# Invariance of deep image quality metrics to affine transformations

Nuria Alabau-Bosque1,2, Paula Daudén-Oliver2, Jorge Vila-Tomás2, Valero
Laparra2, Jesús Malo2
1ValgrAI: Valencian Grad. School Research Network of AI, València, 46022,
Spain
2Image Processing Lab, Universitat de València, Paterna, 46980, Spain

###### Abstract

Deep architectures are the current state-of-the-art in predicting subjective
image quality. Usually, these models are evaluated according to their ability
to correlate with human opinion in databases with a range of distortions that
may appear in digital media. However, these oversee affine transformations
which may represent better the changes in the images actually happening in
natural conditions. Humans can be particularly invariant to these natural
transformations, as opposed to the digital ones.
In this work, we evaluate state-of-the-art deep image quality metrics by
assessing their invariance to affine transformations, specifically: rotation,
translation, scaling, and changes in spectral illumination. Here _invariance_
of a metric refers to the fact that certain distances should be neglected
(considered to be zero) if their values are below a threshold. This is what we
call _invisibility threshold_ of a metric. We propose a methodology to assign
such invisibility thresholds for any perceptual metric. This methodology
involves transformations to a distance space common to any metric, and
psychophysical measurements of thresholds in this common space. By doing so,
we allow the analyzed metrics to be directly comparable with actual human
thresholds. We find that none of the state-of-the-art metrics shows human-like
results under this strong test based on invisibility thresholds.
This means that tuning the models exclusively to predict the visibility of
generic distortions may disregard other properties of human vision as for
instance invariances or invisibility thresholds. The data and code is publicly
available to test other
metrics111https://github.com/Rietta5/InvarianceTestIQA.

###### Index Terms:

Perceptual Metrics, Perception Thresholds, Invariance to Affine
Transformations

##  I Introduction

Deep architectures are the current state-of-the-art in predicting subjective
image quality. These models, usually referred to as _perceptual metrics_ , are
often used as measures to optimize other image processing tasks [1, 2] so
assessing their performance is critical. Usually, these models are evaluated
(or even tuned) according to their ability to correlate with human opinion in
databases including a wide range of generic distortions [3, 4]. In fundamental
terms, this means trying to predict the visibility of generic distortions.
However, focus on generic distortions may be a problem to take into account
other relevant phenomena [5], and human vision may also be described in terms
of perceptual constancies or invariances [6, 7, 8, 9, 10, 11].

In this regard, considering the structural analysis of distortions and high-
level interpretation of human vision, people have suggested that humans are
mainly invariant to transformations which do not change the structure of the
scenes [12]. Affine transformations (rotations, translations, scalings, and
changes in spectral illumination) are examples of distortions that do not
change the structure of the scene. Therefore, humans should be relatively
tolerant to them, and the corresponding models to assess image similarity
should be invariant to these transformations too.

In fact, the spirit of the influential SSIM was focused on measuring changes
of structure [13] to achieve invariances to irrelevant transformations [14].
Moreover, Wang and Simoncelli [12] decomposed generic distortions into
_structural_ and _non-structural_ components so that the part not affecting
the structure (e.g. affine transformations) could be processed, and weighted,
differently. On the other hand, metrics with bio-inspired, explainable
architecture [15, 16, 17, 18, 19, 20, 5, 4] work in multi-scale / multi-
orientation representations where invariances can be introduced by means of
appropriate poolings as in the representations [21]. This sort of poolings are
thought to happen in the visual brain leading to invariances and texture
metamers [22].

However, current state-of-the-art deep-architectures for image quality [3] do
not address the invariance problem in any way, while examples that try to
apply the SSIM concept in deep-nets [23] do not use invariances in simple or
explicit ways. As a result, the analysis of invariance in deep image quality
metrics remains an open question.

In this work, we compare the ability of metrics to be invariant to affine
transformations in the same way as humans are. In particular, we propose to
evaluate the metrics from the point of view of human detection thresholds: by
assessing the invariance of the metrics to image transformations that are
irrelevant (or invisible) to human observers. For example, the classical
literature on visual thresholds determines the intensity of certain affine
transformations which is invisible for humans [24, 25, 26, 27]. The sizes of
invisibility thresholds are related to the more general concept of
_invariance_ to transformations. By definition, transformations whose
intensity is below the threshold are invisible to the observer. Then one can
say that, in this region, the observer is invariant to the transform. Here we
propose a methodology that allows us to assign specific invisibility
thresholds per metric. This proposal uses transduction functions to a common
internal representation based on a subjectively rated database and a
psychophysically measured threshold in this representation.

Then, we can (1) assess if the thresholds for the metrics are comparable to
those found for human observers, and (2) assess if the sensitivities of a
metric for the different distortions follow the same order as the human
sensitivities (for instance humans are more sensible to rotation changes than
to illumination changes). This proposed evaluation of invariance to
distortions is a necessary complement to the conventional evaluation of the
visibility of distortions because, as we will see, none of the studied metrics
presents human-like thresholds nor sensitivities for all the transformations
considered.

The structure of the paper is as follows: Section II describes the proposed
methodology to assess the human-like behavior of metric invariances. The
proposed methodology consists of comparing the detection thresholds for humans
and metrics. This proposal depends on several concepts (transduction
functions, psychophysical thresholds for humans, theoretical thresholds for
metrics… ) that will be detailed in this section too. Section III describes
the experimental setting, and Section IV considers the results on the
thresholds and the sensitivities. Finally, the discussion and conclusions are
presented in Section V and Section VI, respectively.

##  II Proposed Methodology: Thresholds and Sensitivities

Given an original image, $i$, it can be distorted through certain transform
whose intensity depends on a parameter $\theta$, $i^{\prime}=T_{\theta}(i)$.
Image quality metrics are models that try to reproduce the subjective
sensation of distance between the original image and the distorted image,
$d(i,i^{\prime})$. Human observers are unable to distinguish between $i$ and
$i^{\prime}$, i.e. they are invariant to transform $T_{\theta}$, if its
intensity is below a (human) threshold, $\theta_{\tau}^{\textrm{H}}$.

While invariance thresholds in human observers, $\theta_{\tau}^{\textrm{H}}$,
are easy to understand and measure [24, 25, 26, 27], they are not obvious to
define in artificial models of perceptual distance. The reason is simple: for
the usual image quality models any non-zero image distortion leads to non-zero
variations in the distance. In this situation where artificial distances are
real-valued functions one should define a value, _the threshold distance_ ,
$\mathcal{D}_{\tau}$, below which the difference between the images could be
disregarded (or taken as zero). Once this _threshold distance_ is available
(eventually in a common scale for all metrics) one can translate this
threshold to the axis that measures the distortion intensity, $\theta$, and
hence obtain the threshold of the metric, $\theta_{\tau}^{M}$, in the same
units that has been measured for humans, $\theta_{\tau}^{H}$. See an
illustration of this concept in Figure 1.

Figure 1: Illustration of the methodology: comparison of metric vs human
thresholds. For each distortion, parametrized with $\theta$, we evaluate three
different metrics (continuous curves in blue, red, and green) in common units
of subjective distance ($\mathcal{D}$, the normalized Mean Opinion Score).
Given $N$ competing metrics, these curves are the _transduction functions_
that map the physical description of the distortion intensity, $\theta$ into a
common scale of subjective distance, $\mathcal{D}_{M}=g_{M}(\theta)$, with
$M=1,\ldots,3$. The yellow lines show the human _threshold in the internal
distance representation_ , $\mathcal{D}_{\tau}$, experimentally measured in
this work with a psychometric function. We assign the _metric thresholds_ from
the generic human threshold using the inverse of the transduction functions
for the different metrics: $\theta_{\tau}^{M}=g_{M}^{-1}(\mathcal{D}_{\tau})$.
Given the fact that $\mathcal{D}_{\tau}$ has an experimental uncertainty, the
inverse gives us an interval - dotted lines- on the x-axis where the human
threshold should fall for each metric and this particular distortion. For any
number $N$ of metrics we propose to compare the _metric thresholds_ ,
$\theta_{\tau}^{M}$, for $M=1,\ldots,N$, with the actual _human threshold_ in
intensity units, $\theta_{\tau}^{H}$. In this case, $\theta_{\tau}^{H}$ (the
black line), falls within the red interval. Therefore, the metric 2 is the one
with the human-like behavior.

The diagram in Figure 1 displays the relation between the physical description
of the intensity of certain distortions in abscissas (the parameter $\theta$,
e.g. the angle in a rotation transform), and the common internal description
of the perceived distance in ordinates (the distance $\mathcal{D}$, e.g. the
normalized Mean Opinion Score units explained below). This relation between
the physical description of the intensity and common perceptual distance is
what we call _transduction function_. The example displays three transduction
functions, $\mathcal{D}_{M}=g_{M}(\theta)$, with $M=1,2,3$, in blue, red, and
green, for the three corresponding metrics. Transduction functions are
monotonic: the application of a progressively increasing distortion along the
abscissas leads to monotonic increments in distances in ordinates. The
_threshold distance_ , $\mathcal{D}_{\tau}$, in the internal representation is
plotted in orange in Figure 1. This threshold may be uncertain (as represented
by the central value, solid line, and the quartile limits, in dashed lines),
but the empirical transduction functions can be used to put this internal
threshold back in the axis that describes the distortion intensity:
$\theta_{\tau}^{M}=g_{M}^{-1}(\mathcal{D}_{\tau})$. In this way, one can check
if the actual invisibility threshold measured for humans (black line) is
consistent with the threshold interval deduced for each metric. In our
illustration, the metric 2 (red lines) is the only one compatible with human
behavior.

Using the above, on the one hand, we propose to evaluate the alignment between
the metric models and human observers by comparing $\theta_{\tau}^{M}$ versus
$\theta_{\tau}^{H}$. This is a strict comparison that only depends on the
experimental uncertainty of the thresholds. On the other hand, we can define
an alternative (less strict) comparison by considering the order among the
sensitivities for the different distortions in humans and models. In
particular, one can define the sensitivity of a metric for a distortion as the
variation of the transduction function in terms of the energy (or Mean Squared
Error) of the distortion introduced by the transform [28]. The human
sensitivity in detection is classically defined to be proportional the inverse
of the energy required to see the distortion [29].

