# Fractional Motion Estimation for Point Cloud Compression

Haoran Hong⋆ Eduardo Pavez⋆ Antonio Ortega⋆ Ryosuke Watanabe†
Keisuke Nonaka†
⋆University of Southern California This work was funded in part by KDDI
Research, Inc. and by the National Science Foundation (NSF CNS-1956190).  Los
Angeles CA. 90089 USA
†KDDI Research  Inc  Japan
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Motivated by the success of fractional pixel motion in video coding, we
explore the design of motion estimation with fractional-voxel resolution for
compression of color attributes of dynamic 3D point clouds. Our proposed
block-based fractional-voxel motion estimation scheme takes into account the
fundamental differences between point clouds and videos, i.e., the
irregularity of the distribution of voxels within a frame and across frames.
We show that motion compensation can benefit from the higher resolution
reference and more accurate displacements provided by fractional precision.
Our proposed scheme significantly outperforms comparable methods that only use
integer motion. The proposed scheme can be combined with and add sizeable
gains to state-of-the-art systems that use transforms such as Region Adaptive
Graph Fourier Transform and Region Adaptive Haar Transform.

##  1 Introduction

Recent progress in 3D acquisition and reconstruction technology makes the
capture of 3D scenes ubiquitous. In dynamic point clouds, each frame consists
of a list of data points with 3D coordinates and RGB color values. Since point
clouds in raw format would require a huge amount of bandwidth for transmission
there has been a significant interest in point cloud compression techniques,
which has led to MPEG standardization efforts [1], considering both video-
based point cloud compression (V-PCC) and geometry-based point cloud
compression (G-PCC) [1, 2].

Methods for inter-frame (temporal) prediction have been proposed to achieve
efficient compression of dynamic point clouds. These methods can be grouped
into three main categories. In voxel-based schemes [3], where a motion vector
(MV) is estimated for each voxel, a few points in both the prediction and
reference frames are selected as anchors to establish correspondence via
spectral matching, leading to a set of sparse MVs. Then, using a smoothness
constraint, a dense set of MVs can be obtained from the sparse set to provide
motion for all remaining points. In patch-based techniques [4], motion
estimation (ME) is considered as an unsupervised 3D point registration process
wherein a MV is estimated by iterative closest point (ICP) [5] for each patch
generated by K-means clustering. In this paper we focus on block-based
methods, where frames to be predicted are partitioned into several non-
overlapping 3-dimensional blocks of a given size. For each block, the best
matching block in a reference frame is selected according to specific matching
criteria, which can be based purely on geometry, e.g., an ICP-based approach
that generates rigid transforms [6], or can use a combination of geometry and
color attribute information [7]. Recent work has also focused on block-based
motion search speedup, including both efficient search pattern design and
search window reduction [8, 9, 10, 11].

Our work is motivated by the observation that ME with sub-pixel accuracy is an
essential tool for modern video coding [12], while all the aforementioned ME
methods for dynamic point clouds are based on integer-voxel displacements.
There are two main reasons why a direct extension of video-based fractional ME
to 3D contexts is not straightforward. First, point clouds are irregularly
distributed within each frame, i.e., only those voxels that correspond to the
surfaces of objects in the scene contain attribute information. Thus, while
interpolation of attributes at new voxel locations can be based in
conventional methods, we have the additional challenge of choosing only those
new voxel locations that are consistent with object surfaces, even though
those surfaces are not explicitly known. For example, we would like to avoid
creating additional, fractional accuracy voxels inside an object. Second,
voxels are inconsistent across frames, i.e., both the number of voxels and
their distribution in space are different from frame to frame. Thus, since two
matching blocks in consecutive frames will in general have a different number
of voxels containing attribute information, we will need to develop
alternatives to the one-to-one pixel (or sub-pixel) matching commonly used for
conventional video.

