SlowGuess/NeurIPS_2023_OCR huggingface dataset:
using the exponential map to obtain the corresponding perturbation transformation. Please refer to Appendix A for more details about our perturbation transformation.
3.2.2 SE(3) Reverse Process
With the diffusion Markov chain  $\mathbf{H}_0\to \mathbf{H}_1\to \dots \to \mathbf{H}_T$  generated by the SE(3) diffusion process in Sec. 3.2.1, the objective of the SE(3) reverse process is to train a denoising network  $p_{\theta}(\mathbf{H}_{t - 1}\mid \mathcal{X}_t = \mathcal{T}(\mathcal{X},\mathbf{H}_t),\mathcal{M}))$  to progressively refine the noisy transformation towards the optimal one, thus forming a reverse Markov chain  $\mathbf{H}_T\rightarrow \mathbf{H}_{T - 1}\rightarrow \dots \rightarrow \mathbf{H}_0$ . In the context of 3D registration, we design our denoising network to predict a probability distribution over the transformation  $\mathbf{H}_{t - 1}$  given the model point cloud  $\mathcal{M}$  and the transformed source point cloud  $\mathcal{X}_t = \mathcal{T}(\mathcal{X},\mathbf{H}_t) = \{\mathbf{R}_t\mathbf{x}_i + \mathbf{t}_t\}$ . Here,  $\mathbf{R}_t$  and  $\mathbf{t}_t$  denote the rotation matrix and translation vector of transformation  $\mathbf{H}_t$ . To effectively train our denoising network, we derive the following registration-specific variational lower bound of the log likelihood over the training samples as the optimization objective:
$$
\begin{array}{l} \mathbb {E} _ {\mathcal {X}, \mathcal {M}, \mathbf {H} _ {0} \sim p _ {d a t a}} \left[ \ln p _ {\theta} (\mathbf {H} _ {0} \mid \mathcal {X}, \mathcal {M}) \right] \geq \mathbb {E} _ {\mathbf {H} _ {1: T} \sim q _ {\mathcal {X}, \mathcal {M}, \mathbf {H} _ {0} \sim p _ {d a t a}}} \left[ \ln \frac {p _ {\theta} (\mathbf {H} _ {0 : T} \mid \mathcal {X} , \mathcal {M})}{q (\mathbf {H} _ {1 : T} \mid \mathbf {H} _ {0})} \right] \\ = \mathbb {E} _ {\substack {\mathbf {H} _ {1: T} \sim q \\ \mathcal {X}, \mathcal {M}, \mathbf {H} _ {0} \sim p _ {d a t a}}} \left[ \underbrace {\ln p _ {\theta} \left(\mathbf {H} _ {0} \mid \mathcal {X} _ {1} , \mathcal {M}\right)} _ {\text {Residual term}} - \underbrace {\mathrm {D} _ {\mathrm {K L}} \left(q \left(\mathbf {H} _ {T} \mid \mathbf {H} _ {0}\right) \| p \left(\mathbf {H} _ {T}\right)\right)} _ {\text {Prior matching term}} - \right. \tag{7} \\ \sum_ {t = 2} ^ {T} \underbrace {\mathrm {D} _ {\mathrm {K L}} (q (\mathbf {H} _ {t - 1} \mid \mathbf {H} _ {t} , \mathbf {H} _ {0}) | | p _ {\theta} (\mathbf {H} _ {t - 1} \mid \mathcal {X} _ {t} , \mathcal {M}))} _ {\text {D e n o i s i n g m a t c h i n g t e r m}} \Biggr ], \\ \end{array}
$$
where  $p_{data}$  indicates the distribution of the training data, including the pairs of source and model point clouds and their ground-truth rigid transformations. In the following, we elaborate on each loss term of the derived variational lower bound. Please refer to Appendix A for a detailed derivation.
