# Learning to Transfer Texture from Clothing Images to 3D Humans

Aymen Mir1 Thiemo Alldieck1, 2 Gerard Pons-Moll1

1Max Planck Institute for Informatics, Saarland Informatics Campus, Germany
2Computer Graphics Lab, TU Braunschweig, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we present a simple yet effective method to automatically
transfer textures of clothing images (front and back) to 3D garments worn on
top SMPL [42], in real time. We first automatically compute training pairs of
images with aligned 3D garments using a custom non-rigid 3D to 2D registration
method, which is accurate but slow. Using these pairs, we learn a mapping from
pixels to the 3D garment surface. Our idea is to learn dense correspondences
from garment image silhouettes to a 2D-UV map of a 3D garment surface using
shape information alone, completely ignoring texture, which allows us to
generalize to the wide range of web images. Several experiments demonstrate
that our model is more accurate than widely used baselines such as thin-plate-
spline warping and image-to-image translation networks while being orders of
magnitude faster. Our model opens the door for applications such as virtual
try-on, and allows for generation of 3D humans with varied textures which is
necessary for learning. Code will be available at https://virtualhumans.mpi-
inf.mpg.de/pix2surf/.

##  1 Introduction

Our goal is to learn a model capable of transferring texture from two
photographs (front and back) of a garment to the 3D geometry of a garment
template, automatically and in real time. Such a model can be extremely useful
for photo-realistic rendering of humans, varied generation of synthetic data
for learning, virtual try-on, art and design. Despite the many applications,
automatic transfer of clothing texture to 3D garments has received very little
attention. The vast majority of recent methods work in image space directly.
Most works focus on either image based person re-posing [82, 43, 44, 56, 63,
21, 10, 64], or virtual try-on [57, 78, 28, 69, 77, 22, 83]. Re-posing methods
learn to synthesize image pixels to produce novel poses of a person, whereas
virtual try-on methods learn to morph an image of a clothing item to drape it
on a target person. The advantage of these methods is that they can be trained
on large-scale datasets. The critical disadvantage is that they operate in
pixel-space instead of 3D, they can not synthesize difficult poses, and
struggle to produce temporally consistent results. Another line of work
extracts texture by fitting 3D human models (SMPL [42]) to images [15, 6, 4],
but texture quality quickly deteriorates for complex poses. Other works map
texture to 3D meshes [68, 24, 40, 53] by 3D scanning people, but the number
and variety of clothing textures is limited because 3D scanning is time
consuming. To break the lack of 3D data barrier, our idea is to learn a dense
mapping from images of clothing items, which are ubiquitous on the internet,
to the 3D surface of a parametric garment template directly [15]. However,
accurate texture transfer is far from trivial: web images vary in texture,
garment size and style, pose, and background. Nonetheless, in comparison to
clothing worn by humans, web-store images have less variation, which we
exploit for learning our model. Instead of manually annotating garment
landmarks [85, 8], our idea is to collect training pairs by non-rigidly
aligning a parametric 3D garment template to images. We leverage the
parameterized garment models of MGN [15], non-rigidly fit their surface to
image silhouettes. While our alignment usually produces good results, it is
slow ($~{}5-15$ minutes per image), and fails in $~{}5\%$ of the cases.
Consequently, using only the successful fits, we learn a direct mapping from
image pixels which runs in milliseconds, and is more robust than the
optimization based approach. Our key hypothesis is that the _mapping is
determined by the image silhouette shape alone_ , and not by appearance.
Hence, we train a CNN to predict correspondences from a UV map of the garment
surface to pixel locations, given the silhouette shape alone as input. Since
the model learns in-variances to shape and pose as opposed to appearance, it
generalizes to a wide variety of garment images of varied textures. We refer
to the correspondence predictor as Pix2Surf.

Figure 1: Given regular online retail store images, our method can
automatically produce textures for pre-defined garment templates, that can be
used to virtually dress SMPL [42] in 3D.

Pix2Surf allows to digitally map the texture of a clothing image to the 3D
surface of a garment in real time, see Fig. LABEL:fig:teaser. Once the texture
is mapped to 3D, we generalize to novel views, shapes and pose by applying 3D
transformations to the 3D geometry [15] as illustrated in Fig. 1. Pix2Surf
enables, for the first time, virtual try-on from images in 3D directly and in
real time, enabling applications such as VR/AR, gaming, and 3D content
production.

To stimulate further research in this direction, we will make Pix2Surf
publicly available for research purposes. This will allow researchers and
users to add texture to the SMPL+Garments model [15], generate synthetic
humans, and visualize garment images in 3D directly.

##  2 Related Work

Synthesizing images of people in realistic clothing is a long-studied problem
in the Computer Graphics community. Early works enable to represent and
parametrize fabric virtually in 3D [72, 50], later realistic draping and
animation has been achieved [12, 30]. While these works require artist-
designed garments, careful parameter selection, and computational expensive
physics-based simulation, fully-automated and faster methods have been
introduced more recently. In contrast, these methods process sensor data such
as images, depth maps, or 3D point clouds, and produce 3D reconstructions or
photorealistic images with minimal interaction or even fully automatically.
2D image synthesis methods produce images of people holding a given pose or
wearing a desired outfit. These works utilize recent advances in conditional
image-to-image translation using generative adversarial networks (GANs) [25,
32]. E.g. [39] presents a method to produce full-body images of people in
clothing from semantic segmentation of the body and clothing. However, the
produced subject and outfit are not controllable. The method in [82] produces
novel views of given fashion images. While the subject and outfit remain
untouched, the pose can be constrained with a sparse set of camera angles. To
gain more control over the output, a large number of works deal with
synthesizing person images under a desired pose [43, 44, 56, 63, 21, 10, 64,
62]. For the same purpose, in [26], the authors learn warpings between images
and the SMPL texture. Predicting correspondences instead of color values helps
the network to generalize due to much lower variation in the warps than in the
color-space. In Pix2Surf we also infer correspondences [61, 54, 8, 48] but
focus on garments rather than on full bodies. In similar works, [48] utilizes
DensePose [8] to warp image pixels into the SMPL texture, but inpainting is
performed before warping into the target pose. More related to our work,
recent methods focus on exchanging the a subject’s outfit while preserving
[57, 78] or changing [28, 69, 77, 22, 83] his or her pose. Additionally,
special case methods for text-based fashion image manipulation [84] and image-
based outfit improvement [31] have been presented. In contrast to our method,
all these _virtual try-on_ methods work in the image-space and thus perform no
explicit reasoning about the underlying 3D scene. This means they are not
guaranteed to produce consistent output under varying poses and camera angles.
This is approached in [23] via image warping and subsequent refinement of a
previously generated image. While 2D warping improves the quality of
synthesized videos, limitations of 2D methods are still present.
3D reconstruction methods focus on recovering the actual 3D shape of a
captured garment alone, the body shape of a subject wearing the garment, or
both simultaneously. Methods utilizing controlled RGB [35] and RGB-D images
[17] have been presented, that select and refine 3D garment templates based on
image observations. While [35] utilizes a tailor’s dummy, [17] expects the
subject to hold a certain pose. Other methods focus on recovering the shape
and detailed garment wrinkles of clothing item in less controlled settings
[55, 20, 13, 36, 38]. While these methods can produce detailed geometry, none
of these methods focuses on the appearance of the item. Another branch of
research aims at 3D reconstructing the whole human including clothing. This
can be achieved by optimization-based [6, 5, 73] or learning-based methods [4,
40] that utilize silhouettes or semantic segmentation of a short monocular
video clip or recently even from single images [46, 7, 59, 24] and point-
clouds [18]. Other methods utilize Kinect-fusion like approaches [33, 49] to
scan people using RGB-D sensors [60, 41, 79, 19]. Having a 3D reconstruction
of body and clothing, it can be used to non-rigidly track the subject [74,
27]. All these methods fuse body and clothing in a single template. Virtual
try-on applications, however, often require separate meshes [29]. Therefore,
methods that reconstruct the naked body shape or both body shape and clothing
have been developed. The naked body shape alone has been estimated using
several RGB images [11] or more accurately using a sequence of clothed scans
[80]. Body shape and garments have been reconstructed simultaneously and
fully-automatically from a series of scans [53], RGB-D images [66], and
recently even from a small set of RGB images [15]. In [76] the authors present
garment and body-shape recovery from a single image but heavily rely on
physical priors and human interaction. In order to enable dynamic virtual try-
on and clothing re-targeting, joint models of clothing and the human body have
been developed [47, 75, 52, 45]. Again, all these works focus mainly or
exclusively on the geometry of the garment, not on their appearance. Other
works also learn to predict correspondences from depth maps to surfaces [67,
54, 71], image to surfaces [37, 8, 26], but they all address different
problems.
Automatic texturing of 3D models from photos has been presented too, but the
shape has to be first aligned with the input image [51, 70]. This alignment is
expensive and error-prone as silhouette and feature cues may be ambiguous. The
most related work here is [16], which maps texture from clothing items to the
3D SCAPE [9] body model. Their focus is not photo-realistic mapping, but
rather to generate synthetic training data with texture variation to learn
discriminative detectors. Their core texture mapping is based on 2D image
warping – unfortunately, the code for this specific part is not available, and
therefore comparison is not possible. However, qualitatively, our results look
significantly more realistic, and we compare to a very similar baseline based
on shape context (SC) matching and Thin Plate Spline (TPS) warping.
Furthermore, our approach runs in real time. In contrast to all previous work,
our method creates textured 3D garments fully automatically, without requiring
prior alignment at test-time, which allows virtual try-on and novel view
synthesis in real-time.

