#  Multi-resolution intra-predictive coding of 3D point cloud attributes

###### Abstract

We propose an intra frame predictive strategy for compression of 3D point
cloud attributes. Our approach is integrated with the region adaptive graph
Fourier transform (RAGFT), a multi-resolution transform formed by a
composition of localized block transforms, which produces a set of low pass
(approximation) and high pass (detail) coefficients at multiple resolutions.
Since the transform operations are spatially localized, RAGFT coefficients at
a given resolution may still be correlated. To exploit this phenomenon, we
propose an intra-prediction strategy, in which decoded approximation
coefficients are used to predict uncoded detail coefficients. The prediction
residuals are then quantized and entropy coded. For the 8i dataset, we obtain
gains up to $0.5$db as compared to intra predicted point cloud compresion
based on the region adaptive Haar transform (RAHT).

Index Terms—  3D point clouds, intra prediction, multiresolution transform,
graph fourier transform

##  1 Introduction

3D point clouds (3DPC) have become the preferred representation of 3D scenes,
people and objects [1, 2, 3]. They consist of a list of points coordinates in
3D space along with color attributes. Recent advancements in real time 3D
capture, along with potential applications to entertainment and immersive
communications, have prompted research on 3DPC compression [1, 2].

This paper focuses on 3DPC attribute compression. Earlier approaches for
compression of 3DPC attributes were based on transform coding techniques, that
is, transformation followed by quantization and entropy coding, similar to
modern image codecs. For 3DPCs, a popular approach is based on the region
adaptive hierarchical (or Haar) transform (RAHT) [4, 5]. In image and video
coding, intra and inter prediction are often combined with transform coding.
Predictive methods for 3DPC compression have only recently become popular,
which may be explained by the fact that good spatial and temporal predictors
are harder to obtain for 3DPCs, since these i) represent complex surface and
non-surface geometries, ii) have spatial frame to frame irregularity, and iii)
lack temporal consistency. There has been substantial recent work to apply
transform coding to inter frame (temporal) prediction residuals [3, 6, 7, 8,
9]. However, intra frame prediction is less explored. In [10, 11], block based
intra-prediction similar to video was considered. More recently, MPEG adopted
an intra prediction strategy for the RAHT (I-RAHT) [12], which uses a multi-
resolution prediction, instead of traditional directional block based
prediction.

(a) I-RAHT

(b) Proposed

Fig. 1: Comparison of multi-resolution predictors. (1(a)) Predictor attribute
(red dot) is a linear combination of neighbors in lower resolution point cloud
(${\bf V}_{\ell}$). (1(b)) Predictor attribute is a linear combination of
neighbors in current resolution point cloud (${\bf V}_{\ell+1}$).

In I-RAHT, low pass decoded RAHT coefficients are used to predict high pass
RAHT coefficients. To achieve this, a higher resolution predictor point cloud
is constructed from lower resolution decoded coefficients. Each predictor
attribute in the high resolution point cloud is obtained as a linear
combination of the nearest attributes in the lower resolution point cloud.
Then the detail coefficients of this predictor signal are used to predict the
target detail coefficients. While this approach to construct a higher
resolution point cloud is suitable for the RAHT, since it is formed by a
composition of $2\times 2$ orthogonal transforms, it fails to take into
account the more complex geometry of the higher resolution point cloud. This
becomes even more important when considering larger block transforms with more
points used by the region adaptive graph Fourier transform (RAGFT) [13], as
shown in Figure 1.

In this paper we propose an intra-predictive coding framework for RAGFT
(I-RAGFT). We use the same multi-resolution prediction strategy used in
I-RAHT, where decoded approximation coefficients are used to predict and code
detail coefficients, as depicted in Figure 2. However, different from the
prediction algorithm used by I-RAHT, our proposed predictors exploits the
higher resolution point cloud geometry. We start by projecting the low
resolution signal onto the higher resolution geometry by zero padding and
applying a one level inverse RAGFT. The resulting interpolated signal is
piece-wise constant, as depicted by the colored voxels in Figure 1(b). Then a
smoothing graph filter [14, 15, 16, 17] is applied using a graph constructed
on top of the higher resolution point cloud. While our approach is new for
3DPCs, similar ideas have been used to improve image coding by predicting
wavelet coefficients with learning based super resolution algorithms [18].