Denoising Matching Term. This is the fundamental loss term for training our denoising network.  $q(\mathbf{H}_{t - 1} \mid \mathbf{H}_t, \mathbf{H}_0)$  represents the posterior distribution of  $\mathbf{H}_{t - 1}$  conditioned on  $\mathbf{H}_0$  and  $\mathbf{H}_t$ , while  $p_{\theta}(\mathbf{H}_{t - 1} \mid \mathcal{X}_t, \mathcal{M})$  denotes the prior distribution over  $\mathbf{H}_{t - 1}$  predicted by our denoising network. In contrast to the prior distribution, the posterior has access to the optimal transformation  $\mathbf{H}_0$ , enabling it to infer a more reliable distribution for  $\mathbf{H}_{t - 1}$  through Bayes' formula. As a result, we can treat this posterior distribution as ground-truth signal to supervise the prior distribution prediction of the denoising network by minimizing their Kullback-Leibler (KL) divergence. Inspired by the Bayesian posterior in Eq. 3, the random transformation  $\mathbf{H}_{t - 1}^{post}$  following the posterior distribution,  $\mathbf{H}_{t - 1}^{post} \sim q(\mathbf{H}_{t - 1} \mid \mathbf{H}_t, \mathbf{H}_0)$ , can be expressed as
$$
\mathbf {H} _ {t - 1} ^ {\text {p o s t}} = \operatorname {E x p} \left(\underbrace {\frac {\sqrt {\bar {\alpha} _ {t - 1}} \beta_ {t}}{1 - \bar {\alpha} _ {t}}} _ {\lambda_ {0}} \log \left(\mathbf {H} _ {0}\right) + \underbrace {\frac {\sqrt {\alpha_ {t}} \left(1 - \bar {\alpha} _ {t - 1}\right)}{1 - \bar {\alpha} _ {t}}} _ {\lambda_ {1}} \log \left(\mathbf {H} _ {t}\right) + \sqrt {\tilde {\beta} _ {t}} \varepsilon\right), \tag {8}
$$
where we employ the logarithm map to convert the SE(3) transformations  $\mathbf{H}_0$  and  $\mathbf{H}_t$  to their corresponding 6D vector representations in the linear Lie algebra  $\mathfrak{se}(3)$ . This allows us to perform a linear combination with posterior coefficients  $\lambda_0$  and  $\lambda_{1}$ , as in Eq. 3. The randomness of  $\mathbf{H}_{t - 1}^{post}$  arises from the addition of a random variable  $\varepsilon$ , which follows a Gaussian distribution  $\mathcal{N}(\mathbf{0},\mathbf{I})$  over  $\mathbb{R}^6$ . The resulting vector is then converted back from  $\mathfrak{se}(3)$  to SE(3) using the exponential map, yielding  $\mathbf{H}_{t - 1}^{post}$ . Analogously, the random transformation following the prior distribution,  $\mathbf{H}_{t - 1}^{prior} \sim p_{\theta}(\mathbf{H}_{t - 1} \mid \mathcal{X}_t, \mathcal{M})$ , can be represented as
$$
\mathbf {H} _ {t - 1} ^ {p r i o r} = \operatorname {E x p} \left(\log \left(\mu_ {\theta} \left(\mathcal {X} _ {t}, \mathcal {M}\right)\right) + \sqrt {\tilde {\beta} _ {t}} \varepsilon\right), \tag {9}
$$
where  $\mu_{\theta}(\mathcal{X}_t,\mathcal{M})$  denotes the parameterized mean of our prior distribution, and  $\sqrt{\tilde{\beta}_t\varepsilon}$  represents the random term that is identical to that of the posterior. As such, minimizing the KL divergence between the prior and posterior distributions is equivalent to minimizing the error between  $\mathrm{Log}(\mu_{\theta}(\mathcal{X}_t,\mathcal{M}))$  and  $\lambda_0\operatorname {Log}(\mathbf{H}_0) + \lambda_1\operatorname {Log}(\mathbf{H}_t)$ , i.e.,
$$
\mathcal {L} _ {t} (\theta) = \operatorname {l o s s} \left(\lambda_ {0} \log \left(\mathbf {H} _ {0}\right) + \lambda_ {1} \log \left(\mathbf {H} _ {t}\right), \operatorname {L o s s} \left(\mu_ {\theta} \left(\mathcal {X} _ {t}, \mathcal {M}\right)\right)\right). \tag {10}
$$