##  3 Method

\begin{overpic}[width=390.25534pt]{figures/overview_new.jpg}
\footnotesize\put(-4.0,20.0){\rotatebox{90.0}{\parbox{113.81102pt}{\centering{\color[rgb]{.25,.25,.25}{
I. Data collection by non-rigid 3D mesh to image alignment}}\@add@centering}}}
\par\put(-3.0,-0.5){\rotatebox{90.0}{\parbox{113.81102pt}{{\color[rgb]{.25,.25,.25}{
II. Pix2Surf Network}}}}} \par\put(14.0,32.2){Garment} \put(29.0,32.2){Mask}
\par\put(43.5,24.5){Silhouette matching} \put(66.9,24.5){Final match}
\put(79.8,24.5){Aligned mesh} \par\put(3.1,-2.0){Garment}
\put(13.7,-2.0){Segmentation Net} \put(33.7,-2.0){Mask} \put(46.2,-2.0){Image
Coordinates} \put(64.3,-2.0){Mapping Net} \put(76.1,-2.0){Correspondences}
\put(90.5,-2.0){Final Texture} \end{overpic} Figure 2: Overview of method: We
build a large paired dataset of 3D garments and online retail store images by
non-rigidly aligning 3D garment templates with automatically computed image
segmentations (I). This allows us to train our model Pix2Surf (II) to predict
correspondences from the UV map of the garment surfaces to image pixel
locations.

Our key idea is to learn a _mapping_ from images to the UV map of the garment,
_without using texture information, but silhouette shape alone_. We first
explain the parametric 3D garment models (Sec. 3.1) we use to regularize an
automatic mesh to image silhouette fitting procedure (Sec. 3.2). Since fitting
is expensive and error-prone, we learn an efficient neural mapping (Pix2Surf),
which transfers the image texture onto the mesh in real time (Sec. 3.3).

###  3.1 Preliminary: Parametric Garment Model

We leverage publicly available 3D garments templates [15] parameterized as
displacements from the SMPL body model [42]. For every garment category
(T-shirt, short pants, long pants), we define a garment template
$T^{G}\in\mathbb{R}^{m\times{3}}$ as a sub-mesh of the SMPL body template
$T\in\mathbb{R}^{n\times{3}}$. An indicator matrix
$\mathbf{I}\in\mathbb{Z}^{m\times{n}}$ evaluates to $\mathbf{I}_{i,j}=1$ if a
garment vertex $i\in\\{1\ldots m\\}$ is associated with a body shape vertex
$j\in\\{1\ldots n\\}$. This correspondence allows representing garments as
displacements $\mathbf{D}\in\mathbb{R}^{m\times{3}}$ from the unposed SMPL
body. Given shape $\boldsymbol{\beta}$ and pose $\boldsymbol{\theta}$, we can
articulate a garment using SMPL:

$T^{G}(\boldsymbol{\beta},\boldsymbol{\theta},\mathbf{D})=\mathbf{I}~{}T(\boldsymbol{\beta},\boldsymbol{\theta})+\mathbf{D}$
(1)
$G(\boldsymbol{\beta},\boldsymbol{\theta},\mathbf{D})=W(T^{G}(\boldsymbol{\beta},\boldsymbol{\theta},\mathbf{D}),J(\boldsymbol{\beta}),\boldsymbol{\theta},\mathbf{W}),$
(2)

with joints $J(\boldsymbol{\beta})$ and linear blend-skinning $W(\cdot)$ with
weights $\mathbf{W}$. Since in this work we keep $\mathbf{D}$ fixed, we denote
the garment model as $G(\boldsymbol{\theta},\boldsymbol{\beta})$. After the
texture has been transferred to the surface, the geometry can be still be
changed with $\mathbf{D}$.

###  3.2 Non-Rigid Garment Fitting to Retail Images

To find a correspondence map between retail images in the web and 3D garments,
we could non-rigidly deform its 3D surface to fit the image foreground. This
is, however, not robust enough as different retailers photograph garments in
different poses, backgrounds, and clothing itself varies in shape. Hence, we
first automatically segment images, and then leverage the parametric garment
model $G(\boldsymbol{\theta},\boldsymbol{\beta})$ defined in Sec. 3.1 to
regularize the fitting process.

####  3.2.1 Automatic segmentation

We use an automated version of GrabCut [58]. Since garments are typically
photographed over simple backgrounds, we obtain an approximate foreground mask
using thresholding. We then run a closing operation to fill the holes on this
mask, and erode it to obtain a prior for ”absolute foreground”. The difference
region between the mask and its eroded version is marked as ”probable
foreground”. Analogously, we obtain ”absolute background”, and ”probable
background” using dilation. Using these prior maps to initialize GrabCut, we
obtain accurate silhouettes without manual annotation.