The proposed comparisons of metric vs human thresholds and metric vs human
sensitivities are general as long as one can address the following issues:

  1. (a)

The transduction function (red, green, and blue curves in the illustration).
For the $M$-th metric the relation between the physical description of the
image transform and a common metric-independent distance domain has two
components:

    1. (a.1)

The (non-scaled) _response function_ can be empirically computed by generating
images distorted (transformed) with different intensities, $\theta$, and using
the metric expression to compute the corresponding distances from the original
image. This leads to the distances
$d_{M}(\theta)=d_{M}(i,i^{\prime})=d_{M}(i,T_{\theta}(i))$.

    2. (a.2)

A metric _equalization function_ transforming the previous (non-scaled)
distance values into the common scale of the internal distance representation
(what we called normalized MOS units in the illustration). Here we propose to
use auxiliary empirical data (e.g. certain subjectively rated databases) to
scale the range of the different metrics: $\mathcal{D}_{M}=f_{M}(d_{M})$. This
makes the different $\mathcal{D}_{M}$ comparable.

Then, the final (scaled) transduction is the composition of response and
equalization: $\mathcal{D}(\theta)=g_{M}(\theta)=f_{M}(d_{M}(\theta))$.

  2. (b)

The human thresholds, that can be defined in different domains:

    1. (b.1)

_The human threshold in the common internal representation_ ,
$\mathcal{D}_{\tau}$, orange line in the illustration of Fig 1. In principle,
this value is unknown. Here we propose a standard measurement of this
threshold through a psychometric function [30] using distorted images of the
selected subjectively rated database.

    2. (b.2)

_The human threshold in the input physical representation_ ,
$\theta_{\tau}^{H}$, black line in the illustration of Fig 1. In this work, we
explore two options: (i) take the values from the classical literature, which
in general uses substantially different stimuli (synthetic as opposed to
natural), and (ii) re-determine the thresholds in humans by using comparable
natural stimuli and a separate psychometric function for each distortion.

  3. (c)

The metric threshold, which can be expressed in intuitive physical units, or
in the own units of the metric:

    1. (c.1)

In physical units $\theta_{\tau}^{M}$. Here, the blue, red, and green points
in the x-axis of the illustration of Fig 1, computed as
$\theta_{\tau}^{M}=g_{M}^{-1}(\mathcal{D}_{\tau})$. These units are useful to
compare with the equivalent human values.

    2. (c.2)

In the units of the metric, $d_{\tau}^{M}=f_{M}^{-1}(\mathcal{D}_{\tau})$.
This value is particularly interesting, as it indicates a variation in the
distance of each metric for which humans see no difference between the
original and the distorted image. Distortions leading to distances below this
value are invisible to humans and hence should be neglected.

  4. (d)

The sensitivities of humans and metrics for the different distortions. The
sensitivity for a small distortion is usually defined as the inverse of the
energy required to be above the invisibility threshold [29]. In the case of
metrics, this general definition reduces to the derivative of the transduction
function with regard to the energy of the distortion [28].

Below, we elaborate on each of these factors in turn.

###  II-A Transduction: response and equalization

The response function of a metric, $d_{M}$, to a certain image transform,
$T_{\theta}$, is just the average over a set of images,
$\\{i^{s}\\}_{s=1}^{S}$, of the distances for the distorted images for
different transform intensities:

$d_{M}(\theta)=\frac{1}{S}\sum_{s=1}^{S}d_{M}(i^{s},T_{\theta}(i^{s}))$ (1)

which can be empirically computed from controlled distortions of the images of
the dataset. Of course, when considering $M$ different metrics, their response
functions, $d_{M}$, are not given in a common scale.

In this work, we propose to use auxiliary empirical data to determine a common
distance scale, $\mathcal{D}$, for any metric. In particular, subjectively
rated image quality databases (e.g. TID [31]) consist of pairs of original and
distorted images, $\\{(i^{p},{i^{p}}^{\prime})\\}_{p=1}^{P}$, with associated
subjective scores, the so-called Differential Mean Opinion Scores,
$\\{\textrm{DMOS}^{p}\\}_{p=1}^{P}$. The databases contain a wide set of
generic distortions of different intensities thus ranging from invisible
distortions to highly noticeable distortions. In this setting, DMOS values in
the database can be normalized to be in the [0,1] range. In this way, the
extreme values of the normalized DMOS represent an invisible distortion and
the biggest subjective distortion in the database respectively. Therefore, if
the range of distortions in the database is wide, the variations induced by
$T_{\theta}$ will be within the limits of the normalized DMOS, and hence, it
can be used to set the common distance scale $\mathcal{D}=\textrm{norm
DMOS}\in[0,1]$.

Using the normalized DMOS values and the image pairs of a large subjectively
rated database one can fit an equalization function, $f_{M}$, for each metric
to transform the non-scaled response $D_{M}$ into the common scale
$\mathcal{D}$:

$\mathcal{D}=f_{M}(d_{M}(i,i^{\prime}))=a_{M}\cdot
d_{M}(i,i^{\prime})\,^{b_{M}}$ (2)

where an exponential function with $a_{n}>0$ and $0<b_{n}\neq 1$ is chosen
because of the nature of the DMOS, which changes rapidly for low distortion
intensities, low values of $\theta$, and saturates for bigger distortions, big
values of $\theta$. This is due to the maximum number of comparisons performed
when building the database. An example of an equalization function is shown in
Figure 2 as an example.

Figure 2: Example of equalization function fitted on TID13 data, for the
metric PerceptNet, following Eq. 2.

In summary, (a.1) an application of the distance metric to a controlled set of
transformed images, and (a.2) a fit of an equalization function to transform
the arbitrary scale of the distance to a common scale given by the set of
distortions in a wide subjectively rated database, gives us the transduction
function of the metric made of Eqs. 1 and 2:

$\mathcal{D}=g_{n}(\theta)=f_{n}\circ d_{n}(\theta)$ (3)

In our work, we compute these transduction functions for:

  * •

Six distance metrics (four state-of-the-art deep-learning metrics [3, 4, 32,
23] and two convenient references: the Euclidean metric, RMSE, and the
classical SSIM [13]).

  * •

Four affine transformations: translation, rotation, scale, and change of
spectral illumination.

  * •

Four datasets to transform the images and compute the response functions:
MNIST [33], CIFAR10 [34], ImageNet [35], and TID2013 [31].

  * •

One subjectively rated dataset, TID2013 [31], to define the equalization
functions to the common internal distance representation.

###  II-B Human thresholds

In this section, we measure two kinds of human thresholds. First, we use
distorted images from a subjectively rated database to measure from the
internal common representation, $\mathcal{D}_{\tau}$. Then, we measure the
thresholds in physichal units, $\theta_{\tau}$, using natural stimuli.

_B.1. Human thresholds in the common internal representation_

To find out if the metrics behave similarly to humans we need to determine the
human invisibility threshold, $\mathcal{D}_{\tau}$. This is the distance in
normalized DMOS units from which humans can’t tell the difference between $i$
and $i^{\prime}$ for low distortion intensities $\theta$. For that, we need a
database with ratings and opinions of observers, for which we can assign a
threshold value of invisibility. In this case, we use the TID13 database.
[31].

The value of $\mathcal{D}_{\tau}$ could be roughly estimated by visual
inspection of the images presented in Appendix A: images with low values of
normalized DMOS (below 0.3) cannot be discriminated from the original, while
images with big normalized DMOS (above 0.6) are clearly distinct from the
original. However, the more accurate estimation of such a threshold is
obtained from the psychometric function in a constant stimulus experiment [30]
applied to other fields within computer vision and image processing [36, 37,
38, 39, 40].

In this kind of experiments, given a set of distorted images one computes the
probability that an observer sees some distorted image as different from the
original, i.e. $P(\mathcal{D}\geq\mathcal{D}_{\tau})$. This is done by using
the two-alternative forced choice paradigm (2AFC): by randomly presenting the
observer each distorted image together with the original so that the observer
is forced to choose the distorted one. This is repeated $R$ times, so the
probability, $P(\mathcal{D}\geq\mathcal{D}_{\tau})$, is given by the number of
correct responses over $R$. Note that if $\mathcal{D}\ll\mathcal{D}_{\tau}$
the observer will not see the difference and the probability of correct answer
will be 0.5. On the other extreme, if $\mathcal{D}\gg\mathcal{D}_{\tau}$ the
answer will be obvious for the observer, and the probability of a correct
answer will be 1. As a result, the threshold can be defined as the point where
$P(\mathcal{D}\geq\mathcal{D}_{\tau})=0.75$.

We look for the optimal threshold, $\mathcal{D}_{\tau}$, and slope, $k$, that
better fit the experimental data using the following sigmoid [30]:

$p(x)=\frac{1}{2}+\frac{1}{2(1+e^{-k(\mathcal{D}-\mathcal{D}_{\tau})})}$ (4)

Note that this expression enforces that for extremely distorted images
$\lim\limits_{\mathcal{D}\to\infty}p(\mathcal{D}\geq\mathcal{D}_{\tau})=1$,
and that the probability at the threshold, $\mathcal{D}=\mathcal{D}_{\tau}$ is
0.75, as required.

In our experiment, the psychometric function has been evaluated in 20
different values of $\mathcal{D}$, repeated 15 times for 5 observers,
resulting in $20\times 15\times 5=1500$ forced choices in total. The fitted
psychometric function is shown in Figure 3 and shows that the value of the
threshold for humans in the common internal distance representation is

$\mathcal{D}_{\tau}=0.44\pm 0.05$

where the corresponding quartiles give the uncertainty. This threshold
corresponds to the orange lines in Figure 1.

Figure 3: Human threshold in the internal distance representation,
$\mathcal{D}_{\tau}$, from the psychometric function of a constant stimuli
experiment (see details in the text). The detection threshold is
$\mathcal{D}_{\tau}=0.44\pm 0.05$ in normalized DMOS units.