In this paper, we focus on fractional-voxel motion estimation (FvME) under the
assumption that integer-voxel MVs (IvMVs) have already been obtained using an
existing integer-voxel motion estimation (IvME) scheme [7, 8, 9, 10].
Specifically, in this paper we use precomputed IvMVs from a public database
[13]. In our approach, we start by creating fractional voxels between pairs of
neighboring occupied integer voxels. Neighboring voxels are used to favor
consistency with object surfaces, without requiring explicit estimation of the
surfaces. Then, a higher resolution point cloud is obtained by interpolating
attributes at each fractional voxel from the values at nearby integer voxels.
FvME is implemented by searching fractional-voxel MVs (FvMVs) around the
positions given by IvMVs and selecting the fractional displacement leading to
the lowest motion compensation prediction error. Motion-compensated prediction
is implemented by directly copying, as the attribute for a voxel in a block in
the current frame, the attribute of the nearest voxel in the matched blocked
in the reference frame. Our proposed FvME scheme leads to improved performance
over transform-based approaches without inter or intra prediction and is also
significantly better than temporal prediction methods based on the IvMVs from
[13].

##  2 Fractional-Voxel Motion Estimation and Compensation

###  2.1 Motivation

Real-world scenes and objects are captured by multiple calibrated and
synchronized RGB or RGB-D cameras clusters from various viewing angles [14,
15]. After stitching and voxelization, dynamic point clouds are generated on
integer grids. Note that the 3D voxel coordinates are obtained as integer
approximations to the “true” positions of the object in 3D space, while the
optimal displacement between frames is unlikely to be exactly integer. Thus, a
fractional voxel displacement can be better than an integer-one, so that
higher resolution MVs have the potential to provide more accurate motion and
hence more accurate MC prediction.

Furthermore, distortion due to lossy coding in previously reconstructed point
cloud frames can lead to higher prediction errors, while camera noise,
lighting change in the capture environment, object movements, etc., may also
result in noisy color attributes and in imperfect matches during motion search
[16]. Thus, as for conventional video, where it is well known that fractional
motion compensation contributes to noise removal, the process of generating
higher resolution point clouds and attributes at fractional voxel locations
can contribute to denoising and lead to improvements in the quality of the
reference frames.

###  2.2 Occupied fractional voxels

In this section, we define fractional voxels and describe our proposed method
for interpolation. Based on the same design philosophy used for images and
videos, fractional voxels are created at selected intermediate locations
between voxels on the integer resolution grid. We define a fractional voxel of
$1/2$ resolution (1/2-voxel), as a voxel at the mid point between any two
neighboring integer-voxels.

As noted in the introduction, not all integer-voxels are “occupied” in 3D
space, and those that are occupied typically correspond to object surfaces.
Thus, in our proposed method, new fractional voxels are created only at
locations in 3D space that are (approximately) consistent with the surfaces
implied the location of occupied integer voxels and attributes are
interpolated only at these newly created fractional-voxels. We say that two
integer voxels with coordinates $v_{j}$ and $v_{k}$ are neighbors if their
distance is below a threshold $\rho$. Then, a fractional voxel is created only
between neighbors $v_{j}$ and $v_{k}$ (assumed to be close enough so that they
are likely to belong to the same surface) and the corresponding interpolated
color attribute is computed as:

$\begin{split}&C(v_{i})=\frac{1}{2}\left(C(v_{j})+C(v_{k})\right),\\\
&\text{with
}\;L(v_{i})=\frac{1}{2}(L(v_{j})+L(v_{k}))\;\;\text{and}\;\;\text{dist}(v_{j},v_{k})\leq\rho,v_{j},v_{k}\in
V_{i},\end{split}$ (1)

where $v_{i}$ is a voxel in the fractional-voxel set $V_{f}$ with color signal
$C(v_{i})$, $v_{j}$ and $v_{k}$ are voxels in the integer-voxel set $V_{i}$
with color signal $C(v_{j})$ and $C(v_{k})$, respectively. $L(\cdot)$
represents the coordinates of the voxel. $\rho$ is the distance threshold and
$\text{dist}(v_{j},v_{k})$ measures the Euclidean distance between the
coordinates of $v_{j}$ and $v_{k}$.

Note that different pairs of integer voxels may produce the same fractional
voxel. Thus, to remove repeated fractional voxels after interpolation,
attributes that belong to the same fractional voxel and are obtained by
interpolation from different pairs of neighboring voxels are merged by
averaging. Fig. 1(b) shows several examples of possible fractional-voxels
locations, where we can see that interpolation based on neighboring integer
voxels tends to favor increasing the voxel resolution on the (implicit)
surface where the voxels are located.