The loss function 10 indicates that, at each time step  $t$ , given the point clouds  $\mathcal{X}_t = \mathcal{T}(\mathcal{X},\mathbf{H}_t)$  and  $\mathcal{M}$ , the mean  $\mu_{\theta}(\mathcal{X}_t,\mathcal{M})$  needs to be predicted as the transformation  $\mathrm{Exp}(\lambda_0\log (\mathbf{H}_0) + \lambda_1\log (\mathbf{H}_t))$  rather than the relative transformation  $\mathbf{H}_0\mathbf{H}_t^{-1}$  between  $\mathcal{X}_t$  and  $\mathcal{M}$ . Consequently, existing deep registration models such as [54, 62] may not be directly applicable for parameterizing the mean of our prior distribution. Moreover, designing a specific network that takes  $\mathcal{X}_t$  and  $\mathcal{M}$  as input but predicts a non-relative transformation also is a non-trivial task. To handle it, we consider that the transformation  $\mathrm{Exp}(\lambda_0\mathrm{Log}(\mathbf{H}_0) + \lambda_1\mathrm{log}(\mathbf{H}_t))$  can be rewritten as
$$
\left. \operatorname {E x p} \left(\lambda_ {0} \log \left(\mathbf {H} _ {0}\right) + \lambda_ {1} \log \left(\mathbf {H} _ {t}\right)\right) = \operatorname {E x p} \left(\lambda_ {0} \log \left(\underbrace {\left(\mathbf {H} _ {0} \mathbf {H} _ {t} ^ {- 1}\right)} _ {\mathbf {H} _ {t \rightarrow 0}} \mathbf {H} _ {t}\right) + \lambda_ {1} \log \left(\mathbf {H} _ {t}\right)\right). \right. \tag {11}
$$
This inspires us to reformulate  $\mu_{\theta}(\mathcal{X}_t,\mathcal{M})$  using a surrogate registration model  $f_{\theta}(\mathcal{X}_t,\mathcal{M})$  as
$$
\mu_ {\theta} \left(\mathcal {X} _ {t}, \mathcal {M}\right) = \operatorname {E x p} \left(\lambda_ {0} \log \left(f _ {\theta} \left(\mathcal {X} _ {t}, \mathcal {M}\right) \mathbf {H} _ {t}\right) + \lambda_ {1} \log \left(\mathbf {H} _ {t}\right)\right). \tag {12}
$$
Then, minimizing the loss function 10 is equivalent to optimizing the surrogate registration model  $f_{\theta}(\mathcal{X}_t,\mathcal{M})$  to predict the relative transformation  $\mathbf{H}_{t\to 0} = \mathbf{H}_0\mathbf{H}_t^{-1}$  between the point clouds  $\mathcal{X}_t$  and  $\mathcal{M}$ . Therefore, different deep registration models can potentially serve as surrogate registration models in our denoising network. Finally, we optimize the surrogate registration network by minimizing the  $L_{1}$  distance between the source points transformed using the ground-truth transformation  $\mathbf{H}_{t\to 0}$  and the predicted one  $\hat{\mathbf{H}}_{t\to 0} = f_{\theta}(\mathcal{X}_t,\mathcal{M})$ . This is written as
$$
\mathcal {L} _ {t} (\theta) = \operatorname {l o s s} \left(f _ {\theta} \left(\mathcal {X} _ {t}, \mathcal {M}\right), \mathbf {H} _ {t \rightarrow 0}\right) = \frac {1}{N} \sum_ {i} ^ {N} \left\| \mathbf {H} _ {t \rightarrow 0} \left[\begin{array}{c}\mathbf {x} _ {i} ^ {t}\\1\end{array}\right] - \hat {\mathbf {H}} _ {t \rightarrow 0} \left[\begin{array}{c}\mathbf {x} _ {i} ^ {t}\\1\end{array}\right]\right\| _ {1}, \tag {13}
$$
where  $t\in \{2,\dots,T\}$  and  $\mathbf{x}_i^t$  indicates the  $i$ -th point in the transformed source point cloud  $\mathcal{X}_t$ .
Residual and Prior Matching Terms. To maximize the probability  $p_{\theta}(\mathbf{H}_0 \mid \mathcal{X}_1, \mathcal{M})$ , the mean  $\mu_{\theta}(\mathcal{X}_1, \mathcal{M}) = \mathrm{Exp}\left(\lambda_0 \mathrm{Log}(f_{\theta}(\mathcal{X}_1, \mathcal{M}) \mathbf{H}_1) + \lambda_1 \mathrm{Log}(\mathbf{H}_1)\right)$  should be optimized to align closely with  $\mathbf{H}_0$ . As such, the optimization target of  $f_{\theta}(\mathcal{X}_1, \mathcal{M})$  is  $\mathbf{H}_0 \mathbf{H}_1^{-1}$ , and the loss function can be represented as  $\mathcal{L}_1(\theta)$  in 13. In addition, as the prior matching term does not require learning any parameters, it can be regarded as a constant and hence can be omitted.
