# SALVE: A 3D Reconstruction Benchmark of Wounds
from Consumer-grade Videos

Remi Chierchia1,2, Leo Lebrat1,2, David Ahmedt-Aristizabal1,2,
Olivier Salvado1,2, Clinton Fookes1, Rodrigo Santa Cruz1,2
Queensland University of Technology1, CSIRO Data612
<EMAIL_ADDRESS>{leo.lebrat<EMAIL_ADDRESS>

###### Abstract

Managing chronic wounds is a global challenge that can be alleviated by the
adoption of automatic systems for clinical wound assessment from consumer-
grade videos. While 2D image analysis approaches are insufficient for handling
the 3D features of wounds, existing approaches utilizing 3D reconstruction
methods have not been thoroughly evaluated. To address this gap, this paper
presents a comprehensive study on 3D wound reconstruction from consumer-grade
videos. Specifically, we introduce the SALVE dataset, comprising video
recordings of realistic wound phantoms captured with different cameras. Using
this dataset, we assess the accuracy and precision of state-of-the-art methods
for 3D reconstruction, ranging from traditional photogrammetry pipelines to
advanced neural rendering approaches. In our experiments, we observe that
photogrammetry approaches do not provide smooth surfaces suitable for precise
clinical measurements of wounds. Neural rendering approaches show promise in
addressing this issue, advancing the use of this technology in wound care
practices.

Figure 1: Qualitative evaluation of six 3D surface reconstruction methods
using our SALVE dataset. On the left, we present COLMAP, a classic
photogrammetric pipeline, which represents typical methods utilised for 3D
wound reconstruction. Next, we display two Gaussian splatting approaches SuGaR
[13] and 2DGS [14], followed by two NeRF approaches Neuralangelo [24] and
Neus-facto [51]. To highlight surface regularization—important for wound
analysis —we present depth color-coded from blue (deeper regions) to red
(closer regions).

##  1 Introduction

Chronic wounds represent a significant health and economic burden worldwide
[33, 49]. Effective wound treatments depend on multiple wound clinical
measurements, typically performed manually by specialized healthcare
professionals [19, 31]. For instance, wound surface area is typically measured
by performing planimetry of the wound bed [34]. These procedures are not only
invasive and cause patient discomfort but are also prone to errors due to
ambiguous definitions of metrics and variations in professionals’ skill
levels. Moreover, this method is neither cost-effective nor scalable, as it
requires specialized personnel who are often not available in remote areas.

Automatic measurement systems using computer vision and off-the-shelf cameras
offer a promising avenue for wound care [5, 12]. However, most existing
commercial approaches [2, 1] compute wound measurements solely from 2D images,
which are inherently inaccurate for wounds with complex geometry or located on
highly curved body parts such as the heel, toe, and lower leg. Also, 2D wound
measurements are perspective-dependent. As illustrated in Figure 2, wound
measurements such as surface area can vary significantly when computed from
images taken from different view angles. Furthermore, these approaches can not
estimate wound depth, potentially overlooking a crucial aspect of the wound
healing process.

Figure 2: Limitation of 2D Wound Analysis: Images of a wound taken from
different view perspectives will present significantly different wound areas.

In response to these limitations, 3D reconstruction techniques can be
leveraged to generate 3D models from short videos depicting the wound from
multiple viewpoints. Subsequently, measurements can be directly computed in
3D. This approach ensures that measurements conducted on the reconstructed
geometry are independent of the view direction. Additionally, 3D analysis of
wounds allows for the computation of richer wound biomarkers, which could
streamline wound documentation [16]. Leveraging these prospects, several works
have advocated for 3D wound documentation [16, 9, 4, 15, 22, 6, 21, 26, 40].
However, these studies only considered previous generation 3D reconstruction
frameworks, which have recently been surpassed by highly optimized
photogrammetric toolboxes and recent neural rendering alternatives.

More importantly, no studies to date have thoroughly evaluated and compared
the effectiveness of 3D reconstruction methods specifically applied to wound
reconstruction from videos. To address this gap, we introduce a new dataset
SALVE, designed to capture common challenges encountered in clinical settings,
such as complex lighting conditions and varied image quality and resolution of
realistic silicone wounds. Using this dataset, we evaluate robust
photogrammetry pipelines and modern neural rendering approaches for 3D
reconstruction. Our work includes a rigorous evaluation protocol that defines
metrics and procedures to assess the geometric accuracy and precision of the
evaluated reconstruction algorithms.

We believe that this study is crucial to the development of reproducible
research in the area of wound analysis and is a valuable contribution to the
field of computer vision through the introduction of a new challenging task.
Our benchmark highlights novel avenues for the use of neural 3D reconstruction
in a medical setting and exhibits some limitations of current state-of-the-art
(SOTA) methods when applied in this difficult scenario. We highlight the
robustness of different 3D reconstruction algorithms with respect to the
quality of the acquisition device and demonstrate that some approaches show
significant degradation in performance on lower-end devices. We aim for this
research to pave the way towards the development of affordable wound
telehealth solutions and further reduce geographic inequalities in wound care
access.

##  2 Related Work

3D wound analysis has been a topic of interest for at least fifteen years
[44], and its advantages over 2D methods and manual procedures have been
highlighted in multiple clinical studies in wound care [42, 21]. However, to
the best of our knowledge, no study proposing a rigorous and unified
evaluation of the accuracy and repeatability of state-of-the-art methods for
3D reconstruction applied to wound reconstruction from consumer-grade cameras
exists. Most existing works in this field focus on the repeatability and
accuracy of specific wound measurements computed from images, pairs of images,
or videos [52, 28, 36, 26, 6]. These studies typically use a single device and
follow a very specific acquisition protocol, without evaluating the 3D
reconstruction itself. For instance, [52] conducted an inter-laboratory
comparative study between photogrammetric and 3D scanner-based volume
estimation of skin ulcers. Similarly, [26, 28] performed experiments comparing
wound area measurements using 2D and 3D methods. Lastly, [36] extended this
analysis to nine different wound measurements performed by both dermatologists
and non-experts, using automatic and semi-automatic tools for wound
measurements. The findings of these studies are limited to those specific
measurements and cannot be generalized (e.g. different wound shapes, different
devices, other wound etiology, and acquisition protocols,…).

Similar to our work, [44, 8, 40] evaluated the accuracy and repeatability of
3D reconstruction methods for wound analysis by comparing 3D models produced
by these techniques with 3D models obtained with specialized 3D scanners.
However, [44, 40] used a single artificial phantom created by a special
effects artist as their input dataset. [8] used short video recordings of
cutaneous leishmaniasis skin lesions, which are flat-looking and lack
significant 3D features such as depth and complex shapes. These works also
focused solely on a single acquisition device and used very simple
photogrammetric pipelines. In contrast, our evaluation includes three types of
wounds with rich 3D information for proper wound documentation, two
acquisition devices, and state-of-the-art photogrammetric pipelines, as well
as modern neural rendering approaches for 3D reconstruction. Additionally,
unlike existing works, we will make our data and code publicly available for
research purposes.

Figure 3: Materials used for acquiring the proposed dataset, including a
picture of a recording session.

Recently, studies have proposed evaluating 3D wound analysis methods in fully
controlled synthetic 3D environments created with 3D computer graphics
software or game engines. Lebrat et al. released Syn3DWound [22] and Sinha et
al. released DermSynth3D [41]. Although one could utilize it to simulate some
noise characteristics present in realistic settings, modeling the entire
complexity of clinical environments paired with consumer device limitations is
impractical. This could lead to a reality gap in performance in the target
application.

##  3 Materials and Method

###  3.1 SALVE

To unify the evaluation protocol for 3D wound reconstruction and address the
gap between synthetic and real-world acquisition, we propose SALVE, a 3D wound
reconstruction benchmark from consumer-grade camera footage. Given the ethical
considerations associated with human data, and the time required to evaluate
different data acquisition devices and settings, we opted for the use of high-
quality silicone phantoms provided by TraumaSIM 111https://traumasim.com.au/
(a private company specialized in medically accurate simulations of realistic
wounds to assist the training of healthcare professionals). In Figure 3, we
display the different wound types utilized in our study, denoted as SD
(Surgical Wound Dehiscence), PIS3 (Pressure Injury Stage 3), and PIS4
(Pressure Injury Stage 4).

To encompass the variability of the acquisition device, we captured videos
using both a smartphone (iPhone 14 Pro Max, 4K frames 2160 $\times$ 3840) and
a webcam (Logitech 4K Webcam, 1K frames 1080 $\times$ 1920) under realistic
lighting conditions, capturing reflections, shadows, and blurriness. Accurate
3D ground-truth point clouds were obtained using the Revopoint POP 3D scanner
222https://global.revopoint3d.com/pages/face-3d-scanner-pop2. Figure 3 shows
an example of our acquisition materials and setup.

To mitigate noise and blurriness from video recordings, we devised a method
that extracts a collection of sharp frames from each recorded video. We
initially sample frames at 60 frames per second from each video. Then, we
divide these frames into equal-sized consecutive frame sequences. From each
sequence, we select the sharpest frame based on the variance of a 3x3
Laplacian filter applied to it. Using the recorded videos and this frame
selection procedure, we generate various image datasets to support the
evaluations outlined in Section 3.3.

###  3.2 3D Reconstruction Methods

The goal of 3D reconstruction is to estimate the 3D representation of a scene
depicted in a set of images. Traditionally, this task is tackled by
photogrammetry pipelines, which can be summarized in two main steps. The first
step consists of solving a Structure from Motion (SfM) [43, 3] problem, which
produces a sparse reconstruction. By matching features across images, an SfM
algorithm regresses both the 3D structures and the camera matrices. The second
step involves a Multi-View Stereo (MVS) [7, 29] approach, which establishes
denser correspondences between image pairs using stereo matching and patch
warping [10]. Finally, a triangle mesh is generated by applying a meshing
algorithm [18, 23] to the obtained dense reconstruction.

