Received 13 March 2023, accepted 28 April 2023, date of publication 11 May 2023, date of current version 17 May 2023.

Digital Object Identifier 10.1109/ACCESS.2023.3275442

DC-DFFN: Densely Connected Deep Feature
Fusion Network With Sign Agnostic Learning
for Implicit Shape Representation
ABOL BASHER AND JANI BOUTELLIER
School of Technology and Innovation, University of Vaasa, 65200 Vaasa, Finland

Corresponding authors: Abol Basher (abol.basher@uwasa.fi) and Jani Boutellier (jani.boutellier@uwasa.fi)

This work was supported in part by the Academy of Finland Projects Compact and Efficient Deep Neural Networks for Ubiquitous
Computer Vision (CoEfNet) and Robust and Efficient Perception for Autonomous Things (REPEAT).

ABSTRACT Reconstructing 3D surfaces from raw point cloud data is still a challenging and complex
problem in computer vision and graphics. Recently emerged neural implicit representations model 3D
surfaces implicitly in arbitrary resolution and diverse topologies. In this domain, most of the studies have
so far used a single latent code-based variational auto-encoder (VAE) or auto-decoder (AD) architectures,
or architectures similar to UNets. Due to the deep architectures of the existing approaches, gradients
and/or input information can vanish while passing through the layers, which can cause suboptimal learning
at training time and consequently low performance at test time. As a countermeasure, skip connections
and feature fusion have been used in related application fields of convolutional neural networks. In this
study, we embrace this idea and propose a novel densely connected deep feature fusion network, DC-
DFFN, architecture for implicit shape representation. In the experimental results we show that DC-DFFN
outperforms baseline approaches in terms visual reconstruction quality and quantitatively based on several
measures. In addition, the proposed approach provides faster convergence during training compared to the
baseline approaches. The DC-DFFN architecture has been implemented in PyTorch and is available as open
source.

INDEX TERMS Convolutional neural network, implicit representation, dense feature fusion, zero-label set,
surface reconstruction, ShapeNet, D-Faust.

I. INTRODUCTION
Recent advances in learning-based data driven approaches [1],
[2], [3], [4], [5], [6], [7], [8] for reconstructing surfaces from
raw un-oriented point clouds, and triangle soups are showing
huge potential for several practical application fields, for
example AR/VR technology, 3D printing, computer-aided
design, and robotics. Recently emerged neural function-based
implicit representations [1], [2], [3], [5], [8], [9], [10]
can reconstruct a surface with infinite resolution and arbi-
trary topology compared to classical 3D presentations such
as voxels, octrees, point clouds, and meshes, which have
various ingrained issues. For example, voxel-based repre-
sentations have problems related to resolution (memory

The associate editor coordinating the review of this manuscript and

approving it for publication was K. C. Santosh .

requirement increases cubically with resolution), whereas
point cloud-based representations do not have connectivity
among the points, and meshes can have self-intersection
issue [9], [11] and are restricted to a fixed topology.

The recently emerged implicit representations of 3D visual
data to a great extent solve the problems related to classical
representations, but pose new challenges related to com-
plexity and computation time of the involved neural net-
works. Moreover, most of the implicit representation-based
surface reconstruction works [3], [4], [5], [6], [7], [8], [9]
focus only on reconstruction quality, use of novel activation
functions, and optimization methods, hardly paying atten-
tion on network size [12] or training and inference time.
In this study, we propose a densely connected feature fusion-
based encoder-decoder neural architecture to ensure maxi-
mum input information flow through the network for better

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https://orcid.org/0000-0002-6383-493X
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A. Basher, J. Boutellier: DC-DFFN With Sign Agnostic Learning for Implicit Shape Representation

FIGURE 1. Illustration of the proposed architecture’s improved capability of capturing small detail, compared to the state-of-of-the art work
SALD [1]. Zooming in reveals that SALD causes wing and wheel parts to become disconnected, and renders extra holes to the airplane’s hull.

learning and consequently, more expressive reconstruction
quality.

In deep neural architectures, the gradients or input infor-
mation can vanish or wash out while passing through the
layers, which has already been addressed in several works
of 2D image domain [13], [14], [15] applications, such as
object recognition, object detection, and localization. So far,
according to the best knowledge of the authors, none of the
previous studies have considered dense features simultane-
ously to learn the surface and address vanishing gradient
problem in the network architectures used for neural implicit
representations. Besides mitigating the vanishing gradient
problem, fusing features densely will allow themodel to learn
low (blob, edge) and high (object) level features simulta-
neously, when being extracted from different layers of the
network. To this end, we propose a dense feature fusion-based
encoder-decoder network architecture to achieve high fidelity
surface reconstruction.