_B.2. Human threshold in physical units_

In principle, one could take the thresholds $\theta_{\tau}^{H}$ from the
classical literature [24, 25, 26, 27] but, considering the critical effect of
the stimuli used in the experiments and the fact that classical thresholds
were measured with synthetic stimuli, we decided to measure these thresholds
for natural images to ensure consistency with the kind of images used by the
metrics. We measure the thresholds as in Section II-B.1., according to the
constant stimuli setting to derive the classical psychometric function [30].
In particular, images from ImageNet have been used to find the invisibility
thresholds in a more realistic context.

For each transform, this experimental procedure requires the variation of the
stimuli in the parameter space $\theta$, which is trivially one-dimensional
for scale, translation, rotation, and intrinsically two-dimensional for
chromatic shifts (despite they coming from arbitrary variations of a higher
dimensional object such as the spectrum of the illuminant [41]).

On the one hand, regarding the geometrical transforms with one-dimensional
parameter, the results of the thresholds obtained from the psychometric curves
are shown in Figure 4.

Figure 4: Experimental (human) psychometric functions measured with natural
images. According to them, the human thresholds for the geometric affine
transformations are 0.23 $\pm$ 0.10 degrees for translation, 3.6 $\pm 1.5$
degrees for rotation, and 1.03 $\pm 0.02$ for scale.

On the other hand, now we address the results of the color thresholds which
are two-dimensional in nature. As a convenient summary, here we describe the
chromatic content of natural images using a single two-dimensional point in
the so-called chromatic diagram [41]. In particular, we will use the chromatic
coordinates of the spectrum of the illuminant assumed to be the illumination
source of the scene. Moreover, we will describe the color-shifts introduced to
check the invariance of metrics by the new chromatic coordinates of the new
spectrum applied to the scene (as affine transform on the image vector as in
Eq. 8 in Section III-A). Here we assume that the natural images in the
databases are, in principle, illuminated by a generic _white_ spectrum.
Therefore, we describe the chromaticity of these images using the location of
that generic white spectrum in the CIE xy diagram, i.e. in the 2d-point
$(1/3,1/3)$ [41].

The two-dimensional nature of chromatic shifts implies that invisibility
regions are not intervals $[0,\theta_{\tau}^{H}]$, but two-dimensional shapes
(e.g. discrimination ellipses [27]). In order to determine the shape of such
discrimination region around a certain central chromatic location, e.g. the
aforementioned white point, one could measure one-dimensional intervals moving
the images in different chromatic directions away from the selected center.
For instance, Figure 5 represents the white point $(1/3,1/3)$, generic
illuminant, and four linear paths indicating color shifts away from this
center (radial gray lines). Measurements along radial lines can be done with
the constant-stimulus + psychometric function paradigm mentioned above. In our
case, given the ellipsoidal shape of the classical chromatic discrimination
thresholds [27] (also shown in Figure 5), measuring only in four directions is
enough for a trustable fit of an ellipse [42]. The fitted psychometric
functions for these four chromatic directions are shown in Figure 6. The
thresholds in departure in chromatic coordinates from these psychometric
functions determine four two-dimensional experimental thresholds that allow to
fit the ellipse in pink in Figure 5.

Obtaining and assessing the result found (pink ellipse in Figure 5), and also
the assessment of ellipses obtained from the metrics, depends on the measuring
RMSE distances between ellipses. This error is relevant in the fitting process
[42], where theoretical points are compared with the experimental points.
Additionally, assessment implies the comparison of the predicted ellipsoid
with the corresponding region that could be found in classical literature [27,
43]. As the selected reasonable white was not explicitly measured in the
classical literature, we fitted a new ellipse at this point considering the
closest existing ellipses (in green in Figure 5).

Figure 5: New fitted ellipses. In purple, the new ellipse centered on the
absolute white interpolated from the closest ellipses (in green). In red, the
experimental ellipse fitted with our own measured data. The highlighted axes
are the interpolated ellipse’s major and minor axes and two intermediate hues.

Interestingly the ellipse we got by assessing the detection thresholds with
color-shifted natural images is quite consistent with previous results that
used synthetic stimuli and different experimental techniques [27, 43]. On the
one hand, note the similarity in size and orientation between the pink and the
purple ellipses. Note that the ellipse is wider in the yellow-blue (YB)
direction than in the red-green (RG) direction. The authors in [43] reported
that their threshold method with synthetic textured stimuli led to ellipses
$\times 4.5$ than MacAdam ellipses [27] measured with the variance of color
matching paradigms. As Fig 5 shows the MacAdam ellipses with a $\times 5$
visualization factor [41], our experimentally fitted pink ellipse obtained
with thresholds from natural images is very consistent with [43].

Figure 6: Experimental (human) psychometric functions measured with natural
stimuli. For the illuminant changes, we obtained 4 curves where the YB
direction is the major axis, and the RB direction is the minor axis of the
ellipse. In addition, two intermediate shades are also shown for more precise
adjustment.

Table I summarizes the results with synthetic (obtained from classical
literature) and real stimuli (experimentally measured in this work).

TABLE I: Human thresholds for the different related transformations. In the
second column, we have the thresholds according to the classical literature.
In the third column, we have the thresholds obtained in experiments with
humans carried out in the laboratory.

Affine Transformations | Synthetic stimuli | Natural stimuli
---|---|---
Translation | 0.024 degrees [24] | 0.23 $\pm$ 0.10 degrees
Rotation | 3 degrees [25] | 3.6 $\pm$ 1.5 degrees
Scale | 1.03 scale factor [26] | 1.03 $\pm$ 0.02 scale factor
Color discriminant | MacAdam ellipse [27] | Experimental ellipse (Figure 5)

From Table I we see that the rotation and scale thresholds are practically
equivalent, but this is not the case for the translation threshold. This can
be attributed to the fact their experiments with synthetic stimuli [24] can be
considered as being close to a hyperacuity setting, (which returns very low
thresholds) [44]. Our experiments employ natural images and do not enforce the
hyperacuity setting. This results in a considerably higher detection threshold
for this transformation. Recent work we could consider is the thresholds
measured with gabor filters [45]. In this case, the human rotation threshold
was considered to be 2.7 degrees of rotation and the human translation
threshold the equivalent of 0.12 degrees of translation. These measurements
are quite consistent with ours. Taking into account that the methodology of
obtaining the threshold is the same as in our case, the difference in the
thresholds is given exclusively by the type of stimulus used. For this reason,
the detection threshold for translation from other literature will not be
considered in the results because of differences in the procedure for
obtaining it and its interpretation in the case of classical literature; and
for the type of stimulus used in the recent literature.

###  II-C Metric thresholds in physical units

Following the methodology in Figure 1, we can obtain the metric thresholds in
physical units, $\theta_{\tau}^{M}$. For example, the rotation threshold is
expressed in degrees. Through the function $g_{M}(\theta)$, specific to each
metric, a value in physical units is assigned to the threshold in the common
internal representation, computed as
$\theta_{\tau}^{M}=g_{M}^{-1}(\mathcal{D}_{\tau})$. These values can be
compared numerically with the thresholds obtained for humans, as they are
expressed in the same units because they were obtained with the psychometric
functions, Section 3. Therefore, a metric can be said to have human behavior
if the invisibility threshold for a certain affine transformation coincides,
$\theta_{\tau}^{H}=\theta_{\tau}^{M}$. In particular, since
$\mathcal{D}_{\tau}$ has an associated uncertainty, $\theta_{\tau}^{M}$ also
has a confidence interval. In this case, we will check whether the human
threshold falls within the confidence interval of each metric.

This test is particularly demanding because it is strictly quantitative, i.e.
numerical comparison. It is to be expected, that the metrics will not be able
to reproduce these thresholds. This is why an alternative, less demanding,
test is proposed, reflecting the qualitative behavior of the metrics in the
face of these distortions.

###  II-D Sensitivities of metrics and humans for the different distortions

Taking into account that the methodology proposed above is particularly
demanding, we propose another test. Having calculated the thresholds for
metrics and humans, we can define sensitivities for both and compare them.
However, this comparison will be qualitative, not quantitative, so this test
is less demanding for metrics.

The sensitivity for a small distortion is usually defined as the inverse of
the energy required to be above the invisibility threshold, i.e, expressing
the distortion $\theta_{\tau}^{H}$ in RMSE units. In the case of metrics, this
general definition reduces to the derivative of the transduction function
concerning the energy of the distortion, i.e. its slope [28].

$S=\frac{1}{|i-T_{\theta^{H,M}_{\tau}}(i)|_{2}}$ (5)

In this way, we can obtain the sensitivity of a metric for each affine
transform, order them and check if the order matches the human ones. A real
example of this can be found in Figure 7 (the rest of the figures can be found
in Appendix D), where the vertical lines represent the human thresholds
expressed in terms of energy and the different curves describe the behavior of
that metric for the various transformations, where both are calculated as the
mean over the whole database. Given two curves, the most sensitive will be the
one with a higher slope.

In Figure 7, the human thresholds indicate that the human sensitivity order
(indicated by the vertical lines) is: scale, translation, rotation, RG, and YB
illuminants. However, the metric shown returns the following order (given by
the slopes of the curves): scale, rotation, translation, YB, and RG
illuminants. Even if the ordering doesn’t strictly match, it is to be noted
that this metric maintains more sensitivity to geometric than chromatic
distortions.

Figure 7: Example of results comparing the sensitivities. Here, humans are
more sensitive in the following order: scale, translation, rotation,
illuminant RG, and YB. However, this metric, in this case, PerceptNet, returns
the following order: scale, rotation, translation, illuminant YB, and RG.

##  III Experimental Settings

In this section, we review the selected affine transformations, models, and
databases that we will use in the experiments. Code available to test other
metrics publicly222https://github.com/Rietta5/InvarianceTestIQA

###  III-A Affine Transformations

An affine transformation or affine application (also called affinity) between
two affine spaces is a transformation that satisfies:

$F:v\to Mv+b$ (6)

Where $v$ can be any vector and the affine transformation is represented by a
matrix $\mathbf{M}$ and a vector $\mathbf{b}$ satisfying the following
properties: first, it maintains the collinearity (and coplanarity) relations
between points and, second, it maintains the ratios between distances along a
line.