As mentioned earlier, existing works on 3D reconstruction of wounds from
videos follow this approach. For instance, [40] and [6] leverage well-
established frameworks, such as COLMAP [38, 39] and OpenSFM [32],
respectively. However, they rely on accurate feature correspondences between
images, which can be challenging in videos recorded in clinical settings due
to complex lighting, moving shadows, and self-occlusions. These factors often
lead to errors and discontinuities in the reconstructed 3D model.

A novel trend in 3D reconstruction involves using learnable implicit
primitives paired [27] with volumetric rendering [17]. As detailed in [27],
these approaches express the color of a pixel $\hat{C}$ as,

$\hat{C}(\mathbf{r})\\!=\\!\\!\int_{t_{n}}^{t_{f}}\\!T(t)\sigma(\mathbf{r}(t))c(\mathbf{r}(t),\mathbf{d})\text{d}t,$
(1)

where $\mathbf{r}(t)$ is a time-parametrized ray for the pixel $\hat{C}$, $c$
is a learnable function depending on the 3D space location and the view
direction $\mathbf{d}$, responsible for the color information of the neural
radiance field, and $\sigma$ depends only on the spatial coordinate. In this
formulation, the density $\sigma(\mathbf{x})$ can be interpreted as the
probability of a ray being stopped at $\mathbf{x}$, and a 3D model can be
obtained via iso-surface extraction algorithms. However, $\sigma$ is often
implemented as an over-parameterized neural network without surface regularity
constraints, which frequently results in 3D reconstructions with floaters or
non-smooth surfaces [48]. These surface artifacts could compromise the
reliability of clinical wound measurements, making them unsuitable for
longitudinal tracking.

To improve the quality of the reconstructed geometry, multiple works [45, 47]
propose deriving the density function $\sigma$ from a learnable Signed
Distance Field (SDF), enabling surface regularity modelling within the
standard volumetric rendering framework. For instance, in [45], $\sigma$ is
defined to be maximal at the zero-crossings of the SDF $f_{\theta}$,

$\displaystyle\sigma(\mathbf{x})=$
$\displaystyle\max\left(\frac{-H^{\prime}(f_{\theta}(\mathbf{x}))}{H(f_{\theta}(\mathbf{x}))},0\right),$
(2)

where $H$ is the cumulative distribution function of the logistic
distribution, and $\theta$ represents the model parameters to be learned
through volumetric rendering. In this formulation, $f_{\theta}$ is subject to
the eikonal constraint, and further regularizations, such as curvature [24],
can be defined on this implicit representation of objects. The training time
for such approaches is in the order of several hours, which limits their
applicability in some clinical settings. To reduce the training time
associated with the discretization of the rendering Equation (1), a novel
family of methods has emerged. They propose an unstructured explicit
representation of radiance fields. Gaussian Splatting directly optimizes the
position and variance of 3D Gaussians [20]. This representation remains
volumetric, but can be rasterized more efficiently by applying 2D projection
and alpha-compositing.

However, Gaussian splatting does not allow for precise surface extraction.
SuGaR [13] and 2DGS [14] proposed two different formulations amenable to
surface extraction from a 3D sparse representation. Although similar in
concept, they differ in their problem approach. SuGaR enforces additional
constraints on the learned Gaussians to more accurately represent the geometry
of the scene. More specifically, the Gaussians should have limited overlap
with their neighbors, be as opaque as possible, and be as thin as possible in
the normal direction of the surface. Under those assumptions, the neural
density of the field can be approximated using,

$\hat{\sigma}(\mathbf{x})=\exp\left(-\frac{1}{2s^{2}_{g^{\star}}}\langle\mathbf{x}-\mathbf{\mu}_{g^{\star}},\mathbf{d}_{g^{\star}}\rangle^{2}\right),$
(3)

where ${g^{\star}}$ is the index of the closest Gaussian to $\mathbf{x}$,
$s_{g^{\star}}$ is its smallest scaling factor, and $\mathbf{d}_{g^{\star}}$
is the associated direction. By penalizing the 3D Gaussians to achieve such
density, one can significantly enhance the geometric information. Finally, by
deriving a set of points and their normals from the Gaussian’s density, one
can employ Poisson reconstruction to swiftly extract a surface mesh with
improved quality.

On the other hand, 2DGS proposes to remove the extra dimension
$\mathbf{d}_{g^{\star}}$ and replace the 3D Gaussian with a 2D Gaussian, a
multiview-consistent primitive more suited for surface extraction. Both of
these explicit formulations reduce runtimes compared to their continuous SDF
counterpart. However, they may fall short in providing the regularity required
for our application.

#### Benchmarked methods:

We compare three proven photogrammetry pipelines: COLMAP [37] (CM), COLMAP
equipped with LightGlue feature matching [25] (LGCM), and Meshroom [11] (MR).
Additionally, leveraging the computed image poses from LGCM, we benchmark
recent AI-based methods for 3D reconstruction. These include Neural Signed
Field techniques like NeusFacto [51] and NeuralAngelo [24], as well as
discrete explicit approaches such as 3D Gaussian Splatting (3DGS) [20] for
rendering and SuGaR[13] or 2D Gaussian Splatting [14] for 3D reconstruction.
Finally, we experimented with fast methods using Neural Hash encoding such as
InstantNGP [30] for rendering and NeuS2 [47], its SDF-based variant. We also
explored SfM-free approaches like DUSt3R [46], an end-to-end stereo 3D
reconstruction method.

Table 1: Table presenting the accuracy of different 3D reconstruction methods
across three wounds types (PIS3, PIS4, and SD) and two acquisition devices
(iPhone and Logitech). Results highlight the superiority of Meshroom (MR),
Neuralangelo, and Neus-facto across six metrics evaluated: AD,
$\mathcal{W}_{2}$, HD, and $\text{HD}_{90}$ (reported in millimetres), along
with NC and $\mathcal{W}_{2}$-NC values (ranging from 0 for orthogonal to 1
for the same orientation).

Accuracy iPhone Logitech Wound Method AD $\mathcal{W}_{2}$ HD HD90 NC
$\mathcal{W}_{2}$-NC AD $\mathcal{W}_{2}$ HD HD90 NC $\mathcal{W}_{2}$-NC PIS3
CM 0.305 1.702 2.757 0.624 0.812 0.797 30.696 3.106 14.775 2.131 0.745 0.564
LGCM 30.259 31.632 2.858 30.516 0.830 0.823 0.751 4.470 12.131 2.653 0.739
0.549 MR 10.230 11.575 11.272 10.454 20.992 30.991 10.249 11.572 79.029 10.504
20.990 20.990 SuGaR 0.806 4.011 9.657 1.746 0.906 0.822 1.075 2.336 26.317
2.342 0.898 0.869 2DGS 0.757 3.069 6.730 1.695 0.726 0.682 1.402 3.826 37.380
3.170 0.789 0.759 Neuralangelo 20.257 21.614 21.359 20.506 20.992 20.992 0.982
31.990 14.605 32.123 30.977 30.967 Neus-facto 0.282 1.669 31.681 0.569 10.993
10.993 20.263 21.612 11.632 20.621 10.993 10.993 PIS4 CM 30.215 1.572 2.689
30.499 0.853 0.843 30.827 3.668 12.438 2.262 0.806 0.697 LGCM 0.216 1.766
22.313 0.503 0.868 0.859 1.068 6.638 19.263 3.874 0.766 0.559 MR 20.192 31.556
2.664 20.475 20.991 30.984 20.424 21.779 33.503 21.030 20.982 20.980 SuGaR
0.657 2.003 4.608 1.437 0.950 0.942 3.342 7.144 17.882 10.957 0.709 0.462 2DGS
0.747 2.570 6.364 1.835 0.833 0.808 1.057 2.623 7.203 2.409 0.858 0.846
Neuralangelo 10.178 11.546 32.472 10.427 10.992 10.985 0.953 32.076 23.165
32.006 30.972 30.975 Neus-facto 0.217 21.555 12.301 0.539 20.991 10.985 10.256
11.607 12.567 10.639 10.991 10.985 SD CM 30.355 7.547 9.956 0.730 0.887 0.817
0.963 8.467 26.819 2.962 0.775 0.578 LGCM 0.366 37.297 16.488 30.722 0.892
0.819 30.595 37.639 23.129 31.648 0.828 0.687 MR 0.357 7.765 33.292 0.836
30.979 20.924 0.736 7.701 35.767 1.866 30.956 30.933 SuGaR 0.698 8.455 9.625
1.559 0.932 0.823 2.295 11.394 15.475 6.473 0.767 0.597 2DGS 1.195 8.315 8.274
2.484 0.872 0.830 1.846 9.076 10.209 3.818 0.849 0.804 Neuralangelo 10.201
17.149 23.115 10.439 20.980 30.906 20.547 27.460 25.529 21.366 20.957 20.935
Neus-facto 20.230 27.257 13.048 20.510 10.986 10.925 10.428 17.192 15.045
11.034 10.975 10.939

###  3.3 Experiments Description

For 3D wound reconstruction methods to be valuable in research and clinical
settings, they must be both accurate and precise. Accuracy assesses how
closely a reconstruction matches the ground truth, while precision (or
reliability and repeatability) measures the consistency between repeated
reconstructions. This consistency is essential for verifying the robustness to
varying input data and ensuring suitability for tracking wound measurements
over time. Additionally, these methods can serve as powerful tools for wound
visualization. Therefore, we evaluate the aforementioned approaches based on
these three criteria, which are detailed below.

#### Accuracy Evaluation:

From each recorded video in our dataset, we initially curated a set of fifty
frames using the frame selection procedure detailed in Section 3.1. We chose
fifty frames as we observed that adding more frames did not enhance wound
visibility and sometimes compromised the accuracy of SfM pose estimation due
to the inclusion of blurry frames. Using these frames, we generated 3D models
of the wounds using the methods outlined in Section 3.2 and compared their
results to the ground truth obtained with the specialized 3D scanner detailed
in Section 3.1. The outcomes of this experiment are presented in Table 1.