The main contributions of this study are:

• A novel deep feature fusion-based variational auto-
encoder architecture1 for implicit surface reconstruc-
tion, which

• Ensures better learning and shorter convergence time
due to improved information flow through the network,
and

• Better robustness and generalization to unseen shapes,
and

• Reconstruction of high fidelity continuous surfaces and
obtaining state-of-the-art quantitative results.

The proposed study is a natural extension of previous
approaches [1], [8], however transformed into a densely con-
nected feature fusion-based network architecture. The rest
of this paper is organized in the following manner: recently
proposed related studies on explicit and implicit image rep-
resentations are discussed in Section II; the proposed densely
connected deep feature fusion-based encoder and decoder
architectures are illustrated in Section III; qualitative and

1source code available: https://github.com/basher8488881/DC-DFFN

quantitative comparisons with the baseline approaches are
shown in Section IV, and Section VI concludes the paper.

II. RELATED WORK
3D surface reconstruction approaches can be categorised
based on their inherent ways of representing visual data:
(a) explicit representations or classical representations, such
as voxels, point clouds, and meshes, (b) implicit repre-
sentations. In this section, we review traditional analytic
priors-based reconstruction methods, classical and implicit
representation-based approaches. Additionally, we review a
few feature fusion-based studies from the 2D image domain
to illustrate commonly used strategies for constructing effi-
cient network architectures, which are the backbone of this
study.

A. TRADITIONAL RECONSTRUCTION APPROACHES
There are a number of existing methods that are based on ana-
lytic priors for surface reconstruction, for example: Screened
Poisson Surface Reconstruction (SPSR) [16], Moving Least
Squares (MLS) [17], Ball Pivoting Algorithm (BPA) [18] and
Radial Basis Functions (RBF) [19]. SPSR was developed
on top of the previously proposed Poisson Surface Recon-
struction (PSR) algorithm [20], which works based on global
surface smoothness priors, and addresses the limitations asso-
ciated with PSR, for example tendency of over-smoothing
the data. This approach casts the surface reconstruction task
as a spatial Poisson problem and performs reconstruction
in the frequency domain [21]. However, SPSR requires ori-
ented normals of the input points. Similar to SPSR, RBF
also works based on global surface smoothness priors and
performs the reconstruction using radially symmetric basis
functions. MLS is a mesh-independent approach of surface
interpolation. It works considering surfaces in differential
geometry, which includes a local mapping function and a
local reference system for each points of the surface, and
uses the moving least squares concept. BPA, on the other
hand, reconstructs the surface through computing triangles
by interpolating a given point cloud. In BPA, considering a
triangle formation of three points from a point cloud, a sphere

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with a predefined radius is rotated around the edges until it
touches another point.

B. CLASSICAL REPRESENTATIONS OF 3D VISUAL DATA
Voxel-based representations are one of the most popular and
earliest representations for learning-based 3D reconstruction
of shapes and scenes [22], [23], [24]. In this representation,
the 3D space is discretized into a regular grid, making it an
intuitive extension for learning-based algorithms that have
been developed for the 2D image domain, such as deep
(convolutional) neural networks. In its simplest form, voxels
can be used to learn the dense occupancy grid (where each
voxel is occupied or unoccupied), and utilize this information
to render a mesh surface [2], [5], [25]. However, due to cubi-
cally increasing memory requirements and lacking fidelity of
rendered shapes, the usage of occupancy grids is limited to
specific use cases [26], [27], [28].

Point clouds are another classical 3D data representation
that expresses the 3D visual information by sparse data points
and provide several advantages over voxels, for example their
capability of better representing large spaces with fine details.
Point clouds also serve as one possible output representation
of implicit surface modeling [3]. Drawbacks of point clouds
include lack of connectivity information, and high memory
footprint of large surfaces that need to be represented densely.

In contrast, the mesh representation bears more informa-
tion than point cloud-based representations by expressing
connectivity among 3D points. The vertices and faces of a
mesh can be directly regressed using a neural network [29],
[30]. Meshes have a wide range of applications, for exam-
ple in classification and segmentation [31], [32], [33]. More
recently, mesh-based representations have also been used as
the output representation for implicit 3D surface reconstruc-
tion [1], [5], [8], [9], [10], [12].