The difference between our proposed approach and I-RAHT is illustrated in
Figure 1. I-RAHT uses a block-level predictor, since the approximation
coefficients of the nearest neighboring blocks are used (Figure 1(a)), while
our approach uses a point-level predictor, where fine resolution point values
are interpolated, then filtered (across block boundaries), so that only nearby
points (instead of nearby blocks) are used to compute the predictor. We show
through compression experiments that the proposed approach can outperform
I-RAHT, when using uniform quantization and adaptive RLGR entropy coding [19].

The rest of the paper is organized as follows. In Section 2 we introduce the
multi-resolution intra-prediction coding framework. The proposed predictors
for the RAGFT are described in 3. We show numerical experiments and
conclusions in Sections 4 and 5, respectively.

##  2 Multi resolution predictive coding

Consider a point cloud with point coordinates stored in the $N\times 3$ matrix
${\bf V}=[{\bf v}_{i}]$, and attributes ${\bf a}=[a_{i}]$. We will assume the
attributes are processed with a multi-resolution transform, which at each
resolution level takes an attribute vector ${\bf a}_{\ell+1}$, and produces
approximation and detail coefficients at resolution $\ell$

$\begin{bmatrix}{\bf a}_{\ell}\\\ {\bf d}_{\ell}\end{bmatrix}={\bf
T}_{\ell}{\bf a}_{\ell+1}.$ (1)

The transform matrix ${\bf T}_{\ell}$ is an orthonormal matrix, thus ${\bf
T}_{\ell}^{-1}={\bf T}_{\ell}^{\top}$. We assume that the original 3DPC
attributes are stored at the highest resolution, $L$, so that ${\bf
a}_{L}={\bf a}$. After applying (1) $L$ times, we obtain transform
coefficients

$\begin{bmatrix}{\bf a}_{0}^{\top},{\bf d}_{0}^{\top},{\bf
d}_{1}^{\top},\cdots,{\bf d}_{L-1}^{\top}\end{bmatrix}^{\top}.$ (2)

Several orthonormal transforms for 3DPC attributes can be described this way,
including the block based graph Fourier transform [20], RAHT [4] and RAGFT
[13]. Since RAGFT is a composition of spatially localized block transforms,
there may be additional redundancy between transformed coefficients, similar
to what is observed for the RAHT [12]. While previous approaches code the
coefficients in (2) independently ignoring their dependencies across blocks,
in this work we exploit them to improve coding efficiency.

Denote by $\widehat{\hbox{\boldmath$\Phi$}}_{0}={\cal Q}({\bf a}_{o})$, the
quantized low pass coefficients, where ${\cal Q}(\cdot)$ is a quantization
operator, and denote by $\hat{{\bf a}}_{0}={\cal
Q}^{-1}(\widehat{\hbox{\boldmath$\Phi$}}_{0})$ the corresponding decoded
coefficients. Now we define several quantities recursively. The decoded
approximation coefficient at resolution $\ell$ is given by $\hat{{\bf
a}}_{\ell}$, while the corresponding detail coefficient at the same resolution
is ${\bf d}_{\ell}$. We will assume there is function ${\cal
P}_{\ell}(\hat{{\bf a}}_{\ell})=\tilde{{\bf d}}_{\ell}$, that predicts detail
coefficients from approximation coefficients. This predictor will be explained
in detail in the next section. Using ${\cal P}_{\ell}$ we compute a residual
and quantize it obtaining

$\widehat{\hbox{\boldmath$\Delta$}}_{\ell}={\cal Q}({\bf d}_{\ell}-\tilde{{\bf
d}}_{\ell}).$ (3)

If the predictor is good enough, coding
$\widehat{\hbox{\boldmath$\Delta$}}_{\ell}$ is more efficient than coding the
transform coefficients ${\cal Q}({\bf d}_{\ell})$ directly. The decoded
details coefficients are given by

$\hat{{\bf d}}_{\ell}=\tilde{{\bf d}}_{\ell}+{\cal
Q}^{-1}(\widehat{\hbox{\boldmath$\Delta$}}_{\ell}).$ (4)

The decoded approximation and detail coefficients at resolution $\ell$ are
used to obtain approximation coefficients at resolution $\ell+1$ with