#### Precision Evaluation:

We evaluate the consistency of 3D reconstruction methods across different
camera devices (inter-device) and recording attempts (inter-recording). From
each video in our dataset we selected 150 sharp frames using the procedure
outlined in Section 3.1. These frames were then temporally divided into fifty
sub-sequences, each containing three frames. Within each sub-sequence, the
frames were randomly assigned to separate splits, resulting in three data
splits of fifty frames for each video.

For inter-device precision, we compare all pairs of 3D reconstructions
generated by each method using data splits generated from different recording
devices for the same wound. To assess inter-recording precision, we focus only
on data splits derived from videos recorded with the iPhone. Then, we compare
all pairs of 3D reconstructions produced by each method from these data splits
of the same wound. The results of this experiment are presented in Table 2.

Table 2: Table presenting the precision of the best methods from the accuracy
experiment. We report precision for inter-recording in the iPhone acquisitions
and inter-device across the three wound types. Results highlight Neus-facto as
the most robust method among those evaluated.

Precision Inter recording (iPhone) Inter device Wound Method AD
$\mathcal{W}_{2}$ HD HD90 NC $\mathcal{W}_{2}$-NC AD $\mathcal{W}_{2}$ HD HD90
NC $\mathcal{W}_{2}$-NC PIS3 MR 10.089 0.671 2.099 10.138 0.994 0.970 30.224
30.617 32.266 30.441 30.991 30.979 Neuralangelo 30.102 10.405 10.452 30.183
10.997 10.997 0.865 1.578 9.191 1.611 0.981 0.966 Neus-facto 0.126 30.457
30.476 0.229 10.997 10.997 10.197 10.493 10.854 10.382 10.998 10.998 PIS4 MR
0.216 0.542 1.990 0.499 0.988 0.989 30.441 30.736 32.646 30.946 30.984 30.984
Neuralangelo 10.093 10.443 31.382 10.160 30.994 30.992 0.645 1.000 3.256 1.482
0.983 0.982 Neus-facto 30.099 30.449 10.593 30.176 10.995 10.994 10.286 10.561
11.556 10.548 10.992 10.992 SD MR 0.347 10.826 5.954 0.710 0.978 30.976 0.663
1.676 5.041 1.449 0.961 0.952 Neuralangelo 10.279 30.888 12.078 10.533 30.984
0.974 10.330 30.916 12.363 10.657 30.979 30.973 Neus-facto 10.279 0.935 32.122
30.608 10.986 10.987 30.346 10.886 32.516 30.709 10.984 10.984

#### Rendering Quality:

To evaluate the suitability of the 3D reconstruction methods as visualization
tools (excluding the photogrammetric pipelines), we assessed their rendering
capability for view perspectives not present in the input training images. To
this end, we extract 75 frames from each video in our dataset using our frame
selection procedure, divide them into 50 training images and 25 test images
with different poses and visual content, as suggested in [50]. We then train
each AI-based 3D reconstruction model on the training set and evaluate their
rendering quality on the test images. Table 3 presents the results of this
experiment.

#### Evaluation Protocol:

The evaluation protocol involves an initial coarse alignment of the 3D
reconstructions with the ground-truth acquired data. Specifically, ArUco
markers are placed in each scene to retrieve a global transformation and scale
the 3D reconstruction for evaluation. We emphasize that these markers are not
utilized for pose estimation or any other process by the evaluated methods. A
common next step is applying the Iterative Closest Point algorithm (ICP) [35]
for fine alignment. However, the convergence of the ICP algorithm can be
affected by errors in the reconstructed geometry. Some variations of the
algorithm have been proposed to address these issues [35], but these
heuristics are often difficult to implement robustly across all evaluations.
Consequently, we opted to conduct manual alignment using 3D point cloud and
mesh processing software. To ensure a fair and transparent evaluation, we
provide the manually aligned meshes alongside the datasets.

#### Metrics:

To ensure that our evaluation concentrates solely on the wound area, we define
a polygon around the wound in both the reconstructed mesh and the ground-truth
point cloud. Within this polygon, we compute the following 3D reconstruction
metrics: Average Distance (AD), Earth mover’s Distance ($\mathcal{W}_{2}$),
Hausdorff Distance (at percentiles $100^{th}$ (HD) and $90^{th}$ (HD90)), and
Normal Consistency (NC). We also leverage the optimal transport couplings
computed for $\mathcal{W}_{2}$ to measure the normal consistency metric under
optimal transport’s assignment ($\mathcal{W}_{2}$-NC). The metrics AD,
$\mathcal{W}_{2}$, and HD variants are measured in millimeters, while NC and
$\mathcal{W}_{2}$-NC assess the alignment between reconstructed surface
normals and ground-truth normals, achieving a maximum value of one for perfect
alignment. Finally, for assessing rendering quality, we adopt standard metrics
from the NeRF literature: Peak Signal-to-Noise Ratio (PSNR), Structural
Similarity Index Measure (SSIM), and Learned Perceptual Image Patch Similarity
(LPIPS).

##  4 Results & Discussion

###  4.1 Accuracy Evaluation

As shown in Table 1, Meshroom, Neuralangelo, and Neus-facto consistently
outperform other methods in the accuracy evaluation across various metrics,
wound types, and recording devices. In particular, Neuralangelo and Neus-facto
excel with complex wound geometries such as SD and PIS4. Additionally, Neus-
facto exhibits robustness to variations in recording device quality. Even with
Logitech videos more prone to blurriness and with lower resolution compared to
iPhone videos, Neus-facto consistently produces good results. It particularly
excels in NC and $\mathcal{W}_{2}$-NC, highlighting the benefit of the SDF
formulation in generating smooth reconstructed surfaces.

In contrast, Gaussian Splatting methods struggle with sparsely observed
regions, inheriting this limitation from the original 3DGS work. Given our
dataset’s realistic scenario design, these methods failed to achieve
sufficient accuracy, a finding also reflected in the qualitative assessment of
novel-view renderings.

Regarding fast reconstruction methods such as InstantNGP, NeuS2, and DUSt3R,
the generated meshes exhibit poor quality, containing numerous floaters and
defects that do not meet our application’s requirements. Therefore, we have
omitted their results from the main manuscript, but we provide a qualitative
discussion of their results in the supplementary materials. Additionally, the
rendering results of InstantNGP are reported in Table 3.

Figure 4: Figure depicting the reconstructed surfaces, where the color
represents the distance to the closest point in the ground-truth point cloud,
obtained from iPhone and Logitech recordings of SD and PIS3 respectively.
Additional results for different wounds and devices are available in the
supplementary materials.

###  4.2 Precision Evaluation

In Table 2, we examine the repeatability of the reconstructions for the most
accurate methods identified in the previous experiment (i.e., Meshroom,
Neuralangelo, and Neus-facto) between different camera devices (inter-device
column) and recording attempts (inter-recording column). Both Neuralangelo and
Neus-facto demonstrate advantages over Meshroom (the most accurate
photogrammetry method in our evaluation). Remarkably, Neus-facto exhibits the
best performance in the inter-device evaluation, showing the most consistent
reconstructions among the compared methods.

###  4.3 Qualitative Analysis

We conduct a qualitative analysis of the evaluated models using error color-
coded surfaces. In Figure 4, it appears that photogrammetric pipelines often
yield noisy surfaces, primarily due to challenges in stereo matching under
uncontrolled illumination in the videos. Similarly, Gaussian splatting
approaches encounter similar difficulties due to the Gaussians’ initialization
based on the SfM point cloud. While these points reflect the challenges
associated with stereo-matching, Gaussian splatting—which relies on explicit
modelling of the neural surface representation—becomes significantly affected
by noise introduced in the photogrammetry pipelines. In contrast, Neuralanglo
and Neus-facto leverage implicit volumetric rendering based on the SDF
representation of a regular surface, which allows them to regularize noise
that may be present in the training data. This results in accurate,
repeatable, and smooth reconstructions.

Reproducible performance across different imaging devices is a desirable
property for telehealth applications. As detailed in Table 1 and depicted in
Figure 5, NeuralAngelo exhibits a massive performance drop when the quality of
the input images is reduced, we attribute this behavior to the model’s large
size and the reduced photogrammetric consistency between view acquired with a
low-resolution device.

Figure 5: Figure illustrating the influence of image quality on Neuralangelo’s
performance. The best reconstruction was obtained with iPhone acquisitions,
which feature 4K resolution compared to Logitech’s lower image resolution and
quality. From left to right: 3D reconstructed mesh, error color-coded mesh, a
sample image of the wound sourced from the training set. Table 3: Table
presenting the averaged image metrics for the novel-view renderings
experiment. We compare NeRF and Gaussian splatting methods for the iPhone
acquisitions of PIS3, PIS4, and SD.

Method PSNR $\uparrow$ SSIM $\uparrow$ LPIPS $\downarrow$ Time 4k PIS3 PIS4 SD
PIS3 PIS4 SD PIS3 PIS4 SD (Mean) SuGaR 29.719 30.935 30.869 0.959 0.960 0.954
0.056 0.050 0.053 1.234h 2DGS 330.874 31.489 31.852 30.966 0.965 0.959 30.055
0.050 0.050 1.695h 3DGS 30.642 332.595 331.904 20.967 30.967 30.960 20.049
30.045 30.046 57.8m Instant-NGP 30.386 30.979 31.068 0.962 0.962 0.957 0.065
0.059 0.061 6.8m Neuralangelo 133.897 135.560 235.533 10.971 10.973 10.967
10.047 20.039 10.038 17.884h Neus-facto 232.224 235.096 135.552 0.964 20.972
20.966 0.066 10.038 10.038 1.225h

Figure 6: Qualitative evaluation of rendering methods for the SD wound model.
The first row depicts an image in the test set that is well-represented in the
training set. The second row shows a rendering of the wound from an oblique
view, where non-SDF-based methods may exhibit degraded performance (presence
of floaters). The last row displays an image far from the training view
distribution, where the rendering can be challenging due to the scarcity of
training views.