####  3.2.2 Garment fitting

We fit the garment surface to silhouettes in two stages. In the first stage,
we minimize the following objective

$\mathrm{E_{1}(\boldsymbol{\beta},\boldsymbol{\theta},\boldsymbol{t})}=w_{s}E_{s}+w_{\boldsymbol{\beta}}E_{\beta}+w_{\boldsymbol{\theta}}E_{\boldsymbol{\theta}},$
(3)

w.r.t. garment pose, shape and camera translation
$\mathbf{t}\in\mathbb{R}^{3}$. The objective in Eq. (3) consists of a
silhouette $E_{s}$, a shape regularization term $E_{\boldsymbol{\beta}}$ and a
pose prior term $E_{\boldsymbol{\theta}}$, which we explain in the following.
The different terms are balanced using weights $w_{*}$.

Silhouette term: It is defined as:

$\displaystyle E_{s}(\boldsymbol{\beta},\boldsymbol{\theta},\boldsymbol{t})$
$\displaystyle=$
$\displaystyle\Phi(w_{i}\Psi(I_{r}(G(\boldsymbol{\theta},\boldsymbol{\beta}),\boldsymbol{t}))$
(4)
$\displaystyle+w_{o}\hat{\Psi}(1-I_{r}(G(\boldsymbol{\theta},\boldsymbol{\beta}),\boldsymbol{t}))).$

Here, $\Psi$ and $\hat{\Psi}$ are the distance transform, and the inverse
distance transform, respectively, of the silhouette image, $\Phi$ is a
Gaussian pyramid function, and
$I_{r}(G(\boldsymbol{\theta},\boldsymbol{\beta}),\boldsymbol{t}))$ is the
binary garment silhouette image obtained with a differentiable renderer.
Consequently, the objective in Eq. (4) maximizes overlap between the garment
image and the rendered mesh, and penalizes model leackage into the background.

Shape regularization: In order to regularize the fitting process in Eq. 3, we
use a Mahalanobis prior

$E_{\boldsymbol{\beta}}(\boldsymbol{\beta})=\boldsymbol{\beta}^{T}\Sigma^{-1}_{\boldsymbol{\beta}}\boldsymbol{\beta}$
(5)

on the shape parameters, where $\Sigma^{-1}_{\boldsymbol{\beta}}$ is the
diagonal covariance matrix from the SMPL dataset.

Pose prior: The term penalizes deviations of the pose from an A-pose
$\boldsymbol{\theta}_{A}$

$E_{\boldsymbol{\theta}}(\boldsymbol{\theta})=\|\boldsymbol{\theta}-\boldsymbol{\theta}_{A}\|^{2}$
(6)

To minimize Eq. (3), we initialize the pose $\boldsymbol{\theta}$ with an
A-pose, as this approximates the pose of most garment images on the web.
Additionally, we use scheduling: for shirts we first optimize shape and
translation holding pose fixed and optimize all variables jointly afterwards.
For pants and shorts the scheduling order is reversed. Stage 1 provides us
with a coarse match to the silhouette of the garment, but the final mesh is
restricted by the parametric model
$G(\boldsymbol{\theta},\boldsymbol{\beta},\boldsymbol{t})$. To perfectly match
silhouette boundaries, we non-rigidly deform free form vertices
$\mathbf{G}\in\mathbb{R}^{m\times{3}}$ of a mesh initialized with the
optimized garment result of the first stage
$G(\boldsymbol{\theta},\boldsymbol{\beta})$. Specifically, we optimize a more
sophisticated version of Eq. 3:

$\mathrm{E_{2}}(\mathbf{G},\boldsymbol{\theta},\boldsymbol{\beta})=w^{\prime}_{s}E^{\prime}_{s}+w^{\prime}_{c}E^{\prime}_{c}+w^{\prime}_{l}E^{\prime}_{l}+w^{\prime}_{e}E^{\prime}_{e}+w^{\prime}_{b}E^{\prime}_{b}.$
(7)

$E^{\prime}_{s}$ is the same as in Eq. (4), but now we optimize the free form
vertices $\mathbf{G}$ instead of the latent parameters of the model
$G(\boldsymbol{\theta},\boldsymbol{\beta})$. and $E^{\prime}_{c}$,
$E^{\prime}_{l}$, $E^{\prime}_{e}$ and $E^{\prime}_{b^{\prime}}$ are coupling,
laplacian, edge constraint, and boundary smoothing terms, which we explain
next.

Coupling term: It penalizes deviations of the free form vertices $\mathbf{G}$
from the parametric garment model $G(\boldsymbol{\theta},\boldsymbol{\beta})$:

$E_{c}(\mathbf{G},\boldsymbol{\beta},\boldsymbol{\theta})=\|\mathbf{G}-G(\boldsymbol{\theta},\boldsymbol{\beta})\|^{2}$
(8)

Edge Constraint Term: The belt or waistline in shorts and pants retail images
forms an almost perfect _horizontal line_. We exploit this by matching the top
ring (waistline) of pants and shorts 3D meshes to this horizontal line in
image space. Let $\mathbf{G}_{i}\in\mathcal{R}$ denote the set of top ring
vertices, $\pi(\mathbf{G}_{i})_{y}$ denote the $y$ coordinate of vertex
$\mathbf{G}_{i}$ after projection $\pi(\cdot)$ onto the image, and let
$y_{max}$ denote the $y$ coordinate of the horizontal line in the image. We
incorporate the following penalty:

$E_{e}(\mathbf{G})=\sum_{\mathbf{G}_{i}\in{\mathcal{R}}}\|\pi(\mathbf{G}_{i})_{y}-y_{max}\|^{2}$
(9)

Laplacian Term: In order enforce garment smoothness and minimize distortion,
we include a Laplacian term [65]. Given a mesh with adjacency matrix
$\mathbf{A}\in\mathbb{R}^{m\times{m}}$ , the graph Laplacian
$\mathbf{L}\in\mathbb{R}^{m\times{m}}$ is obtained as
$\mathbf{L}=\mathbf{I}-\mathbf{K}^{-1}\mathbf{A}$ where $\mathbf{K}$ is a
diagonal matrix such that $\mathbf{K}_{ii}$ stores the number of neighbors of
vertex $i$. We minimize the mesh Laplacian:

$E_{l}(\mathbf{G})=||\mathbf{L}\mathbf{G}||^{2}_{F}$ (10)

Boundary Smoothing Term: To ensure that the boundaries remain smooth, we
penalize high second order derivatives along the boundary rings, similar to
[53].
The output of the fitting are 3D garment vertices $\mathbf{G}$, which together
with their faces $\mathbf{F}$ define a deformed mesh
$\mathcal{G}=\\{\mathbf{G},\mathbf{F}\\}$ accurately aligned with image
silhouette.

###  3.3 Learning Automatic Texture Transfer

The fitting of the previous section is accurate, but slow, and fails
sometimes. Hence, we run the fitting method on internet images, and manually
remove the unsuccessful fits. From this data, we train an efficient neural
model, referred to as Pix2Surf. Pix2Surf directly transfers texture from
images to the 3D model surface, based on the silhouette shape alone. Next, we
explain the key components of Pix2Surf, namely, input output representation,
and losses used during training.