(a)

(b)

Figure 1: Integer and fractional voxels. Figure 1(a) depicts all possible
candidate integer and 1/2-voxels positions Figure 1(b) shows 3 examples of
occupied integer voxel positions with corresponding fractional voxels obtained
from neighboring integer voxels. Note that these interpolated fractional
voxels are more likely to belong to the same surface as the neighboring
integer voxels they were obtained from.

###  2.3 ME with fractional-voxel accuracy

Due to the inconsistency of voxel distributions in consecutive frames, it is
difficult to establish exact one-to-one correspondences between the voxels in
two matching blocks. To generalize MC prediction for fractional motion in 3D
space, we start by super-resolving the reference frame as described in Section
2.2. As we can see from Fig. 2, the continuity among voxels and their
corresponding attributes is significantly increased underlying surfaces, which
provide better predictors when high resolution motion is available. The low
pass filtering used for interpolation also contributes to attribute noise
removal.

Figure 2: Comparison between the original and super-resolved reference block.

Next, we estimate MVs in fractional precision for MC. The entire ME process is
a coarse-to-fine procedure, including IvME and FvME. Each estimated MV is
obtained as the sum of an IvMV and a FvMV displacement. Assuming the IvMV
$MV_{i}$ is given, the optimal FvMV $MV^{opt}_{f}$ is searched from a set of
candidate fractional displacements. Since we super-resolve the reference frame
in 1/2-voxel precision, each coordinate of a fractional displacement $MV_{f}$
can take values in $\\{-\frac{1}{2},0,\frac{1}{2}\\}$, resulting in $27$
possible displacements. For a given fractional displacement $MV_{f}$, we
predict each attribute in the current block from its nearest voxel in the
translated super-resolved reference block, as depicted in Fig. 3. Then the
displacement with the smallest prediction error is chosen, that is,

$\begin{split}MV^{opt}_{f}&={\hbox{arg}}\min_{MV_{f}}\limits\sum_{v_{i}\in
V(B_{p})}{E_{pred}(C(v_{i}),C(v_{j^{\prime}}))},\\\ \leavevmode\nobreak\
\text{s.t.
}\;&j^{\prime}={\hbox{arg}}\min_{j}\limits(\text{dist}(v_{i},v_{j})),v_{j}\in
V(B_{rMC}^{s}),\\\ &L_{b}(B_{rMC}^{s})=L_{b}(B_{r}^{s})+MV,\\\
&MV=MV_{f}+MV_{i},\end{split}$ (2)

where $B_{r}^{s}$ and $B_{rMC}^{s}$ represent the super-resolved reference
block before and after translation with $MV$, respectively, $v_{i}$ and
$v_{j}$ are voxels with color signals $C(v_{i})$ and $C(v_{j})$ in blocks
$B_{p}$ and $B_{rMC}^{s}$, respectively. $E_{pred}(\cdot,\cdot)$ is the
function for measuring the prediction error. $\text{L}_{b}(\cdot)$ represents
the coordinates of the block while $\text{V}(\cdot)$ represents the set of
voxels within the block.

Figure 3: Motion-compensated prediction.

###  2.4 MC prediction with fractional-voxel accuracy

Finally, we apply MC prediction using the obtained MVs in fractional
precision. Specifically, once the voxels in the reference block are translated
using the integer motion vector $MV_{i}$, they are further shifted by the
obtained optimal fractional displacement $MV^{opt}_{f}$, as shown in (2).
Then, temporal correspondence are established from voxels in the predicted
block $B_{p}$ to their nearest neighbours in the translated super-resolved
reference block $B_{rMC}^{s}$ for motion-compensated prediction. The attribute
of each voxel in the predicted block is predicted by copying the attribute of
its corresponding voxel in the reference frame, that is,

$\begin{split}\forall v_{i}\in B_{p}\text{ ,
}C(v_{i})=C(v_{j^{\prime}})&\leavevmode\nobreak\ \text{s.t.
}j^{\prime}={\hbox{arg}}\min_{j}\limits(\text{dist}(v_{i},v_{j})),v_{j}\in
B_{rMC}^{s}.\end{split}$ (3)

##  3 Experiments

###  3.1 Dataset

In this section, we evaluate the proposed FvME scheme for compression of color
attributes of the dataset of [15], which consists of four sequences:
longdress, redandblack, loot and soldier. Each sequence contains 300 frames.