###  4.4 Rendering Quality

In Table 3, Neuralangelo and Neus-facto significantly outperform Instant-NGP
(SOTA approach for novel view rendering) in terms of image rendering quality.
This superior performance can be attributed to their ability to accurately
represent surfaces, mitigating the floaters commonly found in Instant-NGP and
3DGS, and reducing rendering artifacts present in novel images that are too
different from the training distribution (see Figure 6). Despite Instant-NGP’s
fast runtime, it was excluded from the quantitative and qualitative
evaluations due to its inability to produce surface meshes of good quality.
Consequently, Neus-facto emerged as the overall best, balancing the trade-off
between reconstruction quality, repeatability, and manageable training time.
Rendered images and additional qualitative results are made available in the
supplementary materials.

##  5 Conclusion

This paper presents a comprehensive study on 3D wound reconstruction using
consumer-grade videos. We introduce SALVE, a dataset for 3D wound
reconstruction and visualization, featuring video recordings of three wound
types captured with two different cameras. These videos include photometric
challenges typical in clinical settings, such as transient shadows, motion
blur, and specular reflections. SALVE also provides accurate 3D ground-truth
acquired with a commercial 3D scanner and a rigorous evaluation protocol for
assessing the geometric accuracy and precision of 3D reconstruction methods.
Using this dataset, we explore a wide array of 3D reconstruction methods, from
state-of-the-art neural rendering to traditional photogrammetry. We believe
this study and dataset are crucial for advancing the use of this technology in
wound care practices.

Beyond its clinical significance, this study demonstrates some limitations of
current AI-based approaches, such as repeatability issues with SfM-initialized
methods and the reduced quality of NeuralAngelo’s 3D reconstruction under
degraded image quality. We believe that these findings and benchmarks will
also serve as valuable resources for the computer vision community.

#### Acknowledgments:

This work was supported by the MRFF Rapid Applied Research Translation grant
(RARUR000158), CSIRO AI4M Minimising Antimicrobial Resistance Mission, and
Australian Government Training Research Program (AGRTP) Scholarship.

#### Compliance with Ethical Standards:

This study was performed in line with the principles of the Declaration of
Helsinki. The experimental procedures involving human subjects described in
this paper were approved by CSIRO Health and Medical Human Research Ethics
Committee (CHMHREC) [ethics protocol 2022_015_LR].

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Supplementary Material

In this supplementary material, we present further details on the acquisition
setup and additional qualitative results, including error color-coded meshes
and novel-view renderings. Additionally, we present the non-suitability of the
surfaces produced by DUSt3R [46], Instant-NGP [30], and NeuS2 [47] for wound
assessment applications.

In Section A, we present more details regarding the challenges of the SALVE’s
dataset. Next, in Section B, we describe more precisely the evaluation
protocol, namely a qualitative evaluation of the ground-truth point clouds and
specifications about sampling of the meshes adopted for the experiments of
Section 3.3 of the main paper. Following, with respect to Figure 4 and Figure
6 of the main paper, we present the additional error color-coded meshes
(Figure B.3) and renderings (Figure B.4 for PIS3 and Figure B.5 for PIS4) for
the remaining devices and wound types we omitted in the main paper.

Finally, we present the surfaces produced by DUSt3R, Instant-NGP, and NeuS2 in
Figures B.6, B.7, and B.8, respectively. We also explain why those methods are
not included in the 3D reconstruction benchmark presented in the main
manuscript.

##  A Dataset additional challenges

As depicted in Figure 3 of the main paper, we use a Logitech 4K webcam and an
iPhone 14 Pro Max for video recordings, and a Revopoint POP 3D scanner to
acquire the ground-truth point clouds. Both recording devices utilise default
acquisition settings (e.g. exposure time, frame rate, aperture, white balance,
etc.) to simulate acquisition scenarios as close as possible to a telehealth
application, Figure A.1 shows an example of dynamic lighting conditions
comprised in our dataset. Furthermore, when extracting frames from the
recordings, we select the sharpest frames as outlined in Section 3.1 of the
main paper, however, we might still observe blurred images in the datasets. An
example regarding the Logitech recordings is partially observable from the
image in the second row of Figure A.1. Additionally, SALVE presents moving
shadows cast by the operator during the acquisition, which is a common issue
in environments with multiple illumination sources.

Figure A.1: Two images of the PIS3 wound type captured from slightly different
angles using the Logitech camera show different luminosity levels, as
indicated by their normalized grayscale histograms (on the right).

##  B Evaluation details

In Figure B.2, we illustrate qualitatively the three ground-truth point clouds
acquired with the Revopoint POP 3D scanner.

For the metrics AD, HD, $\text{HD}_{90}$, and NC we uniformly sampled 2
million points from the reconstructed meshes, while for $\mathcal{W}_{2}$ and
$\mathcal{W}_{2}$-NC we sampled around 30 thousand points from the
reconstructed meshes due to the computational complexity of optimal transport
metrics.

Figure B.2: Ground-truth point clouds acquired with the Revopoint POP 3D
scanner and their respective total number of points. From left to right: PIS3,
PIS4, and SD wound types.

Figure B.3: Reconstructed surface results for different methods, wound types
and recording devices. The color represents the distance to the closest point
in the ground-truth point cloud. Figure B.4: Qualitative evaluation of
rendering methods for the PIS3 wound model. The first row presents the
renderings for a view perspective in the test set that is well-represented in
the training set. The second row shows a rendering of the wound from an
oblique view, where we can observe the presence of floaters in non-SDF-based
methods. The last row displays a view perspective under-observed in the
training set. Notably, the PIS3 wound type presents hard reflections that
appear especially in the first two rows. Figure B.5: Qualitative evaluation of
rendering methods for the PIS4 wound model. The first row presents the
renderings for a view perspective in the test set that is well-represented in
training set, however, floaters are already visible for non-SDF-based methods.
The second row shows a rendering of the wound from an oblique view, where more
floaters are present. The last row displays an image from another view angle
where artifacts similar to those in the second row can be observed. Figure
B.6: Six views of the mesh generated by DUSt3R for the PIS3 wound type. From
the middle figure of the second row, we can observe how DUSt3R does not
retrieve accurate geometry. The perimeter of the wound is much higher than the
surrounding regions, not reflecting the real geometry of the scene. DUSt3R,
unlike traditional multi-view stereo approaches, do not follow epipolar
constraints to generate the 3D structure but relies on transformers tailored
to solve reconstructions from a few views in the wild. Figure B.7: Six views
of the mesh generated by Instant-NGP for the PIS3 wound type. As Instant-NGP
is not a method developed for 3D surface reconstruction, its NeRF density is
not regularised. As a result, when extracting a mesh using the Marching Cubes
algorithm, it presents a structure similar to the one reported in the figure.
Given the absence of surface regularizer terms, Instant-NGP solves
photogrammetric consistency by placing density values under the wound’s real
surface, creating a scattered reconstruction. Figure B.8: Six views of the
mesh generated by NeuS2 for the PIS3 wound type. NeuS2 is a fast method for 3D
surface reconstruction, however, we excluded it from our benchmark because it
did not perform consistently well in our SALVE dataset. Although more fine-
tuning might be required for SALVE, we considered Neuralangelo and Neus-facto
as better representing SDF methods in terms of robustness against
photogrammetric complexities. For example, PIS3 presents strong reflections
and we can observe that NeuS2 attempts to achieve photometric consistency and
surface regularization by ”pushing” the reconstructed surface below the wound
level. This behaviour can be attributed to its architecture similar to
Instant-NGP.


11institutetext: Center for Cosmology and Astrophysics, Alikhanian National
Laboratory and Yerevan State University, Yerevan, Armenia 22institutetext:
School of Physics and Astronomy, Monash University, Clayton, Australia
33institutetext: SIA, Sapienza Universita di Roma, Rome, Italy

# Cosmic voids and induced hyperbolicity. III. tracing redshift dependence

M. Samsonyan 11 A.A. Kocharyan 22 A. Stepanian 11 V.G. Gurzadyan 1133

(Received: date / Revised version: date)

###### Abstract

The currently released datasets of the observational surveys reveal the
redshift dependence of the physical features of cosmic voids. We study the
void induced hyperbolicity, that is the deviation of the photon beams
propagating the voids, taking into account the redshift dependence of the void
size indicated by the observational surveys. The cumulative image distortion
parameter is obtained for the case of a sequence of variable size voids and
given underdensity parameters. The derived formulae applied along with those
of redshift distortion ones, enable one to trace the number and the physical
parameters of the line-of-sight voids from the analysis of the distortion in
the galactic surveys.

###### pacs:

98.80.-kCosmology

##  1 Introduction

The role of cosmic voids as of principal features in revealing the large-scale
Universe has been steadily increasing due to the advanced observational
surveys of the recent decades, Au ; Nad ; Pan ; Dav ; Cor ; Se  and references
therein. The release of the data of the GIGANTES project Kr , as of the most
extensive catalog suite containing over 1 billion voids up to the redshift
$z=2$, provided a landscape for testing the models for the cosmic web
formation.

Cosmic voids have also become an efficient tool to test fundamental physical
principles, modified gravity, dark sector models, e.g. Amb ; Cap ; Wil . The
possibility for the Local Group as situated in a local void is discussed e.g.
GC ; Cai ; Ding . The void nature of the Cold Spot of the Cosmic Microwave
Background sky has been concluded spot  based on the estimation of
Kolmogorov’s stochasticity parameter in the Planck’s microwave data for that
region, as indication of the void-induced hyperbolicity of the photon beam
propagation GK1 ; GK2 .

The data in Kr  regarding the void distribution and parameter statistic reveal
the variation of the void size over the redshift. In the present study we will
concentrate on the role of the void size variation in the cumulative image
distortion due to the hyperbolicity of the photon beams propagating the cosmic
voids. We use the methods developed in the theory of dynamical systems An ;
Arn , include those regarding the hyperbolicity of geodesic flows. Certain
features of the void-induced hyperbolicity of photon beams and its observable
consequences are studied in Sam1 ; Sam2 .