Implementation Details. We set the numbers of points in the source and model point clouds to  $N = 512$  and  $M = 1024$  through random sampling. For the SE(3) diffusion process, we adopt a cosine schedule [35] to determine the diffusion coefficients  $\{\beta_t\}$ . The number of diffusion steps  $T$  is set to 200, and the scaling coefficient  $\gamma$  for the perturbation transformation is set to 0.1. For the SE(3) reverse process, the number of reverse steps in the training phase is set to 200, while that in the inference phase is set to 5 to accelerate the inference speed of the diffusion registration. We use the ADAM optimizer [25] with a learning rate of 0.001 to optimize the loss function 13 for 20 epochs with a batch size of 32, and employ PyTorch [39] to implement our framework. All experiments are conducted on a server equipped with an Intel i5 2.2 GHz CPU and one TITAN RTX GPU.
Evaluation Metrics. Following [11], we evaluate the model performance by quantifying the rotation and translation errors between the predicted rotation and translation  $\hat{\mathbf{R}}$  and  $\hat{\mathbf{t}}$ , and the ground-truth ones  $\mathbf{R}^*$  and  $\mathbf{t}^*$ . The evaluation metrics are defined as
$$
\operatorname {R E} (\hat {\mathbf {R}}) = \arccos  \frac {\operatorname {T r} \left(\hat {\mathbf {R}} ^ {\top} \mathbf {R} ^ {*}\right) - 1}{2}, \quad \operatorname {T E} (\hat {\mathbf {t}}) = \left\| \hat {\mathbf {t}} - \mathbf {t} ^ {*} \right\| _ {2} ^ {2}. \tag {14}
$$
As in [12, 11], we summarize these errors via mean average precision (mAP) under varying thresholds.
4.2 Comparison with Existing Methods
Evaluation on TUD-L. We first evaluate our approach on the TUD-L dataset [20], a real-world dataset comprising three household objects. The compared methods encompass four representative
Models	TUD-L	LINEMOD	Occluded-LINEMOD
5°	10°	1cm	2cm	5°	10°	1cm	2cm	5°	10°	1cm	2cm
ICP [4]	0.02	0.02	0.01	0.14	0.00	0.01	0.04	0.27	0.01	0.01	0.07	0.36
FGR [66]	0.00	0.01	0.04	0.25	0.00	0.00	0.05	0.31	0.00	0.00	0.08	0.43
TEASER [59]	0.13	0.17	0.03	0.22	0.01	0.03	0.03	0.21	0.01	0.02	0.04	0.26
S4PCS [34]	0.30	0.50	0.05	0.40	0.02	0.09	0.04	0.31	0.01	0.03	0.06	0.31
IDAM [28]	0.03	0.05	0.02	0.08	0.00	0.01	0.03	0.16	0.00	0.02	0.07	0.26
FMR [22]	0.02	0.09	0.02	0.06	0.00	0.01	0.07	0.17	0.00	0.00	0.09	0.18
RGM [15]	0.00	0.00	0.02	0.03	0.00	0.00	0.07	0.15	0.00	0.00	0.09	0.22
RIENet [44]	0.00	0.00	0.06	0.11	-	-	-	-	-	-	-	-
MN-IDAM [11]	0.36	0.46	0.23	0.47	0.01	0.07	0.13	0.38	0.02	0.08	0.15	0.44
MN-DCP [11]	0.70	0.81	0.71	0.86	0.10	0.27	0.26	0.60	0.07	0.19	0.24	0.57
DCP [54]	0.23	0.62	0.04	0.26	0.06	0.22	0.11	0.27	0.03	0.12	0.11	0.27
Diff-DCP	0.65	0.85	0.73	0.94	0.22	0.51	0.65	0.82	0.10	0.29	0.38	0.57
Improvement ↑	0.42	0.23	0.69	0.68	0.16	0.29	0.54	0.55	0.07	0.17	0.27	0.30
RPMNet [62]	0.73	0.97	0.89	0.94	0.05	0.18	0.22	0.45	0.03	0.13	0.22	0.40
Diff-RPMNet	0.90	0.98	0.98	0.99	0.18	0.47	0.51	0.72	0.12	0.29	0.36	0.52
Improvement ↑	0.17	0.01	0.09	0.05	0.13	0.29	0.29	0.27	0.09	0.16	0.14	0.12
Table 1: Quantitative comparisons on TUD-L [20], LINEMOD [18], and Occluded-LINEMOD [6].