$\hat{{\bf a}}_{\ell+1}={\bf T}^{-1}_{\ell}\begin{bmatrix}\hat{{\bf
a}}_{\ell}\\\ \hat{{\bf d}}_{\ell}\end{bmatrix}.$ (5)

The quantities $\widehat{\hbox{\boldmath$\Delta$}}_{\ell}$ can be computed
recursively as depicted in Figure 2, starting from the lowest resolution
coefficients ${\bf a}_{0}$. A typical transform coding strategy would quantize
and entropy encode (2). In this work we encode

$\begin{bmatrix}\widehat{\hbox{\boldmath$\Phi$}}_{0}^{\top},\widehat{\hbox{\boldmath$\Delta$}}_{0}^{\top},\widehat{\hbox{\boldmath$\Delta$}}_{1}^{\top},\cdots,\widehat{\hbox{\boldmath$\Delta$}}_{L-1}^{\top}\end{bmatrix}^{\top}.$
(6)

Fig. 2:  Multi-resolution intra-predictive coding

##  3 Multi resolution prediction

In this section we describe the function ${\cal P}_{\ell}(\hat{{\bf
a}}_{\ell})=\tilde{{\bf d}_{\ell}}$, used to predict the high pass
coefficients ${\bf d}_{\ell}$.

###  3.1 Graph representation of point clouds

While forming transform coefficients (2), many transforms [4, 13], either
explicitely or implicitely, produce sets of point coordinates at various
resolutions (e.g., by down-sampling), thus for each resolution $\ell$, we have
a point cloud $({\bf V}_{\ell},{\bf a}_{\ell})$, where ${\bf V}_{L}={\bf V}$
and ${\bf a}_{L}={\bf a}$.

For each of these point clouds consider a graph ${\cal G}_{\ell}=({\bf
V}_{\ell},{\bf W}_{\ell},{\cal E}_{\ell})$. The matrix ${\bf W}_{\ell}$ is the
adjacency matrix and ${\bf D}_{\ell}=\operatorname*{diag}(\sum({\bf
W}_{\ell}))$ is the degree matrix. The graph has edge set ${\cal E}_{\ell}$,
where $ij\in{\cal E}_{\ell}$ if point ${\bf v}_{i,\ell}$ is “near” to point
${\bf v}_{j,\ell}$. Edge weights are given by

$({\bf W}_{\ell})_{ij}=\frac{1}{\|{\bf v}_{i,\ell}-{\bf v}_{j,\ell}\|}.$ (7)

###  3.2 Constructing the predictor

The first step in constructing our predictor is interpolation of the low
resolution point cloud $({\bf V}_{\ell},\hat{{\bf a}}_{\ell})$ by zero
padding, and inverse transformation, leading to the attribute signal

${\bf b}_{\ell+1}={\bf T}_{\ell}^{-1}\begin{bmatrix}\hat{{\bf a}}_{\ell}\\\
\mathbf{0}\end{bmatrix},$ (8)

and higher resolution point cloud $({\bf V}_{\ell+1},{\bf b}_{\ell+1})$. We
propose applying a smoothing (low pass) filter to the point cloud $({\bf
V}_{\ell+1},{\bf b}_{\ell+1})$, and then applying the transform ${\bf
T}_{\ell}$. The _predictor_ block from Figure 2 is depicted in Figure 3, with
the caveat that $\tilde{{\bf a}}_{\ell}$ is ignored.

Fig. 3: Proposed predictor $\tilde{{\bf d}}_{\ell}={\cal P}_{\ell}(\hat{{\bf
a}}_{\ell})$.