Note that we assume IvMVs are given and are used to estimate FvMVs. Since
IvMVs derived using different algorithms may lead to different FvMVs with
disparate coding performance, we start from the publicly available 3D motion
vector database [13]. The IvMVs in [13] are selected to minimize a hybrid
distance metric, $\delta=\delta_{g}+0.35\delta_{c}$, which combines
$\delta_{g}$, the average Euclidean distance between the voxels, and
$\delta_{c}$, the average color distance in Y-channel111Note that the
resulting IvMVs aim to select matching blocks with similar geometry
($\delta_{g}$) and color attributes ($\delta_{c}$) but there is no guarantee
that this metric, and in particular the relative weight between the distances
($0.35$) is the optimal choice in terms of coding efficiency. Thus, it will be
shown in our motion compensation experiments, that these IvMVs can sometimes
lead to performance below that of encoding methods that do not use motion
compensation. In these cases performance can be improved by local refinement
of the IvMVs from the database.. We only consider motion for $16\times
16\times 16$ sized blocks. We implement a conventional inter-coding system
where previously decoded frames are used as reference.

###  3.2 Experimental Settings

Following the MPEG Call for Proposal (CfP) for point cloud compression [17],
we evaluate the proposed block-based FvME scheme (Proposed FvME) in groups of
32 frames, with the first frame coded in intra mode, and the rest coded using
inter prediction. The threshold distance between integer voxels for
interpolating fractional voxels is set to $\rho=\sqrt{3}$ in (1). Colors are
transformed from RGB to YUV spaces, and each of the Y, U, and V channels are
processed independently. When searching for the best candidate FvMV in (2), we
use the squared distances to measure prediction errors. All blocks in the
intra-coded frames undergo region adaptive graph Fourier transform (RA-GFT)
[18] while, in the inter-coded frames, all blocks are motion-compensated.
After MC prediction, the residues are transformed using the graph Fourier
transform (GFT) [19]. To compute the GFT, a threshold graph is built inside
every block wherein voxels are connected to each other if their Euclidean
distance is less than or equal to $\sqrt{3}$. If after thresholding, the
resulting graph for a block is not connected, a complete graph is built
instead, which results in the transformed coefficients consisting of a single
approximation coefficient (DC) and multiple detail coefficients (ACs) for each
block. The DC coefficients of all blocks are concatenated and encoded
together. Then, the AC coefficients are coded block by block. This approach is
equivalent to a single level RA-GFT [18]. For all transforms, we perform
uniform quantization and entropy code the coefficients using the adaptive run-
length Golomb-Rice algorithm (RLGR) [20]. As for FvMVs overheads, since there
are 27 FvMVs in total, 8 bits are used to signal each FvMV. For IvMVs, we use
4 bits to signal the value and 1 bit to signal sign for each axis and
therefore, 15 total bits are used to represent an IvMV. The overheads of FvMVs
and IvMVs are entropy coded by Lempel–Ziv–Markov chain algorithm [21].

We considered the following schemes as baselines. IvME using the database
motion (DM) for MC prediction and using DM with additional integer local
refinement (DM+RF) for MC prediction. The local refinement uses different
criteria that aims to minimize color errors only, instead of the hybrid errors
used in the database. DM is refined by additional local search in integer
precision to improve its matching accuracy over the original ones. The locally
refined range for each axis is set to be $[-1,1]$, which entirely encloses
fractional positions searched in the proposed FvME scheme.

To evaluate the benefits of high resolution references and FvMVs, we propose
two inter coding schemes which are using super-resolved reference blocks, with
and without fractional motion vectors for compensated prediction. First, to
evaluate the super-resolution method, we implement a scheme that considers
IvME using integer local refined DM and super-resolved reference blocks for
prediction, which is denoted by “DM+RF+SR”. The difference between DM+RF and
DM+RF+SR is the resolution of the reference block. Then, to evaluate benefits
of FvMVs, we implement a scheme that uses fractional resolution in both
reference blocks and motion vectors, which is denoted by “proposed FvME”. For
a fair comparison between inter coding schemes, all other test conditions are
the same.

Additionally, to make our performance evaluation more complete, we include two
state of the art (all intra) anchor solutions, namely, RA-GFT [18] and region
adaptive Haar transform (RAHT) [22]. For RA-GFT, block size 16 is used. The
residues are entropy coded by RLGR.