The currently available high redshift data on the cosmic voids increase the
interest to the redshift dependence on the void induced observable effects,
including their contribution to the image distortion due to the hyperbolicity
of the photon beams. Therefore, here we will study the cumulative image
distortion taking into account the redshift dependence of the sizes of the
line-of-sight voids.

##  2 The Jacobi equation

The hyperbolicity of the geodesic flows defined in Riemmanian manifold can be
defined analysing the solutions of the Jacobi equation of the geodesic
deviation An ; Arn

$\nabla_{u}\nabla_{u}n+Riem(n,u)u=0,$ (1)

where $n$ is the deviation and the Riemann tensor is

$\displaystyle
Riem(n,u)u=([\nabla_{n},\nabla_{u}]-\nabla_{[n,u]})u=[\nabla_{n},\nabla_{u}]u,$
$\displaystyle\nabla_{n}u-\nabla_{u}n=[n,u]=0.$

At the orthogonality of the deviation and velocity vectors

$<n,u>=0,$ (2)

the Jacobi equation will turn to

$\frac{d^{2}\parallel n\parallel^{2}}{ds^{2}}=-2K_{u,n}\parallel
n\parallel^{2}+2\parallel\nabla_{u}n\parallel^{2},$ (3)

where the two-dimensional curvature is

$K_{u,n}=\frac{<Riem(n,u)u,n>}{\parallel n\parallel^{2}}.$ (4)

When $K_{u,n}$ is negative the solutions of the Jacobi equation diverge not
slower than the exponential function

$\parallel n(s)\parallel\geq\frac{1}{2}\parallel
n(0)\parallel\exp(\sqrt{-2k}s),\qquad s>0$ (5)

if the $K_{u,n}$ is negative

$k=\max_{u,n}\\{K_{u,n}\\}<0.$ (6)

When this condition is fulfilled the geodesic flows are Anosov systems An .

The null geodesic flows describing the photon beams passing through the voids
reveal deviation so that the under-density parameter plays the role of the
negative curvature GK1 ; GK2 . Certain properties of the void-induced
hyperbolicity have been studied in Sam1 ; Sam2 , and here we will reveal the
further observable effects.

##  3 Hyperbolicity vs distortion

It has been shown in GK1 , that for every (d+1)-dimensional Lorentzian
manifold, the twice averaged Jacobi equation has the form

$\frac{d^{2}l}{d\eta^{2}}+\frac{\mathfrak{r}_{u}}{d-1}l=0\,,$ (7)

where $\mathfrak{r}_{u}$ is the d-dimensional spatial Ricci scalar (the
details of justification for these averages can be found in GK1 ). On the
other hand, for FLRW metric with small perturbation $\phi$, the line element
can be written as ($c=1$)

$ds^{2}=-(1+2\phi)dt^{2}+(1-2\phi)a^{2}(t)d\sigma^{2}\,,$ (8)

where depending on sign of the sectional curvature $k$ of the spatial
geometry, $d\sigma^{2}$ represents the spherical, $k=1$, Euclidean, $k=0$ or
hyperbolic, $k=-1$, geometries. Meantime, the perturbation field $\phi$,
defined over the above metric, satisfies the following conditions Holz

$|\phi|\ll 1,\quad\left(\frac{\partial\phi}{\partial t}\right)^{2}\ll
a^{-2}(t)||\phi||^{2}\,.$ (9)

The Ricci scalar is

$\displaystyle\frac{\mathfrak{r}_{u}}{2}$ $\displaystyle=k+\frac{16\pi
G}{3}\,\langle\rho_{0}\rangle a_{0}^{3}\ \Theta\ \tilde{\delta},$ (10)

where

$\tilde{\delta}=\frac{\delta\rho}{a\langle\rho\rangle}=\frac{\rho-\langle\rho\rangle}{a\langle\rho\rangle}$
(11)

and

$\Theta=\frac{3}{4}\left(1+\frac{(\nabla^{2}\phi,\mathbf{u}\otimes\mathbf{u})}{\Delta\phi}\right)\,.$
(12)

$\Theta$ reflects the anisotropy of photon beams, $\Theta=1$ corresponding to
a spherical distribution i.e.
$\langle\mathbf{u}\otimes\mathbf{u}\rangle=\frac{1}{3}\mathbf{\gamma}$.

We can rewrite this using the cosmological parameters Sam1

$\Omega_{k}=-\frac{k}{a^{2}(t)H^{2}},\quad\Omega_{\Lambda}=\frac{\Lambda}{3H^{2}},\quad\Omega_{m}=\frac{8\pi
G\rho}{3H^{2}};\quad\quad H=\frac{\dot{a}(t)}{a(t)}$ (13)

and the Friedmann equations from the other side

$H^{2}=-\frac{k}{a^{2}(t)}+\frac{\Lambda}{3}+\frac{8\pi G\rho}{3}\,.$ (14)

The Eq.(10) becomes

$\displaystyle\frac{\mathfrak{r}_{u}}{2}$
$\displaystyle=(a_{0}H_{0})^{2}\left(-\Omega_{k}+2\Theta\tilde{\delta}\,\Omega_{m}\right)$
(15)

The importance of Eq.(15) is in the fact that, it enables one to study the
stability conditions for the sequence of structures i.e. voids and walls with
different attributed parameters GK2 . Then, the Eq.(7) can be written as

$\frac{d^{2}l}{d\lambda^{2}}+\mathfrak{r}_{\Lambda}l=0\,,$ (16)

where $d\eta=(a_{0}H_{0})^{-1}d\lambda$,
$d\lambda\equiv\left(\Omega_{\Lambda}+(1-\Omega_{\Lambda}+\Omega_{m}z)(1+z)^{2}\right)^{-1/2}dz$.
Now by considering $\Omega_{k}=0$, Eq.(16) becomes

$\frac{d^{2}l}{d\tau^{2}}+\Theta\,\tilde{\delta}l=0,$ (17)

where $\tau=\sqrt{2\Omega_{m}}\lambda$.

##  4 Variable void sizes

We assume $\Theta=1$ and for $0=T_{0}<T_{1}<T_{2}<\cdots
T_{2i-1}<T_{2i}\cdots<T_{2n-1}<T_{2n}=T$

$\tilde{\delta}(\tau)=\begin{cases}-\kappa^{2}&T_{0}<\tau<T_{1}\\\
+\omega^{2}&T_{1}<\tau<T_{2}\\\ \cdots\\\
-\kappa^{2}&T_{2i-2}<\tau<T_{2i-1}\\\ +\omega^{2}&T_{2i-1}<\tau<T_{2i}\\\
\cdots\\\ -\kappa^{2}&T_{2n-2}<\tau<T_{2n-1}\\\
+\omega^{2}&T_{2n-1}<\tau<T_{2n}\end{cases}$ (18)

We know that

$x(T)=f^{T}x(0)=\left[\prod_{i=1}^{n}e^{B\tau_{2i}}e^{A\tau_{2i-1}}\right]x(0),$
(19)

where $\tau_{k}=T_{k}-T_{k-1}$ for $1\leq k\leq 2n$

$A=\begin{pmatrix}0&1\\\ \kappa^{2}&0\end{pmatrix}\ ,\qquad
B=\begin{pmatrix}0&1\\\ -\omega^{2}&0\end{pmatrix}\ ,\qquad
x=\begin{pmatrix}\ell\\\ \dot{\ell}\end{pmatrix}\ .$ (20)

$e^{AT_{A}}=\begin{pmatrix}\cosh(\kappa
T_{A})&\quad\frac{1}{\kappa}\sinh(\kappa T_{A})\\\ \\\ \kappa\sinh(\kappa
T_{A})&\quad\cosh(\kappa T_{A})\end{pmatrix}\ ,\qquad
e^{BT_{B}}=\begin{pmatrix}\cos(\omega T_{B})&\quad\frac{1}{\omega}\sin(\omega
T_{A})\\\ \\\ -\omega\sin(\omega T_{A})&\quad\cos(\omega T_{A})\end{pmatrix}\
,$ (21)

and

$\displaystyle e^{C(T_{A}+T_{B})}\equiv e^{BT_{B}}e^{AT_{A}}$
$\displaystyle=\left(\begin{array}[]{cc}\frac{\kappa}{\omega}\sinh(\kappa
T_{A})\sin(\omega T_{B})+\cosh(\kappa T_{A})\cos(\omega
T_{B})&\quad\frac{1}{\kappa}\sinh(\kappa T_{A})\cos(\omega
T_{B})+\frac{1}{\omega}\cosh(\kappa T_{A})\sin(\omega T_{B})\\\ \\\
\kappa\sinh(\kappa T_{A})\cos(\omega T_{B})-\omega\cosh(\kappa
T_{A})\sin(\omega T_{B})&\quad\cosh(\kappa T_{A})\cos(\omega
T_{B})-\frac{\omega}{\kappa}\sinh(\kappa T_{A})\sin(\omega
T_{B})\end{array}\right)\,,$ (25)
$\displaystyle\text{Det}\left(e^{C(T_{A}+T_{B})}\right)$ $\displaystyle=1\,,$
(26) $\displaystyle\text{Tr}\left(e^{C(T_{A}+T_{B})}\right)$
$\displaystyle=2\cosh(\kappa T_{B})\cos(\omega
T_{A})+\left(\frac{\kappa}{\omega}-\frac{\omega}{\kappa}\right)\sinh(\kappa
T_{B})\sin(\omega T_{A})\,.$ (27)

Our aim is to obtain the distortion parameter $\beta$ for $f^{T}$. Distortion
in this case is defined as follows

$\displaystyle\beta=\text{min}(|\lambda_{-}|,|\lambda_{+}|)\,,$

where

$\lambda_{\pm}=\frac{b\pm\sqrt{b^{2}-4}}{2}\,,\qquad
b=\text{Tr}\left(f^{T}\right)\,.$ (28)