Here, we apply affine transformations in two cases: domain (Equation 7) and
image samples (Equation 8), i.e. modifications within the image vector:

$i^{\prime}(x)=i(Mx+b)$ (7)

$i^{\prime}(x)=Mi(x)+b$ (8)

Some examples of affine transformations that follow Equation 7 are geometric
contraction, expansion, dilation, reflection, rotation, or shear; and some
examples following Equation 8 are changes in contrast, luminance, and
illuminant. In this work, we are going to focus specifically on translation,
rotation scale, and illuminant changes. For each tested affine transformation,
the original images are modified in the following ways:

  * •

Translation: Displacements on the vertical and horizontal axis (and the
combination between them) with an amplitude of 0.3 degrees of translation in
each direction (left, right, up, and down). Given the symmetry in both
displacement directions, we calculate the average and show only the
displacements to the right in the graphs.

  * •

Rotation: Rotations from -10 to 10 degrees, in steps of 0.1 degree. Again,
given the symmetry in both displacement directions, only positive rotations
will be shown in the graphs.

  * •

Scale: Scale factors range from 0.1 to 2, but do not have a fixed step. Only
scale factors that return images of even size are used. This ensures that only
scales that do not force a translation are applied. Only scale factors bigger
than 1 will be shown.

  * •

Illuminant changes: We have desaturated the original images and modified the
illuminant for 20 hues in 8 saturations, i.e. angle with respect to the $x$
axis, and distance with respect to the central white point respectively.
Following the distribution at the locus as indicated in Figure 8.

Figure 8: The considered illuminants are organized as a function of hue and
saturation, i.e. angle concerning the x-axis, and distance concerning the
white point respectively. We consider 20 hues, i.e. 20 different angles, and 8
intensities for each hue.

As a summary, examples of all the affine transformations emulated in this work
can be seen in Figure 9 and an example of all illuminant changes and different
intensities in Figure 11.

Figure 9: Examples of affine transformations applied to the original images.
From left to right it shows the original image, translation (150 pixels
right), rotation (15 degrees), scale (factor of 0.8), and illuminant change
(hue 17, purple, with maximum intensity).

###  III-B Datasets

In terms of dataset selection, we chose four different datasets covering a
wide range of features. On the one hand, we have MNIST [33] for the black and
white images. A simple set both to understand and to modify. On the other
hand, for color images, we have selected CIFAR-10[34] for color images with
low resolution, and ImageNet[35] and TID2013[31] for color images with high
resolution.

Specifically, from each dataset we selected 250 images to reduce the
computational burden of applying all affine transformations and comparing all
metrics; and they were modified so that, when applying the transformations,
the resulting images would not present new artifacts or the central element
would disappear. For the MNIST set images -originally with 28x28 images-, we
simply enlarged the images (56x56) by adding black pixels around the original
images to give us more room to move them. For color images, to avoid the
appearance of black borders when applying some transformations, we decided to
modify them and generate a mosaic. In addition to generating the mosaic, the
images have a mirror effect to make the transitions between the different
images smoother. Once the mosaic is created, the transformation is applied and
a patch is taken from the original size of the database, thus preserving the
size while including the modification. Figure 10 shows an example of the
resulting modified images.

Figure 10: Example of an original image and with the mosaic effect to fill the
image when affine transformations modify it out of the original dimensions.
Once the mosaic is created, the affine transformation is applied and cropped,
as indicated by the red frame, recovering the initial dimensions. An example
of the final result can be seen in Figure 9. Figure 11: Examples of illuminant
change only applied to the Imagenet dataset in the same distribution as Figure
8. There are changes for 20 hues (columns) in 8 intensities (rows).

###  III-C Metrics

We choose different metrics because of their relevance in the context of
perceptual metrics, either because of their age and widespread use, or because
of the good results obtained in other works and applied to other areas of
study. In this work, the following will be used:

  * •

Mean Squared Error (MSE): Measures the average squared difference between
corresponding pixels in two images. It’s a basic metric for image quality
assessment, often used in image denoising and restoration tasks. However, it
doesn’t always align well with human perception.

  * •

Structural Similarity Index (SSIM) [13]: For two images - the original and the
distorted image - it computes three different comparisons: Luminance, contrast
and structure. SSIM provides a more perceptually meaningful measure than MSE.

  * •

Learned Perceptual Image Patch Similarity (LPIPS) [3]: It uses a VGG trained
with Imagenet to pass the images and compute distances in different feature
spaces. Then, it performs a weighted sum so that the correlation with a
perceptual database is maximized.

  * •

Deep Image Structural Similarity (DISTS) [23]: As in LPIPS, it uses the VGG
network, but in addition, it performs SSIM index at different intermediate
layers. This new index combines sensitivity to structural distortions with
tolerance to textures sampled elsewhere in the image.

  * •

PerceptNet [4]: It proposes an architecture to reflect the structure and
stages of the early human visual system by considering a succession of
canonical linear and non-linear operations. The network is trained to maximize
correlation with a perceptual database.

  * •

Perceptual Information Metric (PIM) [32]: Unlike the other metrics, its
training is based on two principles: efficient coding and slowness. On the one
hand, it is compressive and, on the other, it captures persistent temporal
information. This is achieved by training the network with images extracted
from videos over which very short periods of time have passed, which should
make it more robust to small variations in the object’s movements or subtle
changes in lighting.

##  IV Results

Table II summarizes the numerical results of the experiments. For each
geometric affine transformation, we present the discrimination thresholds per
metric, $\theta_{\tau}^{M}$. In the case of illuminant changes, we show the
errors made with respect to the interpolated MacAdam and the experimentally
fitted ellipses. As the ellipses are very similar (Figure 5), the errors are
very similar. In those where they did not coincide completely, the mean of the
errors has been calculated and marked with an asterisk (*) in the Table. The
Figures from which these numerical results are derived are given in Appendix B
and Appendix C to avoid cluttering the main text.

As we are considering both the human thresholds extracted from the literature,
$\theta_{\tau}^{H_{L}}$, and the threshold obtained by ourselves with natural
images, $\theta_{\tau}^{H_{N}}$, a result is marked in bold if
$\theta_{\tau}^{M}$ falls within $\theta_{\tau}^{H_{L}}$ and is underlined if
it falls within $\theta_{\tau}^{H_{N}}$. Note that a result can be marked both
ways at the same time. A metric with many highlighted boxes will be a metric
with a good performance from the point of view of human invariance.

TABLE II: Distortion range of the metrics for each geometric affine
distortion. For geometric transformations, intervals that match the literature
threshold are marked in bold, while those that match the thresholds measured
during the work are underlined. For illuminant changes, metrics that return a
smaller error with respect to the fitted ellipses are marked in the same way
as in geometric ones.

| RMSE | SSIM | LPIPS | DISTS | PerceptNet | PIM | Human
---|---|---|---|---|---|---|---
translation | MNIST | 0.025$\pm$0.003 | 0.030$\pm$0.004 | 0.05$\pm$0.03 | 0.034$\pm$0.008 | 0.031$\pm$0.004 | 0.05$\pm$0.03 | 0.12
| CIFAR-10 | 0.033$\pm$0.004 | 0.032$\pm$0.004 | 0.05$\pm$0.03 | 0.036$\pm$0.019 | 0.05$\pm$0.02 | 0.13$\pm$0.12 |
| TID13 | 0.027$\pm$0.003 | 0.028$\pm$0.003 | 0.034$\pm$0.009 | 0.08$\pm$0.10 | 0.0029$\pm$0.003 | 0.06$\pm$0.04 | 0.23
| ImageNet | 0.036$\pm$0.010 | 0.033$\pm$0.004 | 0.05$\pm$0.04 | 0.13$\pm$0.15 | 0.036$\pm$0.010 | 0.15$\pm$0.13 |
Rotation | MNIST | 2$\pm$1 | 2.9$\pm$1.5 | 3.7$\pm$2 | 1.2$\pm$0.8 | 4.0$\pm$1.7 | 10.7$\pm$8.1 | 3
| CIFAR-10 | 1.1$\pm$0.6 | 1.5$\pm$0.7 | 1.4$\pm$1.2 | 0.7$\pm$0.4 | 3.6$\pm$1.8 | 1.7$\pm$0.9 |
| TID13 | 0.10$\pm$0.03 | 0.11$\pm$0.04 | 0.10$\pm$0.05 | 0.1$\pm$0.3 | 0.11$\pm$0.04 | 0.09$\pm$0.08 | 3.6
| ImageNet | 0.19$\pm$0.09 | 0.15$\pm$0.06 | 0.15$\pm$0.11 | 0.11$\pm$0.05 | 0.22$\pm$0.09 | 0.2$\pm$0.5 |
Scale | MNIST | 1.10$\pm$0.04 | 1.19$\pm$0.12 | 1.2$\pm$0.2 | 1.089$\pm$0.010 | 1.16$\pm$0.07 | 1.3$\pm$0.2 | 1.03
| CIFAR-10 | 1.052$\pm$0.006 | 1.055$\pm$0.007 | 1.061$\pm$0.007 | 1.054$\pm$0.007 | 1.067$\pm$0.018 | 1.06$\pm$0.03 |
| TID13 | 1.047$\pm$0.006 | 1.048$\pm$0.006 | 1.057$\pm$0.007 | 1.062$\pm$0.008 | 1.040$\pm$0.005 | 1.059$\pm$0.007 | 1.03
| ImageNet | 1.0080$\pm$0.0010 | 1.0074$\pm$0.0009 | 1.0086$\pm$0.0011 | 1.00813$\pm$0.00010 | 1.0082$\pm$0.0010 | 1.009$\pm$0.005 |
Illum | MNIST | 0.03$\pm$0.06* | - | - | 0.016$\pm$0.018* | - | 0.05$\pm$0.05* | MacAdam
| CIFAR-10 | 0.007*$\pm$0.009* | 0.03$\pm$0.05* | 0.03$\pm$0.03* |  0.005$\pm$0.013* | 0.05$\pm$0.13* | 0.06$\pm$0.06* |
| TID13 | 0.010$\pm$0.010* | 0.02$\pm$0.02* | 0.02$\pm$0.02 | 0.02$\pm$0.02 | 0.05$\pm$0.09* |  0.005$\pm$0.012* | Our ellipse
| ImageNet | 0.008$\pm$0.010 | 0.02$\pm$0.02* | 0.02$\pm$0.02 | 0.02$\pm$0.02 | 0.05$\pm$0.09* |  0.005$\pm$0.012* |

The results from Table II show that following our strong invariance criteria,
there is no clear winner. No model shows the required robustness to affine
transformations nor type of stimuli.