(a) Longdress

(b) Loot

Fig. 4: Rate distortion curves for color compression.  Fig. 5: Comparison of
reconstructed point clouds at $0.493$ bits per point. I-RAGFT
($PSNR=30.31dB$), I-RAHT ($PSNR=30dB$), and RAGFT ($PSNR=29.11dB$)

###  3.3 Graph filtering of RAGFT coefficients

In the RAGFT [13], each transform coefficient has an importance weight that
depends on the size (number of points) of the region of the point cloud they
represent. The $i$th point of the point cloud of resolution $\ell$ is denoted
by ${\bf v}_{i,\ell}$. The point ${\bf v}_{i,\ell}$ has a set of children at
resolution $\ell+1$ denoted by ${\cal C}_{i,\ell}$. The index set of the
children is denoted by ${\cal I}_{i,\ell}=\\{j:{\bf v}_{j,\ell+1}\in{\cal
C}_{i,\ell}\\}$. We denote the weight of point ${\bf v}_{i,\ell}$ by
$q_{i,\ell}$. Based on this parent-child relation, the weights can be computed
recursively as

$q_{i,\ell}=\sum_{j\in{\cal I}_{i,\ell}}q_{j,\ell+1},$ (9)

and the weights at full resolution are $q_{i,L}=1$ for all $i$. For the RAGFT,
the interpolation equation (8) has a closed form, thus the $j$th entry of
${\bf b}_{\ell+1}$ is equal to

$b_{j,\ell+1}=\sqrt{\frac{q_{j,\ell+1}}{q_{i,\ell}}}\hat{a}_{i,\ell},$ (10)

where point ${\bf v}_{i,\ell}$ is the parent of ${\bf v}_{j,\ell+1}$. Note
that if we define the diagonal matrix of weights by ${\bf Q}_{\ell+1}$, with
$jj$ entry equal to $q_{j,\ell+1}$, then the interpolated signal is

${\bf b}_{\ell+1}={\bf
Q}^{1/2}_{\ell+1}\left(\sum_{i}\frac{\hat{a}_{i,\ell}}{\sqrt{q_{i,\ell}}}\mathbf{1}_{{\cal
I}_{i,\ell}}\right),$ (11)

where $\mathbf{1}_{{\cal I}_{i,\ell}}$ is the indicator of the set ${\cal
I}_{i,\ell}$. Thus after normalization by the square root of the point
weights, the interpolated signal is piece-wise constant. In fact, this signal
is constant within cube shaped regions, because the RAGFT is is a composition
of localized “block” transforms. Figure 1(b) depicts this piece-wise constant
signal. Given that ${\bf b}_{\ell+1}$ has this form, we take into account the
normalization matrix ${\bf Q}_{\ell+1}$, and the piece-wise constant
structure, when designing our smoothing operator. The first step is to
normalize the entries of ${\bf b}_{\ell+1}$ by their point weights using ${\bf
Q}^{-1/2}_{\ell+1}$. The resulting piece-wise constant signal is filtered to
smooth out boundaries between regions. Finally, we scale back each attribute
using the matrix ${\bf Q}^{1/2}_{\ell+1}$. The proposed filtering operation
has the form

$\tilde{{\bf b}}_{\ell+1}={\bf Q}^{1/2}_{\ell+1}{\bf D}^{-1}_{\ell+1}{\bf
W}_{\ell+1}{\bf Q}^{-1/2}_{\ell+1}{\bf b}_{\ell+1}.$ (12)

At resolution $\ell$, the adjacency matrix ${\bf W}_{\ell}$ is constructed
using $k$ nearest neighbors on the point cloud geometry ${\bf V}_{\ell}$. The
signal $\tilde{{\bf b}}_{\ell+1}$ is transformed using (1), resulting in
$\tilde{{\bf a}}_{\ell}$ and $\tilde{{\bf d}}_{\ell}$.

###  3.4 Complexity

The proposed intra predicted RAGFT can be computed by performing a forward and
inverse RAGFT, along with a graph construction and a graph filtering operation
per resolution level. For a point cloud with $N$ points the forward and
inverse RAGFT have complexity ${\cal O}(N)$. If the octree [21, 22] has to be
computed, overall complexity increases to ${\cal O}(N\log(N))$. If the number
of points at resolution $\ell$ is equal to $N_{\ell}$, computation of ${\bf
b}_{\ell}$ using (11) takes ${\cal O}(N_{\ell})$ time. Graph construction with
$k$ nearest neighbors requires ${\cal O}(kN_{\ell}\log(N_{\ell}))$ operations,
while $\tilde{{\bf b}}_{\ell}$ can be computed using sparse matrix vector
products in ${\cal O}(kN_{\ell})$ time. If $k$ is ${\cal O}(1)$, then
prediction operations at all resolutions have complexity ${\cal O}(N\log(N))$,
resulting in an overall ${\cal O}(N\log(N))$ complexity of the intra predicted
RAGFT.