C. IMPLICIT REPRESENTATIONS
Recently emerged implicit representations express the 3D
surface S implicitly using (zero) label sets (Equation 1),

S = {x ∈ R3
|f (x;w) = 0} (1)

of a neural function f : R3
−→ R, where x ∈ R3

is the input data (sampled from a point cloud or triangle
soup, X ∈ R3), and w are the neural network weights that
approximate the surface to X . There are mainly two types
of supervised approaches [1] commonly used to train the
neural network to become an implicit function representation:
(I) regression of known or pre-computed occupancy values
f (p, z) : R3

× Z −→ [1, 0] using an occupancy func-
tion [5], [9] or signed distances f (p, z) : R3

× Z −→ R
using a signed distance function [1], [8], [10] or unsigned
distances f (p, z) : R3

× Z −→ R+

0 using an unsigned
distance function [3], [34], and (II) regression of raw 3D data
using sign agnostic losses [1], [8] by relating points on the
level sets to the neural network model parameters [35] or
supervision with partial differential equations approximating

the signed distance functions [36]. In this study, we adopted
the second the approach of training the proposed network
using a sign agnostic loss function [1], [8]. Our proposed
network outperforms the recently proposed state-of-the-art
method SALD [1] and shows that dense connections in the
network provide improved information flow through the lay-
ers and faster convergence, consequently generating high
fidelity shapes that preserve small details. In addition to
proposing dense connectivity and feature fusion, our archi-
tecture employs 1D convolutional layers with 1 × 1 kernels
that generalize better on complex shapes than the dense layers
used by previous works [1], [8], [36].

D. FEATURE FUSION NETWORK ARCHITECTURES
Feature fusion and skip connections are used to enhance
the performance of (convolutional) neural network models
by mitigating the vanishing gradient problem in deep net-
works [13], [14], [15], [37], [38], [39]. In this concept, fea-
tures from previous layer(s) are fused in the next layer(s)
either by performing summation or concatenation. For exam-
ple in the ResNet architecture [14] the previous layer features
were simply added to the next layer’s output. In contrast,
in the DenseNet architecture [13], all features of the preced-
ing layers are concatenated in the next layer’s output. Finally,
attention-based feature fusion [40] fuses point-wise features
and local features to compensate the loss caused by order-
invariant max-pooling on point clouds, and to improve the
3D semantic segmentation accuracy of point clouds.

In the context of learning from point clouds, feature fusion
and skip connections have been used in a few studies [41],
[42], [43] to improve learning of shapes/scenes. However,
the way features are fused in previous studies is significantly
different from the proposed architecture; previous studies
perform feature fusion similar to UNets [44], whereas in
the proposed architecture the features are fused in a fashion
similar to DenseNet [13] to fuse the features of our proposed
variational auto-encoder architecture.

III. PROPOSED ARCHITECTURE
In the following, we present our densely connected deep
feature fusion network architecture, DC-DFFN, for implicit
3D representation. DC-DFFN is directly trainable on raw
input data, for example raw (un-oriented) point clouds or
triangle soups. Our proposed variational auto-encoder con-
sist of an encoder and a decoder, which are constructed
from novel mPEFE (encoder) and mNSDA (decoder) con-
volutional modules. The proposed feature fusion concept is
applied within the decoder, and within the encoder, but not
between them. We describe the proposed network in the
following sections.

A. ENCODER
The DC-DFFN encoder essentially consists of three
multi-layered permutation equivariant feature extraction
(mPEFE)modules. ThemPEFEmodule includes of two layer
types: (a) a modified PointNet [45] layer (Conv1D-MaxPool)

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FIGURE 2. The proposed DC-DFFN encoder and decoder architectures consist of 3 mPEFE modules, and 5 mNSDA modules, respectively. In the
network figures, DsL{1-3}, B, F{1-3}, M{1-3}, FFL{1-5} (decoder) and N stand for DeepSet layer constructed with densely connected features, batch
size, the features after 1D convolutional operation, extracted features after max pooling operation, densely connected deep feature fusion layer
(decoder), and the number of input points to the network, respectively. In addition, Concat([F1-5], axis=1) states that the features (F1, F2, F3, F4,
F5) are concatenated along channel dimension. The encoder receives raw input data points (N, 3)/(N, 6) (the latter, if surface normals are available)
and unsigned distances as ground truth and outputs a latent vector, µ, and diagonal covariance matrix, 6 = diag exp η, which are later used to form
a probability measure N (µ, 6) to construct the latent code that represents the input object shape. The decoder uses the encoded latent vector as
the input to the network and predicts the signed distances, which are later used to mesh the shape using, e.g., the Marching Cubes algorithm.