The comparison of the order between the human thresholds and the sensitivities
of the metrics can be seen in Table III, where even if no metric shows
complete human behavior, most of them maintain the chromatic ordering.

TABLE III: Results associated with the least demanding test, on the order of
the sensitivities and their comparison with the human order. The first row of
the table shows the order of the humans, which does not change depending on
the database. Displayed in two different colors: In purple, the order of the
geometric transformations and, in pink, the order of the YB and RG
illuminants.

| MNIST | CIFAR-10 | TID13 | ImageNet
---|---|---|---|---
Human | S$>$R$>$T$>$RG$>$YB | S$>$R$>$T$>$RG$>$YB | S$>$T$>$R$>$RG$>$YB | S$>$T$>$R$>$RG$>$YB
SSIM | R$>$S$>$T$>$RG$>$YB | YB$>$T$>$S$>$RG$>$R | YB$>$S$>$R$>$T$>$RG | YB$>$T$>$S$>$RG$>$R
LPIPS | R$>$RG$>$YB$>$S$>$T | R$>$YB$>$RG$>$S$>$T | YB$>$RG$>$R$>$S$>$T | YB$>$R$>$RG$>$T$>$S
DISTS |  RG$>$S$>$YB$>$R$>$T | R$>$RG$>$YB$>$S$>$T | R$>$RG$>$YB$>$S$>$T | R$>$S$>$RG$>$YB$>$T
PerceptNet | S$>$T$>$R$>$RG$>$YB | T$>$S$>$RG$>$R$>$YB | S$>$R$>$T$>$YB$>$RG | T$>$S$>$R$>$YB$>$RG
PIM |  RG$>$YB$>$R$>$S$>$T | R$>$RG$>$YB$>$S$>$T |  RG$>$YB$>$R$>$S$>$T | YB$>$RG$>$R$>$S$>$T

##  V Discussion

As we have seen in Table II, bearing in mind that no model is highlighted for
all related transformations, the conclusion is that there is no metric that
behaves, in terms of invariance, like the human being. We can dissect the
behavior for the different affine transformation as follows:

  * •

Translation: There is no particular model that shows human-like translation
invariance but DISTS and PIM are the best in this regard.

  * •

Rotation: Perceptnet shows the most human-like behavior but it’s very
dependent on the database used.

  * •

Scale: There are isolated highlighted cells. In general, all the models seem
less sensitive than humans except in the ImageNet database, where the
threshold is an order of magnitude smaller than the human threshold.

  * •

Illuminants: Two models rise above the rest. On one side, DISTS has a better
performance with small images, on the other side, PIM performs better with big
images. It caught our attention that PerceptNet’s discrimination ellipse has a
radically different orientation. We attribute this to the fact that the
chromatic transformations it applies at its first layers, make all the images
greener as shown in Figure 12.

Getting into the metrics’ performance regarding the ordering of the
transformations from Table III, PerceptNet is the only metric that almost
maintains the ordering of being more sensitive to geometric over chromatic
transformations. All the metrics are generally more sensitive to YB than RG,
which matches human performance in energy definition. All in all, no metric
can be said to have human-like behavior even with the least strict test.

In addition, a collateral result that can be obtained from our work is
specific invisibility thresholds for each metric. That is thresholds in units
of each metric that indicate variations for which the distance is
imperceptible to the human eye. These values are shown in Table IV.

TABLE IV: Metric-specific invisibility thresholds. Indicates distance variations for which changes between two images are imperceptible to humans. For example, a value equal to or less than 3.09 in PIM implies that these two images are indistinguishable from the observer. | RMSE | SSIM | LPIPS | DISTS | PerceptNet | PIM
---|---|---|---|---|---|---
Invisibility Threshold | 0.020 | 0.016 | 0.017 | 0.017 | 0.008 | 3.09

##  VI Conclussion

As a complement to the usual reproduction of subjective quality ratings, we
argue that the invisibility thresholds of perceptual metrics should also
correspond to the invisibility thresholds of humans. This direct comparison
(particularly interesting for affine transformations) is a strong test for
metrics because human thresholds can be accurately measured with classical
psychophysics experiments. This comparison requires (a) data of human
thresholds and (b) metric thresholds.

Regarding human thresholds, we used both classical and explicitly measured
values of the thresholds obtained using the same kind of natural images
employed in perceptual metrics. Whereas for metric thresholds we propose a
methodology to assign invisibility regions for any particular metric, which
can be expressed in units of the metric themselves or in physical units to
facilitate direct comparison with human psychophysics results.

We also propose a less restrictive test: instead of reproducing the exact
threshold values, we evaluate if the metric just matches the sensitivity
ordering. That means, using sensitivity as the inverse of the threshold
energy, one can check whether the metrics are more sensitive to one distortion
or the other. This ordering can be then compared to the one in humans.

Making the demanding comparison between human and metric thresholds for a
range of state-of-the-art deep image quality metrics shows that none of the
studied metrics (both deep as well as RSMSE and SSIM) succeeds under these
criteria: they do not reproduce all the human thresholds for the different
affine transformations, and they do not reproduce the order of sensitivities
in humans. No metric is capable of reproducing the thresholds of invisibility
and human sensibilities.

This means that tuning the models exclusively to predict quality ratings may
disregard other properties of human vision for instance invariances or
invisibility thresholds.

Figure 12: Example of the operation of the first layers of the PerceptNet
network. For different inputs with different illuminants, the results of the
first layers are all green, which explains their poor performance in the
experiments on illuminant changes.

## Acknowledgment

This work was supported in part by MICIIN/FEDER/UE under Grant
PID2020-118071GB-I00 and PDC2021-121522-C21, in part by Spanish MIU under
Grant FPU21/02256, in part by Generalitat Valenciana under Projects
GV/2021/074, CIPROM/2021/056 and CIAPOT/2021/9 and in part by valgrAI - GVA.
Some computer resources were provided by Artemisa, funded by the European
Union ERDF and Comunitat Valenciana as well as the technical support provided
by the Instituto de Física Corpuscular, IFIC (CSIC-UV).

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## Appendix A.

The following figures show examples from the TID13 database sorted by DMOS for
which the reader can make a first estimate of the threshold value of human
invisibility for the DMOS, i.e., a first estimation of $\mathcal{D}_{\tau}$.
For each reference image (at the top) some distorted images are selected,
sorted in increasing order by the DMOS value. With these images, it can be
tested, in a rather coarse way, which DMOS value tells us what is invisible
and what is not to the observer. Roughly, the value is around 0.45, since the
second image is usually indistinguishable from the original but the third one
always differs.

## Appendix B.

Figure 13 shows the results for the all databases for the geometric affine
transformations (translation, rotation and scale). In addition, the first
subgraph shows again the proposed methodology with the cuts on the y-axis,
marked in gold color; it is the only one that also shows the legend and it is
shared by all the others.

Figure 13: Results of the different metrics in the experiments for the
geometric affine transformations. On these graphs we will apply what we have
seen in Figure 1 to create intervals to compare with the human thresholds. In
the different columns, we have the results differentiated by database (and in
the rows, by affine transformation (translation, rotation and scale).

## Appendix C.

Figures 16, 17, 15, 14 show the results for the all databases for the
illuminant transformations showing the final ellipse in the spectral locus.
The title of each subfigure specifies the errors made with respect to the
MacAdam ellipse at that point in space (left) and with respect to the
experimentally obtained ellipse (right).

Figure 14: Results of the different metrics in the color discrimination
experiments, together with their error for ImageNet data. The error with
respect to the fitted MacAdam ellipse is shown on the left whereas the error
with respect to the experimentally obtained ellipse is shown on the right. The
first subfigure also shows the direction of the 20 hues. The metric that makes
the least error with respect to the MacAdam ellipses is PIM in both cases.

Figure 15: Results of the different metrics in the color discrimination
experiments, together with their error for TID13 data. The error with respect
to the fitted MacAdam ellipse is shown on the left whereas the error with
respect to the experimentally obtained ellipse is shown on the right. The
first subfigure also shows the direction of the 20 hues. The metric that makes
the least error with respect to the MacAdam ellipses is PIM in both cases.

Figure 16: Results of the different metrics in the color discrimination
experiments, together with their error for MNIST data. The error concerning
the fitted MacAdam ellipse is shown on the left whereas the error with respect
to the experimentally obtained ellipse is shown on the right. The first
subfigure also shows the direction of the 20 hues. The metric that makes the
least error with respect to the MacAdam ellipses is DISTS in both cases.

Figure 17: Results of the different metrics in the color discrimination
experiments, together with their error for ImageNet data. The error with
respect to the fitted MacAdam ellipse is shown on the left whereas the error
with respect to the experimentally obtained ellipse is shown on the right. The
first subfigure also shows the direction of the 20 hues. The metric that makes
the least error with respect to the MacAdam ellipses is DISTS in both cases.

## Appendix D.

Figures 18, 19,20, and 21 show the results for the all databases for the
comparison of sensibilities. The vertical lines show the human thresholds for
each transformation, the different curves, the behaviour of each model. The
order of the vertical lines is compared with the value of the slope. A steep
slope implies higher sensitivity.

Figure 18: Sensitivity test results for the MNIST dataset. Table III shows in
detail the order of the sensitivities together with the human ordering, where
it can be seen that no metric follows the order correctly. However, in all
cases the relative order of illuminant changes is respected, i.e. they are
more sensitive to the RG direction than to the YB direction.

Figure 19: Sensitivity test results for the CIFAR10 dataset. Table III shows
in detail the order of the sensitivities together with the human ordering,
where it can be seen that no metric follows the order correctly.

Figure 20: Sensitivity test results for the TID13 dataset. Table III shows in
detail the order of the sensitivities together with the human ordering, where
it can be seen that no metric follows the order correctly.

