T is lowered.
Revised: T is lowered.

Original: Hence, the segmentation error modelled by p out and ω corresponds to over/under segmentation, i.e. when there is a confusion between an organ and another class we don’t make use of in MO-BCPD (e.g. background).
Revised: Hence, the segmentation error modelled by p out and ω corresponds to over/under segmentation, i.e. when there is a confusion between an organ and another class we don’t make use of in MO-BCPD (e.g. background).

Original: Hence, MOBCPD introduces a finer way of handling segmentation error by distinguishing two types of errors: mis-labeling between classes of interest which is modelled by U and over/under-segmentation of classes of interest modeled by ω and p out .
Revised: Hence, MOBCPD introduces a finer way of handling segmentation error by distinguishing two types of errors: mis-labeling between classes of interest which is modelled by U and over/under-segmentation of classes of interest modeled by ω and p out .

Original: In this manuscript, we focused on highlighting the improvements yielded by the MO-BCPD formulation specifically designed for multi-organ point cloud registration.
Revised: In this manuscript, we focused on highlighting the improvements yielded by the MO-BCPD formulation specifically designed for multi-organ point cloud registration.

Original: That being said, some clinical applications would require the deformation on the whole original image volume.
Revised: That being said, some clinical applications would require the deformation on the whole original image volume.

Original: Hence, we present in supplementary material preliminary results on a realistic clinical use case.
Revised: Hence, we present in supplementary material preliminary results on a realistic clinical use case.

Original: In figures 3 and 4 we see that MO-BCPD coupled with the proposed interpolation framework obtain better alignment on the structures of interest than traditional intensity-based baselines.
Revised: In figures 3 and 4 we see that MO-BCPD coupled with the proposed interpolation framework obtain better alignment on the structures of interest than traditional intensity-based baselines.

Original: It is also interesting to note that MO-BCPD also better aligns structures that are particularly challenging to align such as the hepatic vein while not using these structures in the MO-BCPD.
Revised: It is also interesting to note that MO-BCPD also better aligns structures that are particularly challenging to align such as the hepatic vein while not using these structures in the MO-BCPD.

Original: Future work will investigate how MO-BCPD could be used as a fast, accurate initialization for image-based registration algorithms.
Revised: Future work will investigate how MO-BCPD could be used as a fast, accurate initialization for image-based registration algorithms.

Original: From a modeling standpoint, we would also like to further work on segmentation error modeling (in particular over/under segmentation) with a more complex organ specific outlier distribution.
Revised: From a modeling standpoint, we would also like to further work on segmentation error modeling (in particular over/under segmentation) with a more complex organ specific outlier distribution.

Original: Supplementary material: A multi-organ point cloud registration algorithm for abdominal CT registration
Revised: Supplementary material: A multi-organ point cloud registration algorithm for abdominal CT registration

Original: [Figure] [Figure]
Revised: [Figure] [Figure]