For large $n$ the expression for the trace becomes complicated. For example,
when $n=2$, for the trace we get

$\displaystyle b=$
$\displaystyle\frac{1}{k^{2}\omega^{2}}\left(\sinh(k\tau_{1})\sinh(k\tau_{3})[k\omega(k^{2}-\omega^{2})\cosh(k\tau_{3})\sin\omega(\tau_{2}+\tau_{4})+\right.$
(29)
$\displaystyle\left.(k^{4}+\omega^{4})\sin(\tau_{2}\omega)\sin(\tau_{4}\omega)+2k^{2}\omega^{2}\cos(\tau_{2}\omega)\cos(\tau_{4}\omega)\right]+$
$\displaystyle
k\omega\cosh(k\tau_{1})\left[2k\omega\cosh(k\tau_{3})\cos(\tau_{2}+\tau_{4})\omega+\right.\left.\left.(k^{2}-\omega^{2})\sinh(k\tau_{3})\sin(\tau_{2}+\tau_{4})\omega\right]\right).$

First, let us consider two voids and two walls, i.e. when $\tau_{1}=\tau_{3}$,
$\tau_{2}=\tau_{4}$, $\kappa\tau_{1}\ll 1$ and $\omega\tau_{2}\ll 1$. Then we
get

$\displaystyle{}\beta=1-T\sqrt{-\left<\delta_{0}\right>}\,,$ (30)

where

$T=2(\tau_{1}+\tau_{2})\,,\qquad\left<\delta_{0}\right>=\frac{-\kappa^{2}\tau_{1}+\omega^{2}\tau_{2}}{\tau_{1}+\tau_{2}}\,.$
(31)

To describe the variation of the void size over the redshift, we repeat the
same procedure for the case when the next void is slightly larger than the
former one, i.e. when $\tau_{3}$ is slightly larger than $\tau_{1}$,
$\tau_{3}=\tau_{1}+\epsilon$, $\epsilon\ll\tau_{1}$, and for the distortion we
get

$\displaystyle\beta=1-T\sqrt{-\left<\delta_{0}\right>}+\frac{\delta_{void}-\left<\delta_{0}\right>}{\sqrt{-\left<\delta_{0}\right>}}\epsilon\,.$
(32)

For the case of $n$ pairs of voids and walls as described above, one can
easily get

$\displaystyle b$
$\displaystyle=\text{Tr}\left(\prod_{i=1}^{n}e^{B\tau_{2}}e^{A(\tau_{1}+\epsilon_{i})}\right)=\text{Tr}\left(\prod_{i=1}^{n}e^{B\tau_{2}}e^{A\tau_{1}}e^{A\epsilon_{i}}\right)=\text{Tr}\left(\prod_{i=1}^{n}e^{B\tau_{2}}e^{A\tau_{1}}(I+A\epsilon_{i})\right)+o(\epsilon)$
$\displaystyle=\text{Tr}\left(\prod_{i=1}^{n}e^{B\tau_{2}}e^{A\tau_{1}}\right)+\text{Tr}\left(\prod_{i=1}^{n}e^{B\tau_{2}}e^{A\tau_{1}}A\right)\epsilon+o(\epsilon)$
$\displaystyle=\text{Tr}\left(\left[e^{B\tau_{2}}e^{A\tau_{1}}\right]^{n}\right)+\text{Tr}\left(\left[e^{B\tau_{2}}e^{A\tau_{1}}\right]^{n}A\right)\epsilon+o(\epsilon)$
(33)

where $\epsilon=\sum_{i=1}^{n}\epsilon_{i}$ and
$n(\tau_{1}+\tau_{2})+\epsilon=T$.

$\displaystyle\beta$
$\displaystyle=1-n\sqrt{-(\tau_{1}+\tau_{2})\left(-\kappa^{2}\tau_{1}+\omega^{2}\tau_{2}\right)}$
$\displaystyle\quad-\tfrac{1}{2}\left(\frac{1}{\sqrt{-(\tau_{1}+\tau_{2})\left(-\kappa^{2}\tau_{1}+\omega^{2}\tau_{2}\right)}}-n\right)\left(\kappa^{2}(\tau_{2}+\tau_{1})-\left(-\kappa^{2}\tau_{1}+\tau_{2}\omega^{2}\right)\right)\epsilon+o(\epsilon)$
$\displaystyle=1-n(\tau_{1}+\tau_{2})\sqrt{-\left<\delta_{0}\right>}-\left(1-n(\tau_{2}+\tau_{1})\sqrt{-\langle\delta_{0}\rangle}\right)\frac{\left(\delta_{\text{void}}-\left<\delta_{0}\right>\right)}{2\sqrt{-\langle\delta_{0}\rangle}}\epsilon+o(\epsilon)$
$\displaystyle=\left[1-n(\tau_{1}+\tau_{2})\sqrt{-\left<\delta_{0}\right>}\right]\left[1-\frac{\left(\delta_{\text{void}}-\left<\delta_{0}\right>\right)}{2\sqrt{-\langle\delta_{0}\rangle}}\epsilon\right]+o(\epsilon)$
(34)

Figs.1 and 2 illustrate these formulae for given ratio of the void and wall
sizes and $\epsilon$, the size increase parameter of the voids.

Figure 1: The photon beam distortion due to the void hyperbolicity for given
void and wall size ratios and void size variation $\epsilon=0.1$. Figure 2:
The same as in Fig.1 but for fixed void and wall size ratio and
$\epsilon=0.1,0.05,0.02$.

##  5 Conclusions

The GIGANTES dataset Kr  marks the appearance of precision cosmology landscape
from cosmic voids. Such extensive data enable to formulate and study to higher
precision theoretical problems associated to redshift dependence of the voids’
mean physical parameters. Particularly, Kr  provide the median radius
distribution of the voids up to redshift $z=1$, which can be directly compared
with the predictions of the theory of formation of the cosmic web, i.e. of the
sequence of semi-periodical underdense regions, the voids, separated by thin,
overdensity walls.

We studied the hyperbolicity properties of the voids, i.e. their ability for
deviation the photon beams, so that void’s underdensity has the effective role
of the negative curvature in the Jacobi equation. Developing further our
previous studies Sam1 ; Sam2 , we now revealed the cumulative distortion of
the images due to the sequence of line-of-sight voids in the case of varying
sizes of the voids, i.e. as is indicated by the survey Kr . The formula for
the distortion parameter is obtained, quantitatively illustrated in Figs. 1
and 2, for given void/wall size ratio and the increase in the size of the
voids in the sequence.

The tangential image distortion in the galactic surveys are detected long ago
(e.g. Pea ; Guz ; Per ), typically reflecting the contribution of redshift
(Doppler) effects such as e.g. the Kayser effect, and to which (to the
distortion) the hyperbolicity of voids has to contribute as well.

The redshift dependence of the voids’ parameters carries crucial cosmological
information. Along with that, the redshift dependence of the void sizes has to
influence the observational effects associated with the voids at high
redshifts, as revealed in this study obtaining a formula for the quantitative
evaluation of the hyperbolicity by voids of variable sizes at cosmological
distances.

M.S. acknowledges the support of Armenian SCS grant 20RF-142.

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# Law of Large Numbers for Risk Measures 111AMS-Classification 90B50, 91B06,
91B16, 91G99

Freddy Delbaen  Departement für Mathematik, ETH Zürich, Rämistrasse 101, 8092
Zürich, Switzerland  Institut für Mathematik, Universität Zürich,
Winterthurerstrasse 190, 8057 Zürich, Switzerland

(Date: First version April 26, 2020, this version )

###### Abstract.

Under appropriate integrability conditions the risk measure of the sample
measures for a law invariant risk measure converge almost surely to the risk
measure of the sampled random variable. The results follow from general
convergence theorems based on the theory of Orlicz spaces.

##  1\. Introduction and Notation

The notation we use is the standard notation of probability theory.
$(\Omega,{\mathcal{F}},{\mathbb{P}})$ will denote a probability space without
atoms. If ${\mathbb{Q}}$ is a probability defined on ${\mathcal{F}}$ and
absolutely continuous with respect to ${\mathbb{P}}$, we will identify
${\mathbb{Q}}$ with its Radon-Nikodým derivative
$h=d{\mathbb{Q}}/d{\mathbb{P}}$. A basic example of such an atomless space is
$([0,1],{\mathcal{B}},m)$ where ${\mathcal{B}}$ is the Borel sigma algebra and
$m$ is Lebesgue measure. This space will also play a fundamental role in the
sequel.

Because of neganumerophobia we will use a setup closer to premium calculation.
For a convex closed set of probability densities, ${\mathcal{S}}\subset L^{1}$
we denote by $\rho$ the functional (whenever defined)

$\rho(\xi)=\sup\\{{\mathbb{E}}[\xi h\mid h\in{\mathcal{S}}\\}.$

This functional is up to sign changes the same as a coherent monetary utility
function. The interpretation of $\rho$ is the same as for the utility
functions. Among the many uses we mention risk measures and insurance premium
principles defined on random variables. A basic problem is the following.
Suppose we see a large sample of a random variable. Can we define the
functional $\rho$ on such samples and does this procedure converge to the risk
measure of the random variable? The best is to put the problem in the
framework of risk measures that are law invariant (law determined or
rearrangement invariant). That means that when $\eta$ and $\eta^{\prime}$ have
the same distribution, then $\rho(\eta)=\rho(\eta^{\prime})$. Since we will be
working on atomless probability spaces, this property is also equivalent to
the property that ${\mathcal{S}}$ is rearrangement invariant. Law determined
risk measures can be defined for probability distributions on ${\mathbb{R}}$
with compact support (the latter to avoid integrability problems). It
therefore does not matter on which probability space we work. In later
sections we will start with a risk measure defined for random variables
defined on $\Omega$ but then we will “transport” it to random variables
defined on $[0,1]$. For more information on rearrangement invariant measures
see [9] where the general form of these risk measures is given.