Figure 21: Sensitivity test results for the ImageNet dataset. Table III shows
in detail the order of the sensitivities together with the human ordering,
where it can be seen that no metric follows the order correctly.


# Multi-User Data Detection in Massive MIMO with 1-Bit ADCs

###### Abstract

We provide new analytical results on the uplink data detection in massive
multiple-input multiple-output systems with 1-bit analog-to-digital
converters. The statistical properties of the soft-estimated symbols (i.e.,
after linear combining and prior to the data detection process) have been
previously characterized only for a single user equipment (UE) and
uncorrelated Rayleigh fading. In this paper, we consider a multi-UE setting
with correlated Rayleigh fading, where the soft-estimated symbols are obtained
by means of maximum ratio combining based on imperfectly estimated channels.
We derive a closed-form expression of the expected value of the soft-estimated
symbols, which allows to understand the impact of the specific data symbols
transmitted by the interfering UEs. Building on this result, we design
efficient data detection strategies based on the minimum distance criterion,
which are compared in terms of symbol error rate and complexity.

Index Terms—  Massive MIMO, 1-bit ADCs, multi-user data detection.

##  1 Introduction

Increasing the capacity of beyond-5G wireless systems will require exploiting
the wide bandwidths available in the THz spectrum (0.3–3 THz) [1]. This calls
for massive multiple-input multiple-output (MIMO) arrays at the transmitter
and/or at the receiver to compensate for the strong pathloss and penetration
loss therein. In this regard, fully digital architectures provide highly
flexible wideband beamforming and large-scale spatial multiplexing [2].
However, this approach requires a sacrifice in the resolution of the analog-
to-digital/digital-to-analog converters (ADCs/DACs), since their power
consumption scales linearly with the sampling rate and exponentially with the
number of quantization bits [3, 4, 5]. Remarkably, fully digital architectures
with low-resolution ADCs (even down to 1 bit) can significantly outperform
their hybrid analog-digital counterparts in terms of spectral and energy
efficiency [6]. In this respect, 1-bit ADCs/DACs are particularly attractive
as they are the simplest and least power consuming data conversion devices [3,
7]. Such a coarse quantization is suitable with very large bandwidths, for
which high-order modulations may not be needed.

There is a vast literature on 1-bit quantized massive MIMO, ranging from
performance analysis (e.g., [3, 7]) to data detection (e.g., [8, 9]) and
precoding (e.g., [10, 11, 12]). In this paper, we broaden prior analytical
studies on the uplink data detection in massive MIMO systems with 1-bit ADCs.
The statistical properties of the soft-estimated symbols (i.e., after linear
combining and prior to the data detection process) have been characterized in
our prior works [9, 13, 14] only under a simplified system model with a single
user equipment (UE) and uncorrelated Rayleigh fading. However, uncorrelated
channel models cannot account for the sparse scattering at high frequencies.
In this paper, we consider a more general and realistic multi-UE setting with
correlated Rayleigh fading. We assume that the base station (BS) adopts
maximum ratio combining (MRC) based on imperfectly estimated channels, where
the channel estimation is carried out via the Bussgang linear minimum mean
squared error (BLMMSE) estimator [3]. In this context, we derive a closed-form
expression of the expected value of the soft-estimated symbols, which is
relevant to understand the impact of the specific data symbols transmitted by
the interfering UEs. This result is exploited to design efficient data
detection strategies based on the minimum distance criterion, which are
compared in terms of symbol error rate (SER) and complexity.

##  2 System Model

Let us consider a single-cell massive MIMO system where a BS with $M$ antennas
serves $K$ single-antenna UEs in the uplink. We use
$\mathbf{H}\triangleq[\mathbf{h}_{1},\ldots,\mathbf{h}_{K}]\in\mbox{$\mathbb{C}$}^{M\times
K}$ to denote the uplink channel matrix. Considering a general correlated
Rayleigh fading channel model, we have
$\mathbf{h}_{k}\sim\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{C}_{\mathbf{h}_{k}}),\leavevmode\nobreak\
\forall k$, where $\mathbf{C}_{\mathbf{h}_{k}}\in\mbox{$\mathbb{C}$}^{M\times
M}$ is the channel covariance matrix of UE $k$. Furthermore, we define
$\mathbf{h}\triangleq\mathrm{vec}(\mathbf{H})\in\mbox{$\mathbb{C}$}^{MK}$ and,
accordingly, we have
$\mathbf{h}\sim\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{C}_{\mathbf{h}})$,
with
$\mathbf{C}_{\mathbf{h}}\triangleq\mathrm{blkdiag}(\mathbf{C}_{\mathbf{h}_{1}},\ldots,\mathbf{C}_{\mathbf{h}_{K}})\in\mbox{$\mathbb{C}$}^{MK\times
MK}$. For simplicity, and without loss of generality, we assume that all the
UEs are subject to the same SNR $\rho$ during both the channel estimation and
the uplink data transmission (see also [9]). Each BS antenna is connected to
two 1-bit ADCs, one for the in-phase and one for the quadrature component of
the received signal. In this context, we introduce the 1-bit quantization
function $Q(\cdot):\mbox{$\mathbb{C}$}^{A\times B}\rightarrow\mathcal{Q}$,
with $\mathcal{Q}\triangleq\sqrt{\frac{\rho K+1}{2}}\\{\pm 1\pm j\\}^{A\times
B}$ and [4, 9]

$\displaystyle Q(\mathbf{X})\triangleq\sqrt{\frac{\rho
K+1}{2}}\big{(}\mathrm{sgn}(\mathrm{Re}[\mathbf{X}])+j\,\mathrm{sgn}(\mathrm{Im}[\mathbf{X}])\big{)}.$
(1)

###  2.1 Data Detection

Let
$\mathbf{x}\triangleq[x_{1},\ldots,x_{K}]^{\mathrm{T}}\in\mbox{$\mathbb{C}$}^{K}$
denote the data symbol vector comprising the data symbols transmitted by the
UEs. We assume that $\mathbf{x}\in\mathcal{S}^{K}$, where
$\mathcal{S}\triangleq\\{s_{1},\ldots,s_{L}\\}$ represents the set of the $L$
possible data symbols. For instance, $\mathcal{S}$ may correspond to the
16-QAM constellation, as considered in Section 4. During the uplink data
transmission, all the UEs simultaneously transmit their data symbols, and the
signal received at the BS is given by

$\displaystyle\mathbf{y}\triangleq\sqrt{\rho}\mathbf{H}\mathbf{x}+\mathbf{z}\in\mbox{$\mathbb{C}$}^{M}$
(2)

where $\mathbf{z}\in\mbox{$\mathbb{C}$}^{M}$ is the additive white Gaussian
noise (AWGN) vector with i.i.d. $\mathcal{C}\mathcal{N}(0,1)$ elements. The BS
observes the quantized signal

$\displaystyle\mathbf{r}\triangleq Q(\mathbf{y})\in\mbox{$\mathbb{C}$}^{M}$
(3)

where the scaling factor in (1) is such that the variance of $\mathbf{r}$
coincides with that of $\mathbf{y}$. Then, the BS obtains a soft estimate of
$\mathbf{x}$ via linear combining as

$\displaystyle\hat{\mathbf{x}}$
$\displaystyle\triangleq[\hat{x}_{1},\ldots,\hat{x}_{K}]^{\mathrm{T}}$ (4)
$\displaystyle=\mathbf{V}^{\mathrm{H}}\mathbf{r}\in\mbox{$\mathbb{C}$}^{K}$
(5)

where $\mathbf{V}\in\mbox{$\mathbb{C}$}^{M\times K}$ is the combining matrix.
Finally, the data detection process maps each soft-estimated symbol in (4) to
one of the possible data symbols in $\mathcal{S}$.

###  2.2 Channel Estimation

The combining matrix $\mathbf{V}$ used in (5) is designed based on the
estimated channels. In this paper, the channel estimation is carried out via
the BLMMSE estimator [3], which is the state-of-the-art linear estimator with
low-resolution ADCs and reduces to the well-known minimum mean squared error
estimator in the absence of quantization. Let
$\mathbf{P}\triangleq[\mathbf{p}_{1},\ldots,\mathbf{p}_{K}]\in\mbox{$\mathbb{C}$}^{\tau\times
K}$ denote the pilot matrix, where $P_{u,k}$ represents its ($u,k$)th element
and $\tau$ is the pilot length. Assuming $\tau\geq K$, orthogonal pilots among
the UEs, and $|P_{u,k}|^{2}=1,\leavevmode\nobreak\ \forall u,k$, we have
$\mathbf{P}^{\mathrm{H}}\mathbf{P}=\tau\mathbf{I}_{K}$. During the channel
estimation, all the UEs simultaneously transmit their pilots, and the signal
received at the BS is given by

$\displaystyle\mathbf{Y}_{\mathrm{p}}\triangleq\sqrt{\rho}\mathbf{H}\mathbf{P}^{\mathrm{H}}+\mathbf{Z}_{\mathrm{p}}\in\mbox{$\mathbb{C}$}^{M\times\tau}$
(6)

where $\mathbf{Z}_{\mathrm{p}}\in\mbox{$\mathbb{C}$}^{M\times\tau}$ is the
AWGN matrix with i.i.d. $\mathcal{C}\mathcal{N}(0,1)$ elements. At this stage,
we vectorize (6) as

$\displaystyle\mathbf{y}_{\mathrm{p}}$
$\displaystyle\triangleq\mathrm{vec}(\mathbf{Y}_{\mathrm{p}})$ (7)
$\displaystyle=\sqrt{\rho}\bar{\mathbf{P}}^{*}\mathbf{h}+\mathbf{z}_{\mathrm{p}}\in\mbox{$\mathbb{C}$}^{M\tau}$
(8)

with
$\bar{\mathbf{P}}\triangleq\mathbf{P}\otimes\mathbf{I}_{M}\in\mbox{$\mathbb{C}$}^{M\tau\times
MK}$ and
$\mathbf{z}_{\mathrm{p}}\triangleq\mathrm{vec}(\mathbf{Z}_{\mathrm{p}})\in\mbox{$\mathbb{C}$}^{M\tau}$.
Furthermore, we define