###  3.3 Evaluation Metrics

The evaluation metrics are the number of bits per voxel (bpv) and average peak
signal-to-noise ratio over Y component (PSNR-Y),

$PSNR_{Y}=-10\log_{10}\left(\frac{1}{T}\sum_{t=1}^{T}\frac{\|Y_{t}-\hat{Y_{t}}\|_{2}^{2}}{255^{2}N_{t}}\right),bpv=\frac{\sum_{t=1}^{T}b_{t}}{\sum_{t=1}^{T}N_{t}},$
(4)

where $Y_{t}$ and $\hat{Y_{t}}$ represent original and reconstructed signals
on the same voxels of $t$-th frame respectively, $T$ is the total number of
frames, $b_{t}$ is the bits required to encode YUV components of $t$-th,
including IvMVs and FvMVs overhead when necessary, and $N_{t}$ is the total
number of occupied voxels in $t$-th frame. The Bjontegaard-Delta [23] results
for bitrate (BD-rate) are also reported.

###  3.4 Experimental Results and Analysis

Rate distortion (RD) curves are shown in Fig. 4. We first note that using only
the original IvME from the database [13], results in sub-optimal performance
compared to RAHT and RA-GFT. This is in part due to the criteria used in [13]
to choose the optimal MV based on geometry and color information. After local
refinement with integer precision, the performance of IvME (DM+RF) improves
significantly with respect to IvME (DM) but it is still far from being
competitive with other techniques. Further improvements have been shown to be
achievable by using per block intra/inter mode decision [10].

(a) longdress

(b) soldier

(c) redandblack

(d) loot

Figure 4: Rate distortion curves of 8iVFBv2 sequences.

anchors $\backslash$ sequences |  longdress |  soldier |  redandblack |  loot
---|---|---|---|---
IvME(DM+RF) |  $-43.93\%$ |  $-63.96\%$ |  $-57.98\%$ |  $-64.43\%$
RAHT |  $-39.94\%$ |  $-81.91\%$ |  $-51.10\%$ |  $-73.69\%$
RA-GFT(b=16) |  $-16.19\%$ |  $-72.46\%$ |  $-24.37\%$ |  $-61.44\%$

Table 1: The BD-rate performances of the proposed scheme over baselines

After the reference blocks are super-resolved, the performance of the proposed
IvME (DM+RF+SR) is further improved with respect to DM+RF, even without
increasing MV resolution. The DM+RF+SR scheme can be better than the intra
schemes in some cases with the advantage of complexity lower than that of the
proposed FvME. Finally, after we increase MV resolution to 1/2-voxel, further
coding gains are obtained, outperforming intra coding baselines, RA-GFT and
RAHT, with average gains of $2.8dB$ and $4.6dB$, respectively. The method is
always better than DM+RF+SR but at the cost of higher complexity due to
additional motion search. The results show that both interpolated fractional
voxels and high resolution MVs lead to higher coding gain and outperform both
inter coding with IvME and non predictive transform based schemes.

Table 1 summarizes the performance of the proposed method over IvME (DM+RF),
RA-GFT, and RAGT in terms of BD-rate. The proposed FvME can achieve $57\%$
average bitrate reduction over IvME (DM+RF). Compared with the prior arts, the
proposed scheme can achieve $61\%$ and $43\%$ bitrate reduction on average
over RAHT and RA-GFT respectively.

Compared with IvME (DM+RF), the proposed FvME scheme increases the number of
voxels at most eightfold (because of super resolution), and requires
evaluating additional 27 fractional displacements for ME. Therefore, the
complexity of the proposed FvME is larger than the complexity of IvME (DM+RF)
by a constant factor (independent of the point cloud size).

##  4 Conclusions

This paper describes a fractional-voxel motion estimation scheme tailored for
attribute compression in 3D dynamic point clouds. Our scheme defines and
identifies fractional voxels to be interpolated and provides a motion
compensation prediction method by super-resolution and temporal
correspondence. Extensive experiments show superior performance over the prior
art. Our work reveals the benefits given by high resolution references and the
further improvements given by fractional-voxel motion vectors for dynamic
point clouds color compression.

##  5 References

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