The risk measures we will use are sometimes defined on bigger spaces than
$L^{\infty}$, the space of bounded random variables. The best way to study
this is to use Orlicz spaces. For the theory of these spaces we refer to [10].
In this book the reader will find a very general approach. We will restrict
ourselves to Young functions that are more regular than the ones in [10]. For
our presentation this restriction is of no importance but it avoids a lot of –
in our opinion cosmetic –problems. When we say $\Phi$ is a Young function, we
mean that $\Phi$ is an even real valued function defined on ${\mathbb{R}}$
with the help of its derivative $\phi$. This derivative function $\phi$ will
be nondecreasing on ${\mathbb{R}}_{+}$, continuous at $0$, $\phi(x)=0$ if and
only if $x=0$ and $\lim_{x\rightarrow+\infty}\phi(x)=+\infty$. We do not
require more regularity for $\phi$. The conjugate Young function $\Psi$ is
then defined using the right continuous inverse function, $\psi$, of $\phi$.
More precisely for $x\geq 0$: $\psi(x)=\inf\\{u\mid\phi(u)>x\\}$. Hence
$\Psi(x)=\int_{0}^{|x|}\psi(u)\,du$. We realise that the class of Orlicz
spaces so defined, does not contain $L^{1}$ or $L^{\infty}$, so when
appropriate we will make additional remarks. We follow [10] and define

$\displaystyle L^{\Phi}$ $\displaystyle=$ $\displaystyle\left\\{\xi\mid\text{
there is $k>0$ with }{\mathbb{E}}\left[\Phi(k\xi)\right]<\infty\right\\}$
$\displaystyle M^{\Phi}$ $\displaystyle=$ $\displaystyle\left\\{\xi\mid\text{
for all $k>0$ we have
}{\mathbb{E}}\left[\Phi(k\xi)\right]<\infty\right\\}=L^{(\Phi)}\subset
L^{\Phi}.$

These spaces are usually equipped with the Luxemburg norm

$\|\xi\|_{\Phi}=\inf\left\\{\alpha>0\mid{\mathbb{E}}\left[\Phi\left(\frac{\xi}{\alpha}\right)\right]\leq
1\right\\}.$

The dual space of $M^{\Phi}$ is the space $L^{\Psi}$ and we have for $\xi\in
L^{\Phi},\eta\in L^{\Psi}$ that $\xi\eta\in L^{1}$ and
$|{\mathbb{E}}[\xi\eta]|\leq 2\|\xi\|_{\Phi}\|\eta\|_{\Psi}$. The dual space
of $L^{\Phi}$ is not necessarily $L^{\Psi}$. Indeed we have that
$(L^{\Phi})^{*}=L^{\Psi}$ if and only if $L^{\Phi}=M^{\Phi}$ if and only if
$\Phi$ satisfies the $\Delta_{2}$ condition: there is $x_{0}$ and a constant
$C$ such that for all $x\geq x_{0}$ the inequality $\Phi(2x)\leq C\Phi(x)$ is
valid. If this is the case one can modify the function $\Phi$ without changing
the spaces and such that in the inequality we may suppose that $x_{0}=0$.

##  2\. Main Result

In this section we suppose that the scenario set ${\mathcal{S}}$, for the risk
measure $\rho$, is supposed to be a subset of $L^{\Phi}$. The set
${\mathcal{S}}$ is a convex closed subset of $L^{1}$ and hence it must be
bounded in $L^{\Phi}$. Indeed, if $h_{n}\in{\mathcal{S}}$ is such that
$\|h_{n}\|_{\Phi}$ is an unbounded sequence, say $\|h_{n}\|_{\Phi}\geq 4^{n}$,
then the element $h=\sum_{n}\frac{h_{n}}{2^{n}}$ is still in ${\mathcal{S}}$
but we have for each $n$: $\|h\|_{\Phi}\geq\|h_{n}2^{-n}\|_{\Phi}\geq
2^{-n}4^{n}=2^{n}$. This would mean that ${\mathcal{S}}$ is not a subset of
$L^{\Phi}$. The set ${\mathcal{S}}$ is also weak∗, meaning
$\sigma(L^{\Phi},M^{\Psi})$, compact. To see this we must only show that it is
weak∗ closed. This follows from the (obvious) continuity of the injection
$L^{\Phi}\rightarrow L^{1}$ for the topologies
$\sigma(L^{\Phi},M^{\Psi})\text{ and }\sigma(L^{1},L^{\infty})$, just observe
that $L^{\infty}\subset M^{\Phi}$. This continuity also implies that
${\mathcal{S}}$ is closed in $L^{\Phi}$. We remark that when
${\mathcal{S}}\subset L^{1}$ is a convex set of probabilities that is closed
in $L^{\Phi}$, this does not necessarily imply that ${\mathcal{S}}$ must be
closed in $L^{1}$.

Because of this boundedness, the functional $\rho$ with scenario set
${\mathcal{S}}$ can be defined on $L^{\Psi}$. Indeed for $\eta\in L^{\Psi}$ we
have

$|\rho(\eta)|=|\sup\\{{\mathbb{E}}[\eta h]\mid h\in{\mathcal{S}}\\}|\leq
2\|\eta\|_{\Psi}\sup_{h\in{\mathcal{S}}}\|h\|_{\Phi}<\infty.$

As expected this boundedness also implies that $\rho$ is continuous on
$L^{\Psi}$. For completeness we include the obvious proof.

######  Lemma 1.

If $\eta_{n}$ is a sequence in $L^{\Psi}$ tending to $\eta$ for
$\|.\|_{\Psi}$, then $\rho(\eta_{n})\rightarrow\rho(\eta)$. More precisely
$|\rho(\eta)-\rho(\eta^{\prime})|\leq
2\sup_{h\in{\mathcal{S}}}\|h\|_{\Phi}\|\eta-\eta^{\prime}\|_{\Psi}$.

Proof.  We only have to prove the inequality. For given $\varepsilon>0$ and
$\eta$ we choose $h\in{\mathcal{S}}$ with
$\rho(\eta)\leq{\mathbb{E}}[h\eta]+\varepsilon$. This gives

$\rho(\eta)\leq{\mathbb{E}}[h\eta]+\varepsilon={\mathbb{E}}[h\eta^{\prime}]+{\mathbb{E}}[h(\eta-\eta^{\prime})]+\varepsilon\leq\rho(\eta^{\prime})+2\|h\|_{\Phi}\|\eta-\eta^{\prime}\|_{\Psi}+\varepsilon.$

Because $\varepsilon>0$ was arbitrary we get
$\rho(\eta)\leq\rho(\eta^{\prime})+2\sup_{h\in{\mathcal{S}}}\|h\|_{\Phi}\|\eta-\eta^{\prime}\|_{\Psi}$.
We can interchange $\eta$ and $\eta^{\prime}$ and then get the desired
result.∎

From now on we suppose that $\rho$ is rearrangement invariant.

For a probability measure $\mu$ on ${\mathbb{R}}$, we use the notation
$\rho(\mu)$ for the outcome $\rho(\eta)$, where $\mu$ is the distribution of
$\eta$. When ${\mathcal{S}}\subset L^{\Phi}$, this is defined as soon as
$\int_{\mathbb{R}}\Psi(x)d\mu<\infty$, i.e. $\eta\in L^{\Psi}$.

For probability laws on ${\mathbb{R}}$ there is an easy way to transform
convergence of probability distributions into convergence of random variables
(a special case of the Skorohod lemma). Suppose that $\mu_{n}$ converges weak∗
to $\mu$, i.e. for each bounded continuous function $g$ defined on
${\mathbb{R}}$ we have
$\int_{\mathbb{R}}g(x)d\mu_{n}\rightarrow\int_{\mathbb{R}}g(x)d\mu$. The
distribution functions are defined as
${\mathbb{F}}_{n}(x)=\mu_{n}(-\infty,x]$, ${\mathbb{F}}(x)=\mu(-\infty,x]$.
Weak∗ convergence is then equivalent to
${\mathbb{F}}_{n}(x)\rightarrow{\mathbb{F}}(x)$ for each continuity point $x$
of ${\mathbb{F}}$. The quantile functions are defined as
$q_{n}(x)=\inf\\{u\mid{\mathbb{F}}_{n}(u)>x\\}$,
$q(x)=\inf\\{u\mid{\mathbb{F}}(u)>x\\}$. The quantile functions, $q_{n}$, are
defined on the unit interval $(0,1)$ and under the Lebesgue measure they have
the distribution functions $\mu_{n}$. The convergence of $\mu_{n}$ is then
transformed into almost everywhere convergence of the quantile functions.
Convergence $q_{n}\rightarrow q$ in the Orlicz space space $L^{\Psi}$ is not
easily defined in terms of the measures $\mu_{n}$. The following is a partial
answer to this question

######  Proposition 1.

Suppose that $\mu_{n}$ converges weak∗ to $\mu$. If for each $k>0$,
$\int_{\mathbb{R}}\Psi(kx)\mu_{n}(dx)\rightarrow\int_{\mathbb{R}}\Psi(kx)\mu(dx)<\infty$,
we have $\|q_{n}-q\|_{\Psi}\rightarrow 0$. Consequently
$\rho(\mu_{n})\rightarrow\rho(\mu)$.

Proof.  We have $\rho(\mu_{n})=\rho(q_{n}),\rho(\mu)=\rho(q)$ so the last line
follows from the lemma. To prove the norm convergence we have to estimate
$\int_{[0,1]}\Psi(k|q_{n}(u)-q(u)|)\,du$. We use the inequalities

$\int\Psi(k|q_{n}-q|)\leq\int\Psi(2k\max(q_{n},q))\leq\int\Psi(2kq_{n})+\int\Psi(2kq).$

Because $q_{n}\rightarrow q$ a.s. and
$\int\Psi(2kq_{n})\rightarrow\int\Psi(2kq)$, Scheffé lemma implies that there
is $n_{0}$ such that $\Psi(2kq_{n}),n\geq n_{0}$ forms a uniformly integrable
family. Consequently the elements $\int\Psi(k|q_{n}-q|)$ (for $n\geq n_{0}$)
form a uniformly integrable family. Therefore for every $k$:
$\int_{[0,1]}\Psi(k|q_{n}(u)-q(u)|)\,du\rightarrow 0$. This shows that
$\lim_{n}\|q_{n}-q\|_{\Psi}\rightarrow 0$.∎

######  Remark 1.

The hypothesis implies that $q\in M^{\Psi}$ but there is no need to suppose
that $q_{n}\in M^{\Psi}$.