traditional approaches: ICP [4], FGR [66], TEASER [60], and S4PCS [34], alongside eight state-of-the-art learning-based deep registration models: DCP [54], IDAM [28], FMR [22], RPMNet [62], RGM [15], RIENet [44], MN-IDAM [11] and MN-DCP [11]. Although Sec. 3.2.2 revealed that many deep registration models are theoretically applicable to establish our denoising network, the empirical results in Table 1 indicate that some models such as IDAM, FMR, RGM, and RIENet fail to produce meaningful results in real-world challenges, suggesting their limited capability in learning a high-quality denoising network through loss function optimization (see Eq.13). Consequently, we opt to employ DCP and RPMNet, which demonstrate promising performance, to establish our denoising network and generate their corresponding diffusion variants: Diff-DCP and Diff-RPMNet. Table 1 (left block) shows that the real-world challenges posed by TUD-L lead to limited performance exhibited by the compared traditional and deep methods. In contrast, Diff-RPMNet achieves the highest registration accuracy across all rotation and translation criteria. Furthermore, both Diff-DCP and Diff-RPMNet outperform their respective baselines, DCP and RPMNet, by a substantial margin, particularly impressive for the  $5^{\circ}$  @mAP  $(42\% \uparrow)$  and  $5\mathrm{cm}@\mathrm{mAP}(69\% \uparrow)$  improvements achieved by Diff-DCP. Such superior performance can be primarily attributed to two factors: (i) The Bayesian posterior in Eq.8 effectively orchestrates the pose change of the source point cloud at each reverse step, mitigating premature convergence to local optima; (ii) The diffusion process generates a diverse range of poses for the source point cloud, facilitating more comprehensive model training. Table 2 (top block) also strongly confirms these two factors: The baseline DCP model, trained using samples generated by our diffusion process (DiffAug), presents a remarkable precision improvement (factor (ii)). Furthermore, when equipped with the reverse process (Rev.), its performance is further enhanced (factor (i)). Some qualitative results are provided in Fig.2. Please see Appendix B for additional qualitative results.
Evaluation on LINEMOD and Occluded-LINEMOD. We further assess the performance of our method on two widely-used real-world 6D object pose estimation datasets: LINEMOD [18] and Occluded-LINEMOD [6]. The former dataset encompasses 15 texture-less household objects situated in cluttered scenes, while the latter is a subset comprising 8 texture-less objects with varying degrees of occlusion. As shown in Table 1 (middle and right blocks), our Diff-DCP and Diff-RPMNet consistently outperform the compared methods across nearly all rotation and translation mAP criteria, yielding remarkable improvements over their respective baselines, DCP and RPMNet.
5 Ablation Studies and Analysis
Diffusion Process. (1) We first test the performance variations under different noise schedules, namely the linear schedule [19] and cosine schedule [35], using Diff-DCP for our ablation study. As shown in Table 2 (second block), the cosine schedule tends to yield higher precision on the more challenging LINEMOD and Occluded-LINEMOD datasets, whereas the linear schedule performs better on the comparatively easier TUD-L dataset. We attribute this discrepancy to the fact that the transformation denoising during reverse process in challenging datasets (e.g., LINEMOD) necessitates
Figure 2: Qualitative comparisons on TUD-L [20], LINEMOD [18], and Occluded-LINEMOD [6].