####  3.3.1 Pix2Surf: Input and Output Representation

The curated fits of Sec. 3.2 provide dense correspondences from image pixels
$(i,j)$ to the 3D garment surface $\mathcal{G}\subset\mathbb{R}^{3}$. Learning
a mapping to the surface $\mathcal{G}$ embedded in $\mathbb{R}^{3}$ is hard,
and does not allow leveraging fully 2D convolutional neural neutworks. Hence,
we compute a 2D UV-map parameterization (Sec. 3.3.3) of the garment surface,
$u:\mathcal{B}\subset\mathbb{R}^{2}\mapsto\mathcal{G}\subset\mathbb{R}^{3}$,
where $u(\cdot)$ maps $2D$ points $(k,l)$ from the UV space to the surface
$\mathcal{G}$ embedded in $\mathbb{R}^{3}$. In this way, all our targets live
in a UV-space; specifically from the fits, we generate: RGB texture maps
$\mathbf{Y}\in\mathbb{R}^{K\times{L}\times{3}}$ using projective texturing,
and _UV correspondence maps_
$\mathbf{C}\in\mathcal{C}\subset\mathbb{R}^{K\times{L}\times{2}}$, which
store, at every pixel $(k,l)$ of the (front/back) UV map, the $(i,j)$
coordinates of the corresponding image pixel, that is
$\mathbf{C}_{k,l}=(i,j)$. The input representation for the garment images is a
_coordinate mask_ $\mathbf{X}\in\mathcal{X}\subset\mathbb{R}^{M\times{N}\times
2}$, storing at every image pixel location its own coordinates if the pixel
belongs to the foreground $\mathcal{F}$, and $0$ otherwise,
$\mathbf{X}_{ij}=(i,j)\
\forall(i,j)\in\mathcal{F}\,||\,\mathbf{X}_{ij}=(0,0)\,\forall(i,j)\notin\mathcal{F}$.
The foreground mask $\mathcal{F}$ is predicted at test time using a
_segmentation network_ trained using a standard cross-entropy loss–we compute
segmentation labels for training using the automatic – but slow – GrabCut
based method Sec. 3.2.1. For the front view of T-shirts, we additionally ask
annotators to segment out the back portion of the shirt, which is visible in
the front view image. The segmentation network learns to remove this portion
ensuring that the back portion is not mapped to the UV-map. With this we
collect a dataset $\mathcal{D}$ consisting of inputs $\mathbf{X}$, and targets
$\\{\mathbf{Y},\mathbf{C}\\}$.
$\mathcal{D}=\\{\mathbf{X}^{i},\\{\mathbf{Y}^{i},\mathbf{C}^{i}\\}\\}_{i}^{N}$.

####  3.3.2 Pix2Surf: Learning

We could naively attempt to predict the texture maps $\mathbf{Y}$ directly
from images $\mathbf{I}$ using image to image translation, but this is prone
to overfit to image textures as we demonstrate in our experiments (Sec. 4.2) .
Instead, we follow a more geometric approach, and learn a _mapping_
$f(\mathbf{X};\mathbf{w}):\mathcal{X}\mapsto\mathcal{C}$ from coordinate masks
$\mathbf{X}$ to UV correspondence maps $\mathbf{C}$, forcing the network to
reason about the input shape. This effectively, learns to predict, for every
UV map location $(k,l)$, the corresponding pixel coordinates $(i,j)$ in the
image, $\mathbf{C}_{k,l}=(i,j)$. Our key insight and assumption is that this
smooth mapping depends only on silhouette shape $\mathbf{X}$, and not on
texture $\mathbf{I}$. During training, we minimize the following loss

$L_{\mathrm{total}}=\lambda_{\mathrm{reg}}L_{\mathrm{reg}}+\lambda_{\mathrm{perc}}L_{\mathrm{perc}}+\lambda_{\mathrm{recon}}L_{\mathrm{recon}}$
(11)

over the training set $\mathcal{D}$, where each term in the loss is explained
next.
Coordinate Regression Loss:  $L_{\mathrm{reg}}$ evaluates an $L_{2}(\cdot)$
norm of the difference between the network output
$f(\mathbf{X}^{i};\mathbf{w})$ and the pseudo ground truths $\mathbf{C}$
obtained using the silhouette fitting algorithm of Sec. 3.2:

$L_{\mathrm{reg}}=\sum_{i=1}^{N}||f(\mathbf{X}^{i};\mathbf{w})-\mathbf{C}^{i}||^{2}_{2}$
(12)

Reconstruction Loss: To provide more supervision to the network we use a
differentiable sampling kernel [34] to infer a texture map directly from the
correspondence map. We minimize a dense photometric loss between predicted
texture maps and target texture maps $\mathbf{Y}$ obtained with projective
texturing (Sec. 3.3.1):

$L_{\mathrm{recon}}=\sum_{i}^{N}\sum_{k,l}^{KL}||\mathbf{I}\left[f_{k,l}^{1}(\mathbf{X}^{i};\mathbf{w}),f_{k,l}^{2}(\mathbf{X}^{i};\mathbf{w})\right]-\mathbf{Y}^{i}_{k,l}\|_{1}$
(13)

where the original image $\mathbf{I}$ is sampled (using a differentiable
kernel) at locations
$(i,j)=(f_{k,l}^{1}(\mathbf{X}^{i};\mathbf{w}),f_{k,l}^{2}(\mathbf{X}^{i};\mathbf{w}))$
provided by the predicted correspondence map.
Perceptual Loss: $L_{\mathrm{perc}}$ is the perceptual loss as defined in [81]
between
$\mathbf{I}[f(\mathbf{X};\mathbf{w})]\in\mathbb{R}^{K\times{L}\times{3}}$
(tensor notation) and $\mathbf{Y}$.
Once the network predicts a correspondence map for an input image, we use it
and the parallelizable kernel to generate the final image by sampling points
from the input image.

####  3.3.3 Implementation Details

We use Adam optimizer for training both networks. For the segmentation network
we use a UNet with instance normalization and use color jittering in the input
data to improve performance. For Pix2Surf we use a six block ResNet. The
choice of normalization and activation functions is same as [32].
Custom UV Map: Since the artist designed SMPL UV map cuts the garments into
different islands (bad for learning a continuous mapping), we use a custom UV
map for each garment category. We cut the garment surface into front and back
and compute the UV map using Blender This results in two islands (front and
back), which makes the image to UV mapping continuous and hence easier to
learn.

##  4 Experiments

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Figure 3: Textured garments obtained using our method: From the online retail
store images (left), we create textures for three different garment categories
(T-shirt, pants, shorts). We use the textured garments to virtually dress SMPL
(right).

Since we are solving a problem for which there is no ground-truth data, we
evaluate our proposed method qualitatively and with a user study. We show
results for three different garment types (T-shirts, shorts and long-pants),
see Fig. 3. Notice that the texture is nicely mapped without transferring
background. Notice also how the 3D textures are complete without _holes and
seams_. We compare our method against the popular Shape-context with Thin
Plate Spline (TPS) matching baseline, an image-to-image translation which
operates on pixels directly, and image based virtual try on methods.