and (b) a DeepSet [46] layer. The PointNet layer extracts
permutation invariant global features using the max-pooling
layer as a symmetric function for high dimensional feature
embedding learned from unstructured raw point cloud data.
Our PointNet layer implementation uses 1D convolutional
layers, where the original PointNet [45] layers rely on 2D
convolution. The DeepSet layer performs the amalgamation
of global features with high dimensional embedding of local
features extracted by the convolutional layers. A similar
implementation pattern can also be achieved by fully con-
nected layers (used by SAL [8] and SALD [1]) instead of
the 1D convolutional layers (see Appendix A for a more
information). Through the dense interconnectivity between
mPEFE modules the proposed architecture aggregates multi-
layered local features with order invariant global features

within the in-built DeepSet layer. In contrast, the architec-
tures [1] and [8] concatenate only single-layer local features
with an order invariant global features.

Within the mPEFE layers, the 1D convolutional layers are
followed by a pooling layer, a feature fusion layer (modified
DeepSet layer) and a ReLU activation function [47]. The
last convolutional layer, outside mPEFEmodules, is followed
by a pooling layer and a ReLU [47] activation function.
Two fully connected layers are used at the end of the net-
work to formulate the probability measures N (µ, η), where
µ is the latent vector, and η is used to compute the diag-
onal covariance matrix, 6 = diag exp η. Therefore, the
encoder (µ, η) = g(X ,w1) takes X ∈ R3 as input data and
outputs the two 256 dimensional vectors, µ ∈ R256, and
η ∈ R256.

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B. DECODER
Our decoder consists of five symmetric multi-layered neu-
ral signed distance approximation (mNSDA) modules. The
mNSDA module has two main components: (a) signed dis-
tance extraction layers (Conv1D-Softplus with β = 100),
and (b) signed distance blending, i.e. feature blending layer.
The l th module of the decoder receives input data from all
the preceding (l − 1)th modules, fused by a concatenation
operation in the channel dimension. The mNSDAmodule has
some resemblance to the DeepSDF [10] decoder, however
DeepSDF uses a fully connected layer instead of our con-
volutional layer, and a ReLU activation function instead of
our SoftPlus. The mNSDA module promotes maximal infor-
mation flow through the network layers and also prevents the
vanishing gradient problem, mentioned in the DeepSDF [10]
work, from reducing performance. The high level archi-
tectural structure of our decoder is also similar to that of
DeepSDF, except for the dense connectivity introduced in our
architecture.

Withing the decoder there are a total of seven 1D convolu-
tional layers (five of which are inside the mNSDA modules).
The input size of the first layer is (256 + 3/6, 512, N), the
following ([dout(0) , . . . , dout(l−2) ], 512, N), and in the last layer
([dout(0) , dout(1) , . . . , dout(l−1) ], 1, N). Here, doutl is the l

th layer
output of the decoder, and [., .] stands for the concatenation
operation, which concatenates the previous (l − 1)th layer
features of the decoder for the next layer input. N stands for
the number of input points, which is in this case 1282. The
decoder’s input is [x, z] where x ∈ R3, and z ∈ R3 is the
latent vector.

C. DATA PREPARATION
The unsigned distances of given raw input data X are
pre-computed for 500k sample points using the CGAL
library [48] to speed up the training. Moreover, the SALD
loss is computed over the data points and their corresponding
unsigned distance derivatives sampled from distributions D
and D′. Following [1], we set D = D1 ∪ D2, where D1 is
set to be uniformly distributed sampling points {y} from X ,
putting two isotropic Gaussians,N (y, σ 2

1 I ) andN (y, σ 2
2 I ) for

each y. Here, σ1 depends on each sampled point y and is the
distance of the 50th closest point to y, however, σ2 is set to
be a fixed value of 0.3. On the other hand,D2 is estimated by
projecting D1 to surface S.

D. TRAINING AND INFERENCE
We used the SALD loss proposed in [1] with the Adam
optimizer [49] to train our proposed DC-DFFN architecture.
The SALD loss requires gradient incorporation in a differ-
ential manner, which is done based on automatic differen-
tiation [50] forward mode by constructing similar network
layers as in [36]. For the D-Faust dataset, a fixed learning
rate of 0.0005 and 500 training epochs were used for all mod-
els. On the other hand, for the ShapeNet dataset, the initial
learning rate was set to 0.0005 and all models were trained

FIGURE 3. Training loss curves of IGR, SALD, and the proposed DC-DFFN
model. It can be clearly seen that the proposed architecture learns faster
and reaches lower loss than the baseline architectures. The SAL loss
curve is not provided because the numerical scale of SAL loss
significantly differs from the other three methods.

for 1500 epochs. Moreover, a scheduler was set to decrease
the learning rate by a factor of 0.5 after every 1000 epochs
for the Shapenet dataset. For both datasets, the training was
performed on a dual 24GB GeForce RTX 3090 GPU in the
Ubuntu (20.04) Linux environment.