We are now ready to state the main result of this section.

######  Theorem 1.

We use the notation and assumptions from the beginning of this section. Let
$\xi\in M^{\Psi}$. Let
$\xi_{n}\colon(\Omega,{\mathcal{A}},{\mathbb{P}})\rightarrow{\mathbb{R}}$ be
an iid sequence with the same distribution $\mu$ as $\xi$. For each $N$ we
define the sample distribution as
$\mu_{n}=\frac{1}{N}\sum_{n=1}^{N}\delta_{\xi_{n}}$ where $\delta_{y}$ denotes
the Dirac measure concentrated in $y$. Then almost surely
$\rho(\mu_{N})\rightarrow\rho(\mu)$.

Proof.  The usual law of large numbers allows to conclude that almost
everywhere the random measures $\frac{1}{N}\sum_{n=1}^{N}\delta_{\xi_{n}}$
converge weak∗ to $\mu$. The variant of Skorohod’s theorem then permits to
find for almost every $\omega\in\Omega$, increasing functions $q_{N}$ on
$[0,1]$ such that $q_{N}$ has as its distribution
$\mu_{n}=\frac{1}{N}\sum_{n=1}^{N}\delta_{\xi_{n}}$. They converge almost
everywhere (on $[0,1]$) to $q$, the increasing rearrangement of $\xi$. Again
the law of large numbers shows that almost surely for every
$0<K\in{\mathbb{N}}$, $\frac{1}{N}\sum_{n=1}^{N}{\Psi}(K\xi_{n})$ converges to
${\mathbb{E}}[{\Psi}(K\xi)]$ (where the set of measure 1 can be chosen to be
independent of $K$). On $[0,1]$ this translates as
$\int_{[0,1]}\Psi(Kq_{N})\rightarrow\int_{[0,1]}\Psi(Kq)$. We now apply the
previous proposition. ∎

######  Remark 2.

With minor notational modifications, the above proof also works for the case
$\Psi(x)=x$, i.e. the $L^{1}$ space. In particular it also proves the theorem
for the tail expectation. In this case the result was proved by Van Zwet.

##  3\. A more general Result

In the section we generalise the result of the preceding section to the case
of random variables in $L^{\Psi}$ (and not just $M^{\Psi}$). We need an
additional hypothesis on the scenarioset. We recall the result of Ando, [3],
which characterises the $\sigma(L^{\Phi},L^{\Psi})$ relative sequentially
compact sets.

######  Theorem 2.

(Ando) A set $A\subset L^{\Phi}$ is $\sigma(L^{\Phi},L^{\Psi})$ relative
sequentially compact if and only if

$\lim_{\lambda\rightarrow\infty}\sup_{h\in
A}\lambda{\mathbb{E}}\left[\Phi\left(\frac{h}{\lambda}\right)\right]=0.$

In particular if $\Psi$ is $\Delta_{2}$ then bounded sets $A\subset L^{\Phi}$
automatically satisfy the above limit condition.

We use the same notation as in the previous sections. In particular $\rho$ is
defined on $L^{\Psi}$ and is rearrangement invariant.

######  Proposition 2.

Suppose that ${\mathcal{S}}$ satisfies the conditions of Ando’s theorem.
Suppose that $\mu_{n}$ converges weak∗ to $\mu$. If for some $k>0$,
$\int_{\mathbb{R}}\Psi(kx)\mu_{n}(dx)\rightarrow\int_{\mathbb{R}}\Psi(kx)\mu(dx)<\infty$,
we have $\sup_{h\in{\mathcal{S}}}{\mathbb{E}}[h|q_{n}-q|]\rightarrow 0$.
Consequently $\rho(\mu_{n})\rightarrow\rho(\mu)$.

Proof.  By homogeneity we may suppose that $k=2$ and that
$\int_{\mathbb{R}}\Psi(2x)\mu_{n}(dx)\rightarrow\int_{\mathbb{R}}\Psi(2x)\mu(dx)$.
We may also suppose that for all $n$:
$\int_{\mathbb{R}}\Psi(2x)\mu_{n}(dx)<\infty$. We replace the integrals on
${\mathbb{R}}$ by integrals of the quantile functions and get that
$\int_{[0,1]}\Psi(2q_{n})\rightarrow\int_{[0,1]}\Psi(2q)$. Scheffé’s lemma
then implies that the sequence $\Psi(2q_{n})$ is uniformly integrable. The
obvious inequality $\Psi(|q_{n}-q|)\leq\Psi(2q_{n})+\Psi(2q)$ then implies
that the sequence $\Psi(|q_{n}-q|)$ is also uniformly integrable. Hence
$\int_{[0,1]}\Psi(|q_{n}-q|)\rightarrow 0$, since $q_{n}\rightarrow q$ a.s. .
We now have

$\sup_{h\in{\mathcal{S}}}\int_{[0,1]}h|q_{N}-q|\rightarrow 0.$

For given $\epsilon>0$ we take $\lambda$ so that
$\lambda\int_{[0,1]}\Phi(\frac{h}{\lambda})\leq\epsilon$. Then we take $N_{0}$
so that for $N\geq N_{0}$: $\lambda\int_{[0,1]}\Psi(|q_{N}-q|)\leq\epsilon$.
Young’s inequality now yields
$h|q_{N}-q|\leq\lambda\Phi(\frac{h}{\lambda})+\lambda\Psi(|q_{N}-q|)$ and
after integration we get for $N\geq N_{0}$:

$\sup_{h\in{\mathcal{S}}}\int_{[0,1]}h|q_{N}-\eta|\leq 2\epsilon.$

∎

######  Theorem 3.

Suppose that the scenarioset ${\mathcal{S}}$ satisfies the condition of Ando’s
theorem. Let $\xi\in L^{\Psi}$. Let
$\xi_{n}\colon(\Omega,{\mathcal{A}},{\mathbb{P}})\rightarrow{\mathbb{R}}$ be
an iid sequence with the same distribution $\mu$ as $\xi$. For each $N$ we
define the sample distribution as
$\mu_{n}=\frac{1}{N}\sum_{n=1}^{N}\delta_{\xi_{n}}$ where $\delta_{y}$ denotes
the Dirac measure concentrated in $y$. Then almost surely
$\rho(\mu_{N})\rightarrow\rho(\mu)$.

Proof. We already know that almost surely $\mu_{N}$ converges weak∗ to $\mu$.
Without loss of generality we may replace $\xi$ by a multiple of it and hence
we may suppose that ${\mathbb{E}}\left[\Psi(2\xi)\right]\leq 1$. The law of
large numbers implies that:

$\frac{1}{N}\sum_{k=1}^{N}\Psi(2\xi_{k})\rightarrow{\mathbb{E}}[\Psi(2\xi)],$

almost surely and in $L^{1}$. We can now apply the previous proposition.∎

##  4\. The case of Choquet Integration

Choquet integration provides us with one of the most used examples of
rearrangement invariant risk measures. As proved by David Schmeidler, [12],
[8], a rearrangement invariant risk measure that is commonotone is given by
the core of a convex game and hence uses a scenarioset of the form

${\mathcal{S}}=\left\\{h\in L^{1}\mid{\mathbb{E}}[h]=1\text{ and for each
}A\in{\mathcal{F}}:{\mathbb{E}}[h{\rm\bf 1}_{A}]\geq
f({\mathbb{P}}[A])\right\\},$

where $f$ is a convex function $f\colon[0,1]\rightarrow[0,1],f(0)=0,f(1)=1$.
In our context, where we want the risk measure to be defined on a larger space
than $L^{\infty}$ we must use a weakly ($\sigma(L^{1},L^{\infty})$) compact
set ${\mathcal{S}}$. This is equivalent to the left continuity of $f$ at the
point $1$. Ryff’s theorem, [11], [8], then says that ${\mathcal{S}}$ is the
$L^{1}-$closed convex hull of random variables having the same law as the
derivative $f^{\prime}$ of $f$. The statement ${\mathcal{S}}\subset L^{\Phi}$
is equivalent to $f^{\prime}\in L^{\Phi}$. One can show that when
$f^{\prime}\in L^{\Phi}[0,1]$, the closed convex hull taken in $L^{\Phi}$ is
the same as when the closedness is taken in $L^{1}$. The set ${\mathcal{S}}$
is also the set of those nonnegative random variables $h$ such that for each
convex function $\beta\colon{\mathbb{R}}_{+}\rightarrow{\mathbb{R}}$ we have
${\mathbb{E}}[\beta(h)]\leq\int_{[0,1]}\beta(f^{\prime})$ (convex dominance).
Because of this dominance the set ${\mathcal{S}}$ automatically satisfies
Ando’s criterium for $\sigma(L^{\Phi},L^{\Psi})$ sequential compactness. As a
result the topologies $\sigma(L^{\Phi},L^{\Psi})$ and
$\sigma(L^{1},L^{\infty})$ coincide on the set ${\mathcal{S}}$. We get that
the statistical results of the previous section apply to this case. For iid
sequences we find the almost sure convergence

$\xi_{[n:1]}f\left(\frac{1}{n}\right)+\sum_{k=1}^{k=n-1}\xi_{[n:k+1]}\left(f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)\right)=\int_{0}^{1}q_{n}\,f^{\prime}\rightarrow\int_{0}^{1}qf^{\prime}=\rho(\xi)$

where $\xi_{[n:1]},\ldots,\xi_{[n:n]}$ are the observations
$\xi_{1},\ldots,\xi_{n}$ in increasing order. This result can also be obtained
using integration theory.

## References

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