###  4.1 Dataset

To train our models, we create datasets of garment images by scraping the
websites of clothing stores – specifically the websites of Zalando [3], Tom
Tailor [2] and Jack and Jones [1]. The back view of clothing images is
scarcely available on the internet, so we were unable to create a dataset of
back view images large enough for training mapping networks. We leverage the
fact that the distribution of garment silhouettes is similar for both front
and back views, and either augment the dataset used for training the mapping
from the back view image to the back UV map by combining front view images
with back view images or using _only_ front view images.
We create a datasets consisting of 2267 front images of T-shirts. The networks
pertaining to the back are trained using a dataset of 2267 images of which 964
are back view images and the rest are front images. The front shorts dataset
has 2277 items. We use the same dataset to train the networks pertaining to
the back views. For pants we collect a dataset of 3410 front views. We create
a dataset for back views by horizontally flipping the front view images and
their corresponding silhouettes. The back dataset has 3211 items. The failure
of the optimization based registration method explains the discrepancy between
the number of items in the front and back views. Exploiting these front-back
symmetries turns out to work well in practice, and allows to compensate for
the unavailability of back view images.

###  4.2 Shape Context Matching and Image-to-Image Translation Baselines

We compare our method with shape context (SC) matching plus TPS and image-to-
image translation baselines. We implement the shape context matching baseline
as follows. We first render the garment mesh to obtain a silhouette. To make
the baseline as strong as possible, we first pose the garment in the same pose
as the image. We then warp the garment image to match the rendered garment. We
then lift the texture from the warped image onto the mesh using projective
texturing. Our experiments demonstrate that SC matching and TPS [14] is not
powerful enough to precisely match the contours of the two shapes so the final
texture always has artifacts along the sides. See Fig. 4.
We also compare Pix2Surf to a Pix2Pix [32] baseline, which learns to produce a
texture map from an image. To train Pix2Pix, we use the same texture maps we
use to train Pix2Surf. Unlike Pix2Surf, Pix2Pix produces blurry results and
does not preserve details of the input texture. See Fig. 4.
For further evaluation, we have conducted a _user study_ with 30 participants.
We created 20 videos containing images of reference clothing textures and two
rotating textured 3D avatars “wearing” the reference clothing. One is textured
using a baseline method and the other using Pix2Surf. We ask participants to
choose the better looking avatar. In 100% of all comparisons, the avatar
textured with Pix2Surf was preferred over the one textured using the
baselines.

###  4.3 Towards Photo-Realistic Rendering

Provided with the texture map of a 3D avatar without clothes, Pix2Surf allows
to map the texture atop the avatar from potentially an infinite number of
clothing items scraped from online websites, _automatically_ and in _real
time_. See Fig. 5

|  |  \begin{overpic}[width=433.62pt,height=95.3925pt]{figures/comps/pants/dp1/warps/im_19.png} \put(0.0,0.0){\color[rgb]{1,0,0} \polygon(7,64)(7,80)(15,80)(15,64) \polygon(30,65)(30,80)(35,80)(35,65) \polygon(11,50)(15,50)(15,63)(11,63) \polygon(10,20)(10,43)(14,43)(14,20) \polygon(23,43)(27,43)(27,15)(23,15) \par} \par\end{overpic} |  |  |  |  \begin{overpic}[width=433.62pt,height=95.3925pt]{figures/comps/pants/dp2/warps/im_19.png} \put(0.0,0.0){\color[rgb]{1,0,0} \polygon(7,64)(7,80)(15,80)(15,64) \polygon(30,65)(30,80)(35,80)(35,65) \polygon(11,50)(15,50)(15,63)(11,63) \polygon(10,20)(10,43)(14,43)(14,20) \polygon(23,43)(27,43)(27,15)(23,15) } \end{overpic} |  |  |  \begin{overpic}[width=433.62pt,height=95.3925pt]{figures/comps/pants/dp3/warps/im_19.png} \put(0.0,0.0){\color[rgb]{1,0,0} \polygon(7,64)(7,80)(15,80)(15,64) \polygon(30,65)(30,80)(35,80)(35,65) \polygon(11,50)(15,50)(15,63)(11,63) \polygon(10,20)(10,43)(14,43)(14,20) \polygon(23,43)(27,43)(27,15)(23,15) \par} \par\end{overpic} |
---|---|---|---|---|---|---|---|---|---|---

|  |  \begin{overpic}[width=433.62pt,height=95.3925pt]{figures/comps/shorts/dp1/warps/im_19.png} \put(0.0,0.0){\color[rgb]{1,0,0} \polygon(7,64)(7,80)(15,80)(15,64) \polygon(30,65)(30,80)(35,80)(35,65) \polygon(11,50)(15,50)(15,63)(11,63) \polygon(8,30)(8,45)(12,45)(12,30) } \par\end{overpic} |  |  |  |  \begin{overpic}[width=433.62pt,height=95.3925pt]{figures/comps/shorts/dp2/warps/im_19.png} \put(0.0,0.0){\color[rgb]{1,0,0} \polygon(7,64)(7,80)(15,80)(15,64) \polygon(30,65)(30,80)(35,80)(35,65) \polygon(11,50)(15,50)(15,63)(11,63) \polygon(8,30)(8,45)(12,45)(12,30) } \par\end{overpic} |  |  |  \begin{overpic}[width=433.62pt,height=95.3925pt]{figures/comps/shorts/dp3/warps/im_19.png} \put(0.0,0.0){\color[rgb]{1,0,0} \polygon(7,64)(7,80)(15,80)(15,64) \polygon(30,65)(30,80)(35,80)(35,65) \polygon(11,50)(15,50)(15,63)(11,63) \polygon(8,30)(8,45)(12,45)(12,30) } \par\end{overpic} |
---|---|---|---|---|---|---|---|---|---|---

Figure 4: We compare our results against textures obtained via SC matching and
an image-to-image translation baseline. SC matching produces textures with
artefacts and Pix2Pix produces blurry results. Left: Pix2Pix, Middle: SC
matching, Right: Pix2Surf

|  |  |  |
---|---|---|---|---

|  |  |  |
---|---|---|---|---

Figure 5: Photo-realistic virtual try-on. Given the garment images and a 3D
avatar with texture, we show our automatically textured garments on top.

Using these photo-realistic renderings, we are able to compare our method with
Image based virtual try-on method such as VITON [28] and CP-VTON [69]. Since
our method requires a 3D avatar as input, to be able to compare, we first
render a photo-realistic image using our method, and use VITON to change the
upper garment of the rendering. Fig. 6 shows that VITON and CP-VTON work
reasonably well for some poses, but fail badly for extreme poses while our
method does not. We note that these methods are related but conceptually
different compared to our method, so a direct fair comparison is not possible.
These methods are also not explicitly trained to handle such poses, but the
figure illustrates a general limitation of image based virtual try-on
approaches to generalize to novel view-points and poses. In stark contrast, we
only need to generalize to silhouette shape variation, and once the texture is
mapped to the 3D avatar, novel poses and viewpoints are trivially generated
applying transformation in 3D.

|  |  |  |
---|---|---|---|---

o X[c]X[c]X[c]X[c]X[c] Input image | Target Cloth | CP-VTON | VITON | Ours
---|---|---|---|---

Figure 6: Comparison with VITON and CP-VTON

##  5 Conclusions and Future Work

We have presented a simple yet effective model that learns to transfer
textures from web images of clothing to 3D garments worn by virtual humans.
Our experiments show that our non-linear optimization method is accurate
enough to compute a training set of clothing images aligned with 3D mesh
projections, from which we learn a direct mapping with a neural model
(Pix2Surf). While the optimization method takes up ten minutes to converge,
Pix2Surf runs in real time, which is crucial for many applications such as
virtual try-on. Our key idea is to learn the correspondence map from image
pixels to a 2D UV parameterization of the surface, _based on silhouette shape
alone instead of texture_ , which makes the model invariant to the highly
varying textures of clothing and consequently generalize better. We show
Pix2Surf performs significantly better than classical approaches such as 2D
TPS warping (while being orders of magnitude faster), and direct image-to-
image translation approaches.