Models	TUD-L	LINEMOD	Occluded-LINEMOD
5°	10°	1cm	2cm	5°	10°	1cm	2cm	5°	10°	1cm	2cm
DCP	0.23	0.62	0.04	0.26	0.06	0.22	0.11	0.27	0.03	0.12	0.11	0.27
DCP+DiffAug	0.48	0.82	0.56	0.88	0.16	0.42	0.52	0.71	0.07	0.24	0.33	0.51
DCP+DiffAug+Rev.*	0.65	0.85	0.73	0.94	0.22	0.51	0.65	0.82	0.10	0.29	0.38	0.57
Linear schedule	0.78	0.93	0.93	0.97	0.16	0.44	0.56	0.75	0.10	0.26	0.35	0.54
Cosine schedule*	0.65	0.85	0.73	0.94	0.22	0.51	0.65	0.82	0.10	0.29	0.38	0.57
Random infer.	0.63	0.86	0.68	0.92	0.22	0.50	0.62	0.80	0.11	0.29	0.38	0.55
Deterministic infer.*	0.65	0.85	0.73	0.94	0.22	0.51	0.65	0.82	0.10	0.29	0.38	0.57
Train. steps T = 50	0.61	0.83	0.65	0.87	0.19	0.47	0.54	0.73	0.10	0.27	0.35	0.55
Train. steps T = 100	0.70	0.92	0.87	0.95	0.18	0.46	0.61	0.80	0.12	0.28	0.42	0.57
Train. steps T = 200*	0.65	0.85	0.73	0.94	0.22	0.51	0.65	0.82	0.10	0.29	0.38	0.57
Table 2: Ablation studies on TUD-L [20], LINEMOD [18], and Occluded-LINEMOD [6]. the rich sample diversity offered by the cosine schedule. In contrast, an abundance of sample diversity on TUD-L would lead to reduced sample efficiency (many generated pose samples of the source point cloud are useless), thus resulting in a degradation of model performance. (2) Additionally, we investigate the registration precision of Diff-DCP across different training steps. Table 2 (bottom block) illustrates that, compared to a small number of steps, such as  $T = 50$ , employing a larger number of diffusion steps, such as 100 and 200, leads to higher precision. This observation stems from the fact that a greater number of diffusion steps significantly enhances sample diversity. Notably, Diff-DCP with  $T = 200$  cannot achieve the best performance on TUD-L. This further validates our claim that too rich sample diversity on the easier TUD-L does not bring higher performance due to low sample efficiency.
Reverse Process. (1) We evaluate our model using two inference strategies: deterministic inference and random inference. In deterministic inference, the denoised transformation at each time step is equal to the predicted mean from Eq. 12, while in random inference, the mean is slightly perturbed by the sampled noise as described in Eq. 9. As shown in Table 2 (third block), deterministic inference without noise tends to exhibit greater stability and achieve lower estimation errors. (2) To verify the effectiveness of the Bayesian posterior (Eq. 8) in guiding the pose change of the source point cloud at each reverse step, we plot the mean error changes of the estimated transformations at different denoising time steps. Fig. 3 confirms that, scheduled by the Bayesian posterior, the denoised transformation gradually approaches to the optimal transformation. Furthermore, the table in Fig. 3 indicates that employing a larger number of reverse steps gradually increases the inference time. Therefore, in our implementation, we set the inference step to 5 to improve the registration efficiency.
Figure 3: Variations in prediction error and runtime for different time steps during the reverse process.
#steps	1	2	3	4	5
Sec.	0.04	0.07	0.11	0.14	0.17
In this paper, we have proposed a novel and effective SE(3) diffusion model-based point cloud registration framework for robust 6D object pose estimation in real-world scenarios. The framework
formulates point cloud registration as a denoising diffusion process, enabling progressive refinement of the pose of the source point cloud for precise alignment with the model point cloud. To facilitate the diffusion and reverse processes over the SE(3) manifold, we have introduced the Lie algebra  $\mathfrak{se}(3)$  associated with SE(3) to constrain the transformation transitions. Furthermore, we have derived a registration-specific variational lower bound to effectively optimize our denoising network. By reformulating our denoising network with a surrogate registration model, different deep registration networks can theoretically be employed within our approach. Our extensive experiments on challenging real-world datasets have validated the effectiveness of our framework. We discuss the broader impact, limitations, and future work in Appendix C.
The authors would like to thank the editor and the anonymous reviewers for their critical and constructive comments and suggestions. This work was supported by the National Science Fund of China under (Grant Nos. U1713208, 62361166670, 62276144) and the Swiss Innovation Agency (Innosuisse).
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