$\displaystyle\mathbf{C}_{\mathbf{y}_{\mathrm{p}}}$
$\displaystyle\triangleq\mathbb{E}[\mathbf{y}_{\mathrm{p}}\mathbf{y}_{\mathrm{p}}^{\mathrm{H}}]$
(9) $\displaystyle=\rho\
\bar{\mathbf{P}}^{*}\mathbf{C}_{\mathbf{h}}\bar{\mathbf{P}}^{\mathrm{T}}+\mathbf{I}_{M\tau}\in\mbox{$\mathbb{C}$}^{M\tau\times
M\tau}$ (10)

and

$\displaystyle\mathbf{A}_{\mathrm{p}}\triangleq\sqrt{\frac{2}{\pi}(\rho
K+1)}\mathrm{Diag}(\mathbf{C}_{\mathbf{y}_{\mathrm{p}}})^{-\frac{1}{2}}\in\mbox{$\mathbb{C}$}^{M\tau\times
M\tau}.$ (11)

The BS observes the quantized signal

$\displaystyle\mathbf{r}_{\mathrm{p}}$ $\displaystyle\triangleq
Q(\mathbf{y}_{\mathrm{p}})\in\mbox{$\mathbb{C}$}^{M\tau}$ (12)

and obtains the estimate of $\mathbf{h}$ via the BLMMSE estimator as

$\displaystyle\hat{\mathbf{h}}$
$\displaystyle\triangleq\sqrt{\rho}\mathbf{C}_{\mathbf{h}}\bar{\mathbf{P}}^{\mathrm{T}}\mathbf{A}_{\mathrm{p}}\mathbf{C}_{\mathbf{r}_{\mathrm{p}}}^{-1}\mathbf{r}_{\mathrm{p}}\in\mbox{$\mathbb{C}$}^{MK}$
(13)

with
$\mathbf{C}_{\mathbf{r}_{\mathrm{p}}}\triangleq\mathbb{E}[\mathbf{r}_{\mathrm{p}}\mathbf{r}_{\mathrm{p}}^{\mathrm{H}}]$.
Finally, the estimate of $\mathbf{H}$ is given by
$\hat{\mathbf{H}}\triangleq[\hat{\mathbf{h}}_{1},\ldots,\hat{\mathbf{h}}_{K}]$,
with

$\displaystyle\hat{\mathbf{h}}_{k}\triangleq\sqrt{\rho}\mathbf{C}_{\mathbf{h}_{k}}\bar{\mathbf{p}}_{k}^{\mathrm{T}}\mathbf{A}_{\mathrm{p}}\mathbf{C}_{\mathbf{r}_{\mathrm{p}}}^{-1}\mathbf{r}_{\mathrm{p}}\in\mbox{$\mathbb{C}$}^{M}$
(14)

and
$\bar{\mathbf{p}}_{k}\triangleq\mathbf{p}_{k}\otimes\mathbf{I}_{M}\in\mbox{$\mathbb{C}$}^{M\tau\times
M}$.

##  3 Data Detection Analysis

In our prior work [9], we focused on a single-UE setting with uncorrelated
Rayleigh fading and characterized the statistical properties of the soft-
estimated symbols. In this paper, we consider a more general and realistic
multi-UE setting with correlated Rayleigh fading and provide a closed-form
expression of the expected value of the soft-estimated symbols. This result,
presented in Section 3.1, allows to understand the impact of the specific data
symbols transmitted by the interfering UEs. Furthermore, it can be exploited
to design efficient data detection strategies based on the minimum distance
criterion, as described in Section 3.2.

###  3.1 Expectation of the Soft-Estimated Symbols

As in [9], we consider that the MRC receiver is adopted at the BS. Hence, the
combining matrix is given by $\mathbf{V}=\hat{\mathbf{H}}$ and the soft-
estimated symbol for UE $k$ can be expressed as
$\hat{x}_{k}=\hat{\mathbf{h}}_{k}^{\mathrm{H}}\mathbf{r}$ (cf. (5)), with
$\hat{\mathbf{h}}_{k}$ and $\mathbf{r}$ given in (14) and (3), respectively.
Let us define
$\mathbf{C}_{\mathbf{r}\mathbf{r}_{\mathrm{p}}}\triangleq\mathbb{E}[\mathbf{r}\mathbf{r}_{\mathrm{p}}^{\mathrm{H}}]$,
which represents the cross-correlation matrix between the quantized signals
received during the uplink data transmission and the channel estimation.
Moreover, we introduce the function
$\Omega(x)\triangleq\frac{2}{\pi}\arcsin(x)$ and the following preliminary
definitions:

$\displaystyle\alpha_{m}$
$\displaystyle\triangleq\bigg{[}\rho\sum_{k=1}^{K}\mathbf{C}_{\mathbf{h}_{k}}+\mathbf{I}_{M}\bigg{]}_{m,m},$
(15) $\displaystyle\beta_{m}$
$\displaystyle\triangleq\bigg{[}\rho\sum_{k=1}^{K}\mathbf{C}_{\mathbf{h}_{k}}|x_{k}|^{2}+\mathbf{I}_{M}\bigg{]}_{m,m},$
(16) $\displaystyle\zeta_{m,n,u,v}$
$\displaystyle\triangleq\frac{\rho}{\sqrt{\alpha_{m}\alpha_{n}}}\bigg{[}\sum_{k=1}^{K}\mathbf{C}_{\mathbf{h}_{k}}^{\mathrm{T}}P_{u,k}P_{v,k}^{*}\bigg{]}_{m,n},$
(17) $\displaystyle\eta_{m,n,u}$
$\displaystyle\triangleq\frac{\rho}{\sqrt{\alpha_{n}\beta_{m}}}\bigg{[}\sum_{k=1}^{K}\mathbf{C}_{\mathbf{h}_{k}}x_{k}P_{u,k}\bigg{]}_{m,n}.$
(18)

The following theorem provides a closed-form expression of the expected value
of the soft-estimated symbol for UE $k$ for a given data symbol vector
$\mathbf{x}$. This is denoted by
$\mathsf{E}_{k}\triangleq\mathbb{E}[\hat{x}_{k}]$, where the expectation is
taken over $\mathbf{H}$, $\mathbf{z}$, and $\mathbf{z}_{\mathrm{p}}$.

######  Theorem 1.

Assuming that the MRC receiver is adopted at the BS, for a given data symbol
vector $\mathbf{x}$, the expected value of the soft-estimated symbol for UE
$k$ is given by

$\displaystyle\mathsf{E}_{k}=\sqrt{\rho}\mathrm{tr}(\mathbf{C}_{\mathbf{r}_{\mathrm{p}}}^{-1}\mathbf{A}_{\mathrm{p}}\bar{\mathbf{p}}_{k}^{*}\mathbf{C}_{\mathbf{h}_{k}}\mathbf{C}_{\mathbf{r}\mathbf{r}_{\mathrm{p}}})$
(19)

where the ($(u-1)M+m,(v-1)M+n$)th element of
$\mathbf{C}_{\mathbf{r}_{\mathrm{p}}}$ can be written as in (21) at the top of
the next page and the ($m,(u-1)M+n$)th element of
$\mathbf{C}_{\mathbf{r}\mathbf{r}_{\mathrm{p}}}$ can be written as

$\displaystyle[\mathbf{C}_{\mathbf{r}\mathbf{r}_{\mathrm{p}}}]_{m,(u-1)M+n}$
$\displaystyle=(\rho K+1)\Big{(}\Omega\big{(}\mathrm{Re}[\eta_{m,n,u}]\big{)}$
$\displaystyle\phantom{=}\
+j\,\Omega\big{(}\mathrm{Im}[\eta_{m,n,u}]\big{)}\Big{)}.$ (20)

$\displaystyle[\mathbf{C}_{\mathbf{r}_{\mathrm{p}}}]_{(u-1)M+m,(v-1)M+n}$
$\displaystyle=\begin{cases}\rho K+1,&\quad\textrm{if}\leavevmode\nobreak\
m=n\leavevmode\nobreak\ \textrm{and}\leavevmode\nobreak\ u=v,\\\ (\rho
K+1)\Big{(}\Omega\big{(}\mathrm{Re}[\zeta_{m,n,u,v}]\big{)}-j\,\Omega\big{(}\mathrm{Im}[\zeta_{m,n,u,v}]\big{)}\Big{)},&\quad\textrm{otherwise}\end{cases}$
(21)

The proof of Theorem 1 follows similar (and more involved) steps as in [9,
App. D]. It is omitted due to the space limitations and will be provided in
the extended version of this paper. The expression in (19) for UE $k$ clearly
depends on the specific data symbols transmitted by the interfering UEs. In
the following, we build on Theorem 1 to design efficient data detection
strategies, which will be compared in Section 4.

###  3.2 Data Detection Strategies

In this section, we exploit Theorem 1 and the minimum distance criterion to
map each soft-estimated symbol in (4) to one of the possible data symbols in
$\mathcal{S}$. In this respect, we present three data detection strategies: 1)
exhaustive, 2) heuristic, and 3) genie-aided data detection. In the following,
we use $l_{k}^{\star}\in\\{1,\ldots,L\\}$ to denote the index of the detected
symbol for UE $k$.

  * $\bullet$

Strategy 1: exhaustive data detection. This strategy uses the statistical
information of the interfering UEs to detect the symbol for the target UE. Let
$\mathcal{E}_{k}\triangleq\\{\mathsf{E}_{k},\forall\mathbf{x}\in\mathcal{S}^{K}\\}$
denote the set of the expected values of the soft-estimated symbols for UE $k$
obtained from all the possible data symbol vectors, with
$|\mathcal{E}_{k}|=L^{K}$. The soft-estimated symbol for UE $k$ is mapped to
one of the elements in $\mathcal{E}_{k}$ as

$\displaystyle\mathsf{E}_{k}^{\star}=\operatornamewithlimits{argmin}_{\mathsf{E}_{k}\in\mathcal{E}_{k}}|\hat{x}_{k}-\mathsf{E}_{k}|$
(22)

from which $l_{k}^{\star}$ is readily obtained. This strategy amounts to
performing an exhaustive search over all the $L^{K}$ possible values of
$\mathsf{E}_{k}$ in (19). Hence, its complexity increases exponentially with
$K$.