We believe our model represents an important step towards learning a
generative model of textures directly in 3D. We plan to address this in the
future since it is lacking in current models like SMPL [42]. We focused on
texture, and assume garment geometry is given, but we think it should be
possible to infer geometry from images. Since clothing is laid out on a flat
surface on web photographs, geometry inference will require modelling how 3D
garments deform when they are flattened out on a surface.  Acknowledgments: We
thank Bharat Lal Bhatnagar and Garvita Tiwari for in-depth discussions and
assistance in our data preparation. This work is partly funded by the Deutsche
Forschungsgemeinschaft (DFG) - 409792180 (Emmy Noether Programme, project:
Real Virtual Humans).

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##  Appendix A Non-Rigid Garment Fitting

In all our experiments, we use the neutral gendered SMPL model. For each
garment class we only use a subset of angles in the input axis-angle
representation of pose as our optimization variables. For shirts, we use only
the pose parameters corresponding to the shoulder joints - numbered
$13,14,16,17$ in the canonical ordering. For shorts, joint corresponding to
the hip joint - joints $1$ and $2$ are used. Joints $1,2,3,4$ which correspond
to the hip and knee joints are used while registering pants.
The weights used in the optimization based registration are given below. For
registration the front of T-shirts, we set
$w_{\boldsymbol{\beta}}=3.0,w_{s}=100,w_{\boldsymbol{\theta}}=1.0,w^{\prime}_{s}=2.0,w^{\prime}_{c}=0.3,w^{\prime}_{b}=2.0,w^{\prime}_{l}=30.0,w^{\prime}_{e}=0$.
The same weights when registration the back are:
$w_{\boldsymbol{\beta}}=5.0,w_{s}=100,w_{\boldsymbol{\theta}}=0,w^{\prime}_{s}=2.0,w^{\prime}_{c}=0.3,w^{\prime}_{b}=2.0,w^{\prime}_{l}=30.0,w^{\prime}_{e}=0$.
For the front view of shorts, we set $w_{\boldsymbol{\beta}}=4.0,w_{s}=100$
$w_{\boldsymbol{\theta}}=30.0$ in the first stage when shape is not being
optimized and $w_{\boldsymbol{\theta}}=20$ when it is.
$w^{\prime}_{s}=100,w^{\prime}_{c}=4.0,w^{\prime}_{b}=2.0,w^{\prime}_{l}=45.0,w^{\prime}_{e}=0.001$.
For the back $w_{\boldsymbol{\beta}}=3.0,w_{s}=160.0$
$w_{\boldsymbol{\theta}}=40.0$ when shape is not an optimization variable and
$w_{\boldsymbol{\theta}}=30$ when it is.
$w^{\prime}_{s}=100,w^{\prime}_{c}=3.8,w^{\prime}_{b}=6.0,w^{\prime}_{l}=75.0,w^{\prime}_{e}=0.01$.
For both front and back of pants, we set
$w_{\boldsymbol{\beta}}=4.0,w_{s}=100$ $w_{\boldsymbol{\theta}}=40.0$ in the
stage when just the pose is optimized $w_{\boldsymbol{\theta}}=5$ when both
shape and pose are optimization variables.
$w^{\prime}_{s}=100,w^{\prime}_{c}=0.5,w^{\prime}_{b}=2.0,w^{\prime}_{l}=45.0,w^{\prime}_{e}=0.1$.

##  Appendix B Learning Automatic Texture Transfer

For all our experiments, we use an Adam optimizer with $\beta 1=0.5$ and
$\beta 2=0.999$. We use a constant learning rate of $0.0001$. All mapping
networks are trained for $200$ epochs and all segmentation networks are
trained for $20$ epochs. During the training of the segmentation networks we
use heavy colour jittering to prevent the network from overfitting to the
textures in the training data. The weights are set as
$\lambda_{reg}=\lambda_{recon}=\lambda_{perc}=1$

##  Appendix C Baseline Implementation

For the SC-matching TPS based baseline, all parameters are set to their
default values. The pix2pix baseline is implemented with the weight
$\lambda_{gan}=1$. We observed that increasing the weight of the adversarial
loss worsens performance.

##  Appendix D Custom UV Maps

All the UV maps used are created in blender. We obtained best performance when
the shape of the UV maps are similar to the shapes of the input garments. The
exact UV maps used can be seen in Fig. 7

Figure 7: Custom UV maps

##  Appendix E Results

In Fig. 8 and Fig. 9 more textures mapped by Pix2Surf are displayed atop SMPL.

Once the garment textures are mapped to their corresponding texture maps, they
can be rendered atop SMPL in different shapes and poses as seen in Fig. 10.

|  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---

|  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---

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|  |  |  |  |  |  |  |  |  |
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|  |  |  |  |  |  |  |  |  |
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Figure 8: Textures mapped by Pix2Surf rendered atop SMPL

|  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---

|  |  |  |  |  |  |  |  |  |
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|  |  |  |  |  |  |  |  |  |
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|  |  |  |  |  |  |  |  |  |
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|  |  |  |  |  |  |  |  |  |
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|  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---

Figure 9: Textures mapped by Pix2Surf rendered atop SMPL Figure 10: Once
textures are mapped to their corresponding texture maps, they can be rendered
atop SMPL in different shapes and poses


# Study of strongly intense quantities and robust variances in multi-particle
production at LHC energies

###### Abstract

The strongly intense quantities and robust variances in processes of multi-
particle production in pp and AA interactions at LHC energies was studied. The
Monte Carlo and analytic modelling of these quantities in the framework of a
quark-gluon string model were implies. The string fusion effects were also
taken into account by implementing of a lattice (grid) in the impact parameter
plane. Strongly intensive variable $\Sigma(n_{F},\ n_{B})$ was calculated for
different energies for two values of the width of the observation rapidity
windows as a function of the distance between the centres of this windows.
Scaled variance $\omega_{n}$ and robust variance $R_{n}$ for different
energies and for different width of the observation rapidity window was
calculated by MC simulations.

S. Belokurova111E-mail<EMAIL_ADDRESS>
PACS: 12.40.−y

## Introduction

As is well known, at the present the quantum chromodynamics does not enable to
describe numerically the soft part of multi-particle production. The different
versions of the QCD-inspired quark-gluon string model are used for a
description of this component of hadronic interaction at high energy. One of
the most popular approaches is the string model [1, 2, 3, 4]. In this model at
first stage the color quark-gluon strings are formed. At second stage the
hadronization of these strings produces the observed hadrons.