In the inference phase, the implicit representation of
test samples was meshed using the Marching Cubes algo-
rithm [25]. For quantitative comparisons Chamfer distances
and intersection-over-union (IOU) between the reconstructed
surface against the ground truth (for both datasets) and input
raw scans (for D-Faust dataset) were computed.

IV. EXPERIMENTAL RESULT
The proposed DC-DFFN architecture is evaluated on two
challenging benchmark datasets, and compared to recently
proposed three state-of-the-art approaches [1], [8], [36].

A. DATASETS
1) D-FAUST [51]
The D-Faust dataset contains 41k raw 3D scans (triangle
soups) of 10 Human subjects including 5 female and 5 male
subjects, in multiple poses. The scans have various defects,
such as noise, holes, missing body parts, and occasional arti-
facts caused by reflections. In training and testing, we include
only 1 out of 5 samples from the total 41k scans due to the
dense temporal sampling of the dataset. We establish three
types of experiments on the D-Faust dataset [51] following
the experimental setup used in [8]: (a) shape space learning
where 10 human subjects in various poses (129 different
actions) are used for training and testing, (b) generalization
on unseen human shapes where 8 human (4 females and
4 males) subjects are used to train the model, and 2 human
(1 male and 1 female) subjects are used to test the perfor-
mance of the trained model, (c) generalization to unseen
poses, where randomly selected two human poses (from
10 human subjects) were used to test the model, and the
rest of the data (10 humans) were used to train the model.

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FIGURE 4. Visual and quantitative results of a single test sample for shape space learning. From the quantitative and qualitative results, it is clearly
visible that the proposed DC-DFFN generates high fidelity reconstruction and achieves better IOU and CD scores than the baseline architectures.

TABLE 1. Quantitative results on shape space learning for the D-Faust dataset. The Chamfer distances are presented in percentiles (5th, 50th, and 95th)
and mean scores, Chamfer distances multiplied by 103. ↓: lower value is better; ↑ higher value is better.

We consider the same train and test split as [8]. However,
we have removed those poses from the test shapes that contain
scanning artifacts such as floor or side walls. The cleaned
test split shapes have been used during the inference for all
methods. Therefore, for each experimental setting the number
of test shapes before and after the removal are: (a) (2044 →

2003), (b) (1920 → 1869), and (c) (652 → 651). However,
the results with the original test splits (with artifacts) can be
found in Appendix B.

2) ShapeNet [52]
The ShapeNet dataset contains non-manifold meshes with
inconsistent orientation. We consider four different object
classes in our experiments: (1) Car (3533), (2) Sofa (3173),
(3) Guitar (797), and (4) Airplane (4045). The performance

of the proposed DC-DFFN architecture. For ShapeNet, train
and test split files (75/25) were created locally.

B. METRICS
For performance evaluation, we consider Chamfer distance
(CD) and volumetric intersection over union (IOU).

Volumetric IOU is the quotient of the volume of the
generated and the ground truth meshes’ union and their inter-
section. As the baseline implicit reconstruction methods and
our proposed DC-DFFN architecture produce only the mesh
file, we create voxelized volumes of the test-time ground
truth meshes and of the generated meshes. In order to obtain
unbiased estimates of the union and intersection volumes,
we randomly sample 100k data points from the ground truth
and generated meshes, and determine whether the points are
occupied or not occupied.

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FIGURE 5. Visual comparison of a single test sample for unseen human shape learning. Quantitative scores are for reconstructed mesh against the
ground truth. ↓: lower value is better; ↑: higher value is better.

TABLE 2. Generalization performance on unseen human shape reconstruction for the D-Faust dataset. The chamfer distances are presented in
percentiles (5th, 50th, and 95th) and mean scores, Chamfer distances multiplied by 103. ↓: lower value is better; ↑ higher value is better.

Chamfer distance is computed as the mean distance of
points from the generated mesh to the ground truth mesh and
in the opposite direction as well. Additionally, we compute
the Chamfer distances between the generated mesh and the
input scan. Similar to the evaluation approach taken in [5],
we define completeness as the computed mean Chamfer dis-
tance from the direction of registration, Rg, (ground truth) /
raw input scan (Sc) to the generated mesh (Gn) (Rg→Gn,
and Sc→Gn), whereas the opposite direction (Gn→Rg, and
Gn→Sc) is defined as accuracy.