  * $\bullet$

Strategy 2: heuristic data detection. This strategy considers the expected
values of the soft-estimated symbols for the target UE averaged over all the
possible data symbols transmitted by the interfering UEs. Let
$\mathbf{x}_{-k}\triangleq[x_{1},\ldots,x_{k-1},x_{k+1},\ldots
x_{K}]^{\mathrm{T}}\in\mbox{$\mathbb{C}$}^{K-1}$ and let
$\mathcal{E}_{k,l}\triangleq\\{\mathsf{E}_{k}:x_{k}=s_{l},\leavevmode\nobreak\
\forall\mathbf{x}_{-k}\in\mathcal{S}^{K-1}\\}\subset\mathcal{E}_{k}$ be the
set containing the elements in $\mathcal{E}_{k}$ corresponding to
$x_{k}=s_{l}$, with $|\mathcal{E}_{k,l}|=L^{K-1}$. Furthermore, let us define

$\displaystyle\bar{\mathsf{E}}_{k,l}\triangleq\frac{1}{L^{K-1}}\sum_{t\in\mathcal{E}_{k,l}}t$
(23)

which represents the average of the expected values of the soft-estimated
symbols for UE $k$ when $x_{k}=s_{l}$ (see the green markers in Fig. 3). Then,
the index of the detected symbol for UE $k$ is obtained as

$\displaystyle
l_{k}^{\star}=\operatornamewithlimits{argmin}_{l\in\\{1,\ldots,L\\}}|\hat{x}_{k}-\bar{\mathsf{E}}_{k,l}|.$
(24)

This strategy can be seen as a low-complexity, heuristic version of Strategy
1, which reduces the size of the search space from $L^{K}$ to $L$.

  * $\bullet$

Strategy 3: genie-aided data detection. This strategy is obtained from
Strategy 1 by assuming that a genie instantaneously provides the data symbols
transmitted by the interfering UEs to detect the symbol for the target UE.
Hence, for UE $k$, $\mathbf{x}_{-k}$ is assumed to be perfectly known, which
reduces the size of the search space from $L^{K}$ to $L$. Evidently, this
strategy cannot be implemented in practice and is considered only to evaluate
how the knowledge of the data symbols transmitted by the interfering UEs
impacts the data detection performance for the target UE.

Other practical data detection strategies (e.g., resulting from combining
Strategy 1 and Strategy 3 above) will be considered in the extended version of
this paper.

$-120$$-80$$-40$$0$$40$$80$$120$$-120$$-80$$-40$$0$$40$$80$$120$IQ$\hat{x}_{k}$$\mathsf{E}_{k}$
Fig. 1: 3-UE scenario ($K=3$) with $\tau=61$: soft-estimated symbols (black
markers) and their expected values (red markers) for UE $1$ when
$x_{2}=\frac{1}{\sqrt{10}}(-3+j\,3)$ and $x_{3}=\frac{1}{\sqrt{10}}(-3+j)$.

##  4 Numerical Results

In this section, we utilize Theorem 1 and the data detection strategies
described in Section 3.2 to evaluate the impact of the specific data symbols
transmitted by the interfering UEs in a massive MIMO system with 1-bit ADCs.
We assume that the BS is equipped with $M=128$ antennas and adopts the MRC
receiver. The set of data symbols $\mathcal{S}$ corresponds to the 16-QAM
constellation, i.e., $\mathcal{S}=\frac{1}{\sqrt{10}}\big{\\{}\pm 1\pm j,\pm
1\pm j\,3,\pm 3\pm j,\pm 3\pm j\,3\big{\\}}$, which is normalized such that
$\frac{1}{L}\sum_{l=1}^{L}|s_{l}|^{2}=1$. We consider a 2-UE scenario ($K=2$)
and a 3-UE scenario ($K=3$). The channel covariance matrices are generated
according to the one-ring channel model [15] with angular spread of
$30^{\circ}$ for each UE and angular separation between the UEs of
$120^{\circ}$ and $60^{\circ}$ for the 2-UE and 3-UE scenarios, respectively.
All the UEs are subject to the same (normalized) pathloss, such that
$\mathrm{tr}(\mathbf{C}_{\mathbf{h}_{k}})=M,\leavevmode\nobreak\ \forall k$;
unless otherwise stated, we consider $\rho=0$ dB. The channels are estimated
as described in Section 2.2 with orthogonal pilots chosen as Zadoff-Chu
sequences, which are widely adopted in the 4G LTE and 5G NR standards [16];
unless otherwise stated, we fix $\tau=61$.

$-120$$-80$$-40$$0$$40$$80$$120$$-120$$-80$$-40$$0$$40$$80$$120$IQ$\mathsf{E}_{k}$$78.2$$78.3$$78.4$$78.2$$78.3$$78.4$
Fig. 2: 2-UE scenario ($K=2$) with $\tau=61$: expected values of the soft-
estimated symbols for UE $1$ when UE $2$ transmits all the possible data
symbols from $\mathcal{S}$.
$-120$$-80$$-40$$0$$40$$80$$120$$-120$$-80$$-40$$0$$40$$80$$120$IQ$\mathsf{E}_{k}$$-120$$-80$$-40$$0$$40$$80$$120$$-120$$-80$$-40$$0$$40$$80$$120$IQ$\mathsf{E}_{k}$$\bar{\mathsf{E}}_{k,l}$$78.2$$78.3$$78.4$$78.2$$78.3$$78.4$
Fig. 3: 3-UE scenario ($K=3$) with $\tau=61$: expected values of the soft-
estimated symbols (red markers) and their mean values for UE $1$ (green
markers) when UE $2$ and UE $3$ transmit all the possible data symbols from
$\mathcal{S}$.

Considering the 3-UE scenario, Fig. 1 provides the scatter plot of the soft-
estimated symbols for UE $1$ when this transmits all the possible data symbols
and the interfering UEs transmit fixed data symbols. Here, the soft-estimated
symbols (black markers) originate from independent channel and AWGN
realizations. We observe that, for each data symbol transmitted by UE $1$, the
mean value of the soft-estimated symbols is in agreement with the
corresponding expected value obtained as in (19) (red markers).

Considering the 2-UE scenario, Fig. 2 depicts the expected values of the soft-
estimated symbols for UE $1$ when both UEs transmit all the possible data
symbols. Note that there are $16^{2}=256$ different pairs of data symbols
transmitted by the two UEs, each corresponding to a different value of
$\mathsf{E}_{k}$ in (19). However, in Fig. 2, only $3\times 16$ points can be
clearly distinguished, which implies that there is significant overlap among
many of the $256$ values of $\mathsf{E}_{1}$. This stems from the fact that
there are three different amplitude levels in the 16-QAM constellation and,
for a given data symbol transmitted by UE $1$, the data symbols with the same
amplitude transmitted by UE $2$ produce nearly the same value of
$\mathsf{E}_{1}$ (as shown in the zoomed window).

Fig. 3 extends the insights of Fig. 2 to the 3-UE scenario. Here, there are
$16^{3}=4096$ different triplets of data symbols transmitted by the three UEs,
each corresponding to a different value of $\mathsf{E}_{k}$ in (19). As in the
2-UE scenario, for a given data symbol transmitted by UE $1$, the data symbols
with the same amplitude transmitted by UE $2$ and UE $3$ produce nearly the
same value of $\mathsf{E}_{1}$. Interestingly, the dispersion of such values
of $\mathsf{E}_{k}$ reduces as the pilot length increases, since the channel
estimates become more accurate. Moreover, the green markers in Fig. 3
correspond to the average of the expected values of the soft-estimated symbols
for UE $1$ when this transmits a specific data symbol (see Strategy 2 in
Section 3.2).

Lastly, we evaluate the performance of the data detection strategies described
in Section 3.2, which are based on Theorem 1 and the minimum distance
criterion. Considering the 2-UE scenario, Fig. 4 plots the SER obtained with
the different data detection strategies as a function of the SNR $\rho$, with
$\tau=31$. In this context, the SER is computed by averaging over $10^{4}$
independent channel and AWGN realizations, and considering all the possible
data symbols. As in the single-UE analysis in [9], we observe that the SER
curves feature an optimal SNR operating point: at low SNR, the AWGN is
dominant; at high SNR, the soft-estimated symbols corresponding to data
symbols with the same phase are hardly distinguishable. In between these
regimes, the right amount of AWGN produces a useful scrambling of the 1-bit
quantized signals at the $M$ antennas. As expected, Strategy 1 outperforms
Strategy 2, since the latter corresponds to a heuristic single-UE data
detection after averaging over all the possible data symbols transmitted by
the interfering UE (see the green markers in Fig. 3). Nonetheless, even
Strategy 2 yields an acceptable performance at the optimal SNR operating point
and beyond, partly due to the additional useful scrambling produced by the
interfering UE. Furthermore, Strategy 3 outperforms all the other strategies,
but Strategy 1 is remarkably close at the optimal SNR operating point.

$-10$$0$$10$$20$$30$$40$$50$00.050.10.150.20.250.30.35$\rho$ [dB]SERStrategy
1: exhaustiveStrategy 2: heuristicStrategy 3: genie-aided Fig. 4: 2-UE
scenario ($K=2$) with $\tau=31$: SER versus the SNR obtained with the data
detection strategies presented in Section 3.2.

##  5 Conclusions

We studied the uplink data detection in massive MIMO system with 1-bit ADCs
considering a multi-UE setting with correlated Rayleigh fading, where the
soft-estimated symbols are obtained by means of MRC based on imperfectly
estimated channels. We derived a closed-form expression of the expected value
of the soft-estimated symbols, which is relevant to understand the impact of
the specific data symbols transmitted by the interfering UEs. Building on this
result, we designed efficient data detection strategies based on the minimum
distance criterion, which are compared in terms of SER and complexity.
Motivated by the superior performance of the genie-aided data detection, which
requires the knowledge of the data symbols transmitted by the interfering UEs,
future work will focus on developing practical methods for joint data
detection.

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