## Generation of the string configuraton

At first we formulate the MC algorithm based on the string model. In our
approach we will consider that each cut pomeron corresponds to formation of
two strings [4]. To take into account string fusion one should know not only
the mean number of pomerons in $pp$ collisions at a given impact parameter
$b$, but also the event by event distribution of the number of pomerons around
this mean value. This distribution at a given value of the impact parameter
$b$ at $N\geq 1$was chosen in the following form:

$\widetilde{P}(N,b)={P(N,b)}/[1-P(0,b)],$ (1)

where $P(N,b)$ is the poissonian distribution with some parameter
$\overline{N}(b)$:

$P(N,b)=e^{-\overline{N}(b)}{\overline{N}(b)^{N}}/{N!}\ \ .$ (2)

The difference of our distribution $\widetilde{P}(N,b)$ (1) from the
poissonian one (2) is only in excluding of the case $N=0$:
$\widetilde{P}(0,b)=0$, which corresponds to the absence of the non-
diffractive scattering.

According to [12], we suppose that in the proton-proton collision at the
impact parameter $b$ the string density in transverse plane at a point
$\vec{s}$ is proportional to

$w_{str}(\vec{s},\vec{b})\sim\frac{1}{\sigma_{\\!pp}(b)}T(\vec{s}-\vec{b}/2)T(\vec{s}+\vec{b}/2)\
,$ (3)

where the $T(\vec{s})$ is the partonic profile function of nucleon. We will
use for the partonic profile function of nucleon the simplest gaussian
distribution:

$T(s)=\frac{e^{-s^{2}/\alpha^{2}}}{\pi\alpha^{2}}\ .$ (4)

Substituting (4) in (3) one gets

$w_{str}(\vec{s},\vec{b})\sim\frac{1}{\sigma_{\\!pp}(b)}e^{-2s^{2}/\alpha^{2}}e^{-b^{2}/2\alpha^{2}}\
.$ (5)

Simultaneously in this approach we have
$\overline{N}(b)=N_{0}e^{-b^{2}/2\alpha^{2}}$, where the parameter $N_{0}$
depends on initial energy.

As has been shown in [12], in the framework of this assumptions the average
number of pomerons $\left\langle N_{pom}(E)\right\rangle$, the scaled variance
of number of pomerons $\omega_{N_{pom}}(E)$, the cross-section of non-
diffractive $pp$ interaction $\sigma_{\\!pp}$ and the probability $P(N)$ to
have $N$ cut pomerons in a non-diffractive pp collision has the following
form:

$\left\langle
N_{pom}(E)\right\rangle=\frac{N_{0}}{E_{1}(N_{0})+\gamma+\ln{N_{0}}}\ ,\
E_{1}(x)=\int_{x}^{\infty}\frac{e^{-t}}{t}dt,$ (6)
$\omega_{N_{pom}}(E)=1+\frac{N_{0}}{2}-\left\langle N\right\rangle_{pom}(E)\
,$ (7)
$\sigma_{\\!pp}=2\pi\alpha^{2}\left[E_{1}(N_{0})+\gamma+\ln{N_{0}}\right],$
(8)
$P(N)=\frac{2\pi\alpha^{2}}{\sigma_{\\!pp}N}\left[1-e^{-N_{0}}\sum_{l=0}^{N-1}N_{0}^{l}/l!\right].$
(9)

The last formula shows that this approach is equivalent to the Gribov-Regge
approach, as it was noted in [12]. This enables to connect the parameters
$N_{0}$ and $\alpha$ of string fusion model, which describe the dependence of
the mean number of pomerons on the impact parameter $b$ with the parameters of
the pomeron trajectory and its couplings to hadrons:

$\alpha=\sqrt{\frac{2\lambda}{C}}/5.05\ fm,\
N_{0}=\frac{2\gamma_{pp}C}{\lambda}\exp(\Delta\xi),\
\lambda=R_{pp}^{2}+\alpha^{\prime}\xi,\ \xi=\ln(s/1GeV^{2})\ .$ (10)

Here $\Delta$ and $\alpha^{\prime}$ are the intercept and the slope of the
pomeron trajectory. The parameters $\gamma$ and $R_{pp}$ characterize the
coupling of the pomeron trajectory with the initial hadrons. The quasi-eikonal
parameter $C$ is related to the small-mass diffraction dissociation of
incoming hadrons.

For the case of $pp$ collisions the following numerical values of the
parameters were chosen to describe the multiplicity and the non-diffractive
cross section:

$\Delta=0.2,\ \ \ \alpha^{\prime}=0.05\ GeV^{-2},\ \ \ \gamma_{pp}=1.035\
GeV^{-2},\ \ \ R_{pp}^{2}=3.3\ GeV^{-2},\ \ \ C=1.5.$ (11)

The string density in the transverse plane increases with the growth of
initial energy or going to collisions of nuclei, and it is necessary also to
take into account an interaction between the strings [5, 6], which leads to
the formation of fused strings [7, 8]. To simplify the account of string
fusion processes in our calculations we use the version of the model with the
transverse lattice (grid) [9, 10, 11]. In the model the transverse plane is
divided into cells, which area is equal to the string transverse cross-
section. It is supposed that the strings with the centers in the same cell are
fused.

In the framework of the string fusion model [7, 8] the dependence of the
average number of particles formed from decay of the fused strings in the cell
on the number of strings, $\eta_{i}$, in the rapidity observation window of
width $\delta y$ have the following form:

$\overline{n}\left(\eta_{i}\right)=\mu_{0}\delta y\sqrt{\eta_{i}},$ (12)

where $\mu_{0}$ is the average number of a particles produced from the
hadronizations of the one string in the window of width $\delta y=1$. In our
calculations the following numerical value of the string radius was chosen:

$r_{str}=0.2\ fm.$ (13)

We assume that the number of particles produced from the hadronizations of the
strings in i-th cell in the rapidity observation window of width $\delta y$ is
distributed over the negative binomial distribution (NBD) with mean value (12)
and scaled variance:

$\omega_{\mu}(\delta y,\ \eta)=1+\delta y\mu_{0}^{\eta}J_{FF}^{\eta},$ (14)

see the paper [14], where

$J_{FF}^{\eta}=\frac{1}{(\delta y_{F})^{2}}\int_{\delta
y_{F}}dy_{1}\int_{\delta y_{F}}dy_{2}\Lambda_{\eta}(y_{1}-y_{2})$ (15)

and $\Lambda_{\eta}(\Delta y)$ is the two-particle (pair) correlation
function, which was chosen in the simplest form

$\Lambda_{\eta}(\Delta y)=\Lambda_{0}^{\eta}e^{-\frac{|\Delta
y|}{y_{corr}^{\eta}}},$ (16)

$y_{corr}^{\eta}$ is a characteristic correlation length in the rapidity
space. In accordance with the physical picture of the string fusion in the
model we assume that the dependence of the parameters on the string density,
$\eta$, is as follows

$y_{corr}^{\eta}=\frac{y_{1}}{\sqrt{\eta}},\
\mu_{0}^{\eta}=\mu_{0}\sqrt{\eta}.$ (17)