C. BASELINES
We compare the proposed DC-DFFN architecture to several
related generative approaches that are capable of learning the
shape space directly from raw 3D data.

SAL [8]: SAL is a generative implicit 3D reconstruction
approach that learns the shape space from raw unsigned
geometric data in a sign agnostic manner. We compare
the proposed work against SAL using the D-Faust dataset,
as SAL has inherent difficulties [8] in reconstructing thin
shapes that are common in the ShapeNet dataset.

SALD [1]: SALD is a state-of-the-art approach for recon-
structing 3D surfaces, which uses a sign agnostic regression
loss function with derivatives, and learns the shape space
directly from raw unsigned geometric data. The proposed
approach is compared against SALD in all of our experi-
ments.

IGR [36]: As shown in [36], a simple loss function can
possess the implicit geometric regularization (IGR) prop-
erty, which allows to generate smooth and high fidelity

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FIGURE 6. Visual comparison of baseline methods and the proposed DC-DFFN on unseen pose learning with IOU and Chamfer distance scores. ↓:
lower value is better; ↑: higher value is better.

TABLE 3. Comparison on unseen pose learning for D-Faust test samples. The Chamfer distances are presented in percentiles (5th, 50th, and 95th) and
mean scores, Chamfer distances multiplied by 103. ↓: lower value is better; ↑ higher value is better.

surfaces directly from raw (un-oriented point cloud or tri-
angle soup) input data, by directing the neural network to
vanish on the input data, and ensuring unit norm gradi-
ent. Previously, IGR has achieved state-of-the-art quantita-
tive results and high fidelity reconstruction [36]. The pro-
posed approach is compared against IGR using the D-Faust
dataset.

D. SHAPE SPACE LEARNING
In this experiment, randomly selected 70% of 10 D-Faust
human subjects are used to train the proposed and the baseline
architectures with 500 epochs. The remaining data, 30% of
the samples, are used to test the trained models. Additionally,
randomly drawn 1282 points from pre-computed 500k sample
points are used to train and test the proposed architecture. For

baseline architectures, the number of randomly drawn points
is as follows: SAL — 1282, SALD — 922, and IGR — 1282

as given in the respective original implementations. The final
shape reconstruction has been generated with a resolution of
1003 for all architectures.
The quantitative and qualitative results of this experiment

are shown in Table 1, and Fig. 4, respectively. From the quan-
titative results, it can be clearly seen that the proposed archi-
tecture achieves superior results compared to the baseline
methods, with one exception: SAL outperforms the proposed
approach in 95% percentile using the completeness (Sc→Gn)
measure. Moreover, the proposed architecture generates sur-
face reconstruction with superior accuracy in small details
compared to the baselines, which can be seen in the IOU
results and in Fig. 1.

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FIGURE 7. Visual quality comparison of a single test sample from each ShapeNet [52] test class. ↓: lower value is better, ↑: higher value is better.

E. GENERALIZATION TO UNSEEN HUMANS
In this experiment on the D-Faust dataset, we test our the pro-
posed architecture’s generalization capability on previously
unknown subjects. The training samples have been drawn
from randomly selected 8 human subjects (4 females and
4 males subjects), whereas the remaining 2 human subjects
(1 male and 1 female subject) were used to test the architec-
tures. The same number of randomly drawn data points (SAL:
1282, SALD: 922, IGR:1282, andDC-DFFN:1282) were used
to train and test the models.

The evaluation results in Table 2 show that the proposed
architecture has outperformed the baseline architectures in
most cases in each metric. As exceptions, Table 2 shows that
IGR outperforms the proposed architecture in two Cham-
fer distance cases (5%, and 50% percentiles, Rg→Gn and
Sc→Gn, respectively). However, in the case of mean Cham-
fer distance and IOU, DC-DFFN outperforms IGR by a
large margin. Additionally, the proposed architecture can
preserves structural detail better than the baseline architec-
tures. Compared to the shape space learning experiment
(Section IV-D), all architectures however provide overall
worse results, as expected.

F. GENERALIZATION TO UNSEEN POSES
In this experiment on the D-Faust dataset, two poses have
randomly been selected from 10 humans subjects for testing,
and the rest of the data is used to train the models. Similar to
previous settings, the number of data points used are: SAL—
1282, SALD — 922, IGR — 1282, and DC-DFFN — 1282,
drawn from the pre-computed sample data points during the
training and testing the models.