##  4 Numerical results

We integrate intra prediction with the RAGFT with $2\times 2\times 2$ blocks
at all resolution levels, thus each localized block transform processes at
most $8$ points. The proposed predictors are implemented using graph filters,
that use $K$ nearest neighbor graphs, with $K=7$ neighbors per point. We
denote this approach by “I-RAGFT”. We also implement predictors similar to
those used by the G-PCC implementation of I-RAHT using KNN from the lower
resolution graph (see Fig. 1(a)). For this approach we set $K=5$, found after
optimizing for best performance. This approach is denoted by “I-RAGFT LowRes”.
We also compare against the RAGFT with high resolution block size equal to
$16$, which achieves highest coding performance compared to RAHT, and the
RAGFT with high resolution block size equal to $2$, which provides a mild
improvement over RAHT (see [13] for details). For RAGFT and intra RAGFT, we
uniformly quantize coefficients and entropy code them using adaptive run-
length Golomb-Rice (RLGR) algorithm [19].

The state of the art in non-video-based compression of point cloud attributes,
called G-PCC [5] uses I-RAHT. Several techniques are implemented in G-PCC to
improve coding efficiency beyond that obtained through intra prediction. Some
of them include jointly encoding YUV coefficients that are equal to zero, and
adaptive quantization schemes of AC coefficients. These techniques could also
be applied to the RAGFT but their implementation goes beyond the scope of this
paper and we leave them for future work. In order to obtain a performance
comparison between our method and the current state-of-the-art, under the same
conditions, we changed the source code of G-PCC modifying both the quantizer
and the entropy encoder schemes. The adaptive quantizer was replaced with a
uniform quantizer and the entropy encoder was modified to an adaptive RLGR
algorithm. Moreover, we encode YUV coefficients independently. In Figure 4 we
report rate distortion curves for the “longdress” and “loot” point clouds of
the “8iVFBv2” dataset [23].

Incorporating intra prediction into the RAGFT improves coding performance
significantly. The difference between “RAGFT $b=2$” and “I-RAGFT LowRes” is
about $1.5$db, while the gain obtained by using better predictors (“I-RAGFT
Proposed”) can reach up to $2.5$db. In [13], the RAGFT implemented with larger
block sizes ($b_{L}=16$) led to the best results, since spatial redundancy can
be removed more efficiently using graph transforms on larger blocks. Our
results show that by combining low resolution intra prediction and RAGFT with
small block size (“I-RAGFT $b=2$”), we can outperform the best RAGFT at low
bitrates. However this approach is still inferior to I-RAHT. The proposed
predictor based on the higher resolution point cloud, combined with the RAGFT
achieves the best rate distortion performance among all methods considered,
and outperforms I-RAHT by up to $0.5$db.

In Figure 5 we compare decoded attributes obtained after compression using
I-RAGFT, I-RAHT and RAGFT ($b_{L}=16$), at the same rate of $0.493bpp$.
Transform coding with the RAGFT produces strong blocking artifacts, similar to
those associated to JPEG encoding. Both intra coding approaches have improved
visual quality with respect to RAGFT, while also using smaller transforms,
thus avoiding blocking artifacts. I-RAHT produces an over smoothed
reconstruction, which can be accredited to the less localized filtering
operations. Since the I-RAGFT uses higher resolution predictors, image details
are better preserved, however it suffers from other localized artifacts.

##  5 Conclusion

We studied the use of intra prediction for compression of 3D point cloud
attributes with the RAGFT. As with the RAHT, we showed that it is possible to
remove redundancy between RAGFT coefficients at different resolutions. In the
RAHT, a high resolution point cloud is predicted from a lower resolution point
cloud. While this approach is efficient for predicting RAHT coefficients, it
is less effective for the RAGFT. This is because the RAGFT uses larger block
transforms, that makes prediction more challenging. To overcome this issue, we
proposed a different prediction based on interpolation and graph signal
filtering, that takes into account the higher resolution geometry, thus
adapting better to the RAGFT. Compression experiments show the proposed
approach outperforms intra predictive coding of RAHT coefficients at all rates
by up to $0.5$db.

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