For the correlation function chosen in the simplest form, (16), the integral
$J_{FF}^{\eta}$ can be calculated explicitly:

$J_{FF}^{\eta}=\frac{2\Lambda_{0}^{\eta}}{(\delta
y)^{2}}y_{corr}^{\eta}\left(\delta y-y_{corr}^{\eta}\left(1-e^{-\frac{\delta
y}{y_{corr}^{\eta}}}\right)\right).$ (18)

Parameters $y_{1}$ and $\Lambda_{0}^{\eta}$ was chosen to obtain a
correspondence with the results for $\Sigma(n_{F},\ n_{B})$ obtained in [16,
17, 18] using the pair correlation function extracted in [14] in the
approximation of identical strings from ALICE [13] experimental data. The
value of the parameter $\mu_{0}$ was chosen to describe $dN/dy$ distribution
at different energies taken from [15, 21, 22, 23, 24]:

$\mu_{0}=0.7,\ \ y_{1}=2.7,\ \ \Lambda_{0}^{\eta}=0.8.$ (19)

Based on the foregoing, the MC algorithm was elaborated. The developed
algorithm was used to generate the events at the following energies: 60 Gev,
900 Gev, 7 TeV, 13 TeV.

## Calculation of the $\Sigma(n_{F},\ n_{B})$, $\omega_{n}$, $R_{n}$

The definitions of of the strongly intensive variable $\Sigma(n_{F},\ n_{B})$,
scaled variance $\omega_{n}$ and robust variance $R_{n}$ are as follows [19,
20]:

$\Sigma(n_{F},\ n_{B})\equiv\frac{\left\langle
n_{F}\right\rangle\omega_{n_{B}}+\left\langle
n_{B}\right\rangle\omega_{n_{F}}-2cov(n_{F},\ n_{B})}{\left\langle
n_{F}\right\rangle+\left\langle n_{B}\right\rangle},$ (20)
$\omega_{n}\equiv\frac{\left\langle n^{2}\right\rangle-\left\langle
n\right\rangle^{2}}{\left\langle n\right\rangle},\
R_{n}\equiv\frac{\omega_{n}-1}{\left\langle n\right\rangle}=\frac{\left\langle
n(n+1)\right\rangle}{\left\langle n\right\rangle^{2}}-1.$ (21)

As it was shown in [25], for the strongly intensive variable $\Sigma(n_{F},\
n_{B})$ (20) the following expression can be obtained:

$\Sigma(n_{F},\ n_{B})=\sum_{\eta=1}^{\infty}\frac{\left\langle
n\right\rangle_{\eta}}{\left\langle n\right\rangle}\Sigma_{\eta}(\mu_{F},\
\mu_{B}),\ \Sigma_{\eta}(\mu_{F},\ \mu_{B})=1+\mu_{0}^{\eta}\delta
y\left[J_{FF}^{\eta}-J_{FB}^{\eta}\right],$ (22)

where $\Sigma_{\eta}(\mu_{F},\ \mu_{B})$ is the variable $\Sigma$ for a cell
with $\eta$ strings, $\left\langle n\right\rangle_{\eta}$ is the average
numbers of particles produced from the decay of all string clusters with
$\eta$ strings, $\left\langle n\right\rangle$ — the multiplicity,

$J_{FB}^{\eta}=\frac{1}{\delta y_{F}\delta y_{B}}\int_{\delta
y_{F}}dy_{1}\int_{\delta y_{B}}dy_{2}\ \Lambda_{\eta}(y_{1}-y_{2}).$ (23)

For the correlation function of the simplest form (16), we have

$J_{FB}^{\eta}=\frac{\Lambda_{0}^{\eta}\left(y_{corr}^{\eta}\right)^{2}}{(\delta
y)^{2}}e^{\frac{-\Delta y}{y_{corr}^{\eta}}}\left(e^{\frac{\delta
y}{y_{corr}^{\eta}}}-2+e^{\frac{-\delta y}{y_{corr}^{\eta}}}\right),$ (24)

where $\delta y=\delta y_{F}=\delta y_{B}$ is the rapidity observation window,
$\Delta y$ is the rapidity distance between the centers of observation windows
(formula (24) was obtained in the case $\Delta y>\delta y$).

## Results

For the calculation of the $\Sigma(n_{F},\ n_{B})$, $\omega_{n}$ and $R_{n}$
formulas (22) and (21) was used. $\omega_{n}$ and $R_{n}$ were studied as a
function of rapidity width of the observation windows $\delta y$ for min.bias
$pp$ interactions at different energies. This dependence is shown in the fig.
1.

|
---|---
Figure 1: Results for scaled variance $\omega_{n}$ and robust variance $R_{n}$
calculated with help of (21) as a function of the rapidity width of the
observation window $\delta y$ for min.bias $pp$ interactions at energies 60 -
13000 GeV

$\Sigma(n_{F},\ n_{B})$ were studied as a function of the rapidity distance
between the observation windows $\Delta y$ for min.bias $pp$ interactions at
different energies for rapidity width of the observation windows $\delta
y=0.2$ and $\delta y=0.4$. Results for this study is shown on the fig. 2. As
one can see $\Sigma(n_{F},\ n_{B})$ increase with initial energy of pp
collision.

|
---|---
Figure 2: Results for the strongly intensive variable $\Sigma(n_{F},\ n_{B})$ calculated with help of (22) as a function of the rapidity distance between the observation windows $\Delta y$ for min.bias $pp$ interactions at energies 60 - 13000 GeV for rapidity width of the observation windows $\delta y=0.2$ and $\delta y=0.4$. |
---|---
|
Figure 3: Results for the strongly intensive variable $\Sigma(n_{F},\ n_{B})$ calculated with help of (22) as a function of the rapidity distance between the observation windows $\Delta y$ for different centralities of $pp$ interactions at energies 60 - 13000 GeV for rapidity width of the observation windows $\delta y=0.2$. |
---|---
|
Figure 4: The same as in fig.3 but for rapidity width of the observation
windows $\delta y=0.4$.

The dependence of the strongly intensive variable $\Sigma(n_{F},\ n_{B})$ on
the collision centrality was also investigated at different energies. As one
can see in the figs. 3, 4 $\Sigma(n_{F},\ n_{B})$ grows with increasing
centrality of the pp collision.

## Summary

The quark-gluon string model approach and the MC algorithm for the analysis of
high energy pp collisions were developed.

The strongly intensive variable $\Sigma(n_{F},\ n_{B})$ was calculated for
different energies for two values of the width of the observation rapidity
windows as a function of the distance between centers of these windows. It has
been shown that $\Sigma(n_{F},\ n_{B})$ increases with both initial energy of
pp collision and centrality.

As is clear from formula (22), in both cases this is caused by the growth of
the proportion of string clusters with a larger number of merged strings.
Recall that, as it was shown in [16, 17, 18, 25], the value of the variable
$\Sigma(n_{F},\ n_{B})$ depends only on the properties of sources and the
proportion in which they are formed in a collision.

Using the obtained MC simulation results the scaled variance $\omega_{n}$ and
robust variance $R_{n}$ in pp collisions for different energies and for
different width of the observation rapidity window were also calculated.

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