We results are shown in Table 3, and Fig. 6: on aver-
age, DC-DFFN achieves better or comparable results in all
metrics. As an exception, SAL outperforms the proposed
architecture in terms of Chamfer distance (95% percentile)
and completeness (Rg→Gn and Sc→Gn) scores. However,
in terms of volumetric IOU DC-DFFN outperforms all other
approaches without exceptions.

G. GENERALIZATION TO OBJECT SHAPES
Beyond learning human shapes, also experiments on learn-
ing object shapes were performed to evaluate the proposed
architecture using the ShapeNet dataset comprises of non-
manifold/non-oriented meshes that depict various objects.

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TABLE 4. ShapeNet quantitative results. The Chamfer distances are presented in percentiles (5th, 50th, and 95th) and mean scores, Chamfer distances
multiplied by 103. Moreover, ↓ means lower is better, whereas ↑ means higher is better.

TABLE 5. Comparison against inferior DC-DFFN variants on unseen human reconstruction. The Chamfer distances are presented in percentiles (5th, 50th,
and 95th) and mean scores, where the Chamfer distance numbers are multiplied by 103. All variants were with 500 epochs. ↓: lower value is better; ↑:
higher value is better.

Compared to human shapes, objects exhibit sharp corners,
holes and thin structure. For baseline architectures, we have
resorted to the train and test settings as described in the
original works, except for data split files where we used 75%
of the samples for training and 25% samples for testing. The
network architectures were trained with 1500 epochs on each
class (Car, Guitar, Airplanes, and Sofa). The IGRmethod was
not included in this evaluation, as IGR expects consistently
oriented normals, which were unavailable for the ShapeNet
dataset.

The results are shown in Table 4 and Fig. 7, respec-
tively. DC-DFFN significantly outperforms the state-of-
the-art SALD architecture, except for the Guitar class,
where measured Chamfer distances favor SALD in 5%
and 95% percentile cases. The qualitative results of Fig. 7
show that DC-DFFN can capture thin structure (airplane
wings) and sharp corners (sofa armrests) much better than
SALD.

V. DISCUSSION
In this study, we proposed the feature fusion-based variational
auto-encoder network DC-DFFN. The novel characteristics
in the architecture design improve training speed (see Fig. 3),
improve performance at inference time (see Tables 1, 2, 3,
and 4), and provide better generalisation in the 3D shape
space compared to reviewed baseline approaches.

In the test data split of D-Faust, we decided to remove the
samples (See Section IV-A) that include scanning artifacts
(see also Appendix B and Fig. 9) for improving the inter-
pretability of the results. However, this decision can also be
questioned, as keeping the scanning artifacts could on the
other hand evaluate the architectures’ capability of rejecting
outliers.

However, Table 6, and Table 7 in Appendix B show
the D-Faust results with artifacts included, and confirms
that the proposed DC-DFFN architecture outperforms in gen-
eral the baseline architectures in the presence of artifacts as

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A. Basher, J. Boutellier: DC-DFFN With Sign Agnostic Learning for Implicit Shape Representation

FIGURE 8. The qualitative results are shown for other alternative variant of the feature fusion networks and the proposed
DC-DFFN on the experimental setup illustrated in section IV-E of main paper for D-Faust dataset. Additionally, we reported the
computed Chamfer distance for each model, where ↓ means lower is better.

TABLE 6. Results on D-Faust shape space learning including samples with artifacts. The Chamfer distances are presented in percentiles (5th, 50th, and
95th) and mean scores, Chamfer distances multiplied by 103. ↓: lower value is better; ↑ higher value is better.

TABLE 7. Generalization to D-Faust unseen humans including samples with artifacts. The Chamfer distances are presented in percentiles (5th, 50th, and
95th) and mean scores, Chamfer distances multiplied by 103. ↓: lower value is better; ↑ higher value is better.

well. In individual cases (See Fig. 9) the baseline SALD
approach is sometimes better in removing outliers.

In all experimental setups for both D-Faust and ShapeNet
datasets, the proposed DC-DFFN significantly outperformed

the baseline approaches in almost all the cases. In a
few cases the baseline approaches provided better Cham-
fer distance results with a small margin. Considering
volumetric IOU, in contrast, DC-DFFN outperformed the

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FIGURE 9. Visual reconstruction result comparison for a sample with scanning artifacts. ↓: lower value is better.

FIGURE 10. Training Loss curve of SALD and DC-DFFN models on
ShapeNet dataset (class: lamp). In this dataset, we compared our
approach only with SALD. From the learning curve, it can be seen that
DC-DFFN is learning faster than the baseline SALD model, however, both
architectures are not yet reached to the saddle point at 1500 epochs.
Based on the complexities of the data, the required number of training
epochs will vary for all models. However, DC-DFFN converges faster than
all other baseline approaches presented in this study.

baseline approaches in all experimental setups in both
datasets.

Although DC-DFFN performs better than the baseline
approaches, however, the proposed architecture still suffers
in reconstructing the thin structures to some extent.

VI. CONCLUSION
In this paper we have proposed the densely connected deep
feature fusion network architecture DC-DFFN for neural
implicit shape learning and reconstruction from raw input
data. In the experimental section the proposed work is shown
to learn faster, generalize better and outperform the baseline
works quantitatively by a clear margin in all experiments.
Additionally, the visual results show that the proposed archi-
tecture can especially capture small detail better than the
previousworks. As the broader impact of ourworkwe see that
in the future DC-DFFN has potential to serve as the prevailing
neural architecture for upcoming studies on shape learning
from raw 3D data.

APPENDIX A
ARCHITECTURE ALTERNATIVES
In the process of designing DC-DFFN, two alternative
implementation variants of the proposed architecture were
developed, out of which the proposed architecture was iden-
tified as the best one. The other alternative variants were:
(I) Densely connected feature fusion network with multilay-
ered latent codes, DC-DFFN-MLLCs, and (II) Dense layer
feature fusion with dense neural network, DFF-DFFN-Lin.
Results of these architecture variants are shown in one exper-
imental setting in Table 5.

A. DC-DFFN-MLLCs
In the DC-DFFN-MLLCs architecture variant, one latent
code is extracted after every Conv1D-MaxPool-DeepSet-
Relu block, and finally, concatenated in the channel
dimension. In this case, the final latent code shape is
(B, 1024, N), where N=1282 and B is the batch size.
The assumption was that multiple multi-layered latent codes
would contain more information than a single latent code
and provide better reconstruction quality. However, in prac-
tice this architecture variant was performing worse than
DC-DFFN.

1) DC-DFFN-LIN
In this variant, the 1D convolutional layers (Kernel: 1× 1) of
the encoder and decoder were replaced by fully connected
layers, keeping the rest of the network settings similar to
DC-DFFN-MLLCs. Eventually, DC-DFFN-Lin performed
comparatively worse than the multiple latent codes-based
architecture, DC-DFFN-MLLCs, constructed with of 1D
convolutional layers. Even clearer, DC-DFFN-MLLCs per-
formed significantly worse than the proposed DC-DFFN.

APPENDIX B
RECONSTRUCTION OF SAMPLES WITH ARTIFACTS
In the main paper experiments, the D-Faust samples that con-
tain scanning artifacts in the test samples were removed from
the experiments shown in Section IV-D and Section IV-E.
Here, in Table 6, Table 6 and Fig. 9 we present as additional
quantitative results the D-Faust results without any removed

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FIGURE 11. Additional qualitative results are shown for ShapeNet dataste (Class: lamp). We reported the computed Chamfer distance for both models,
where ↓ means lower is better.

samples, making the test sets identical to the ones used
in [1] and [8].

APPENDIX C
ADDITIONAL RESULTS
Additional qualitative results are shown in Fig. 11 from
ShapeNet dataset (class: Lamp). Moreover, the training loss
curve on lamp class is shown in Fig. 10 for SALD, and the
proposed DC-DFFN models. It can be seen from the learning
curve that the proposed architecture learns faster than the
baseline SALD architecture.

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ABOL BASHER received the B.Sc. degree in
electrical and electronics engineering from the
Mymensingh Engineering College, University of
Dhaka, Bangladesh, in 2015, and the M.E. degree
in computer engineering from Chosun Univer-
sity, Gwangju, South Korea, in 2020. He is cur-
rently pursuing the Ph.D. degree in computer
science with the University of Vaasa, Finland.
He is a Project Researcher with the Digital Econ-
omy Research Platform, University of Vaasa. His

research interests include 3D data representation, computer vision, machine
learning, and medical image processing.

JANI BOUTELLIER received the Ph.D. degree,
in 2009. He is currently an Associate Profes-
sor with the University of Vaasa, Finland. He is
leading projects that concentrate on efficient neu-
ral networks and 3D computer vision. He has
coauthored more than 70 peer-reviewed articles.
His research interests include parallel computing,
model-based design, and signal processing. He is
a member of the IEEE Signal Processing Society
ASPS Technical Committee. He is an Associate

Editor of the Journal of Signal Processing Systems (Springer).

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