Spatial Simulation of the M

uller-Lyer Illusion Genesis with

Convolutional Neural Networks

Anton N. Mamaev

and Ivan A. Gorbunov

Department of Psychology, St. Petersburg University, Makarova Embarkment 6, Saint Petersburg, Russian Federation

Keywords:

Optical Illusions, Depth Perception, Image-source Relationships, Computational Cognitive Science, Com-

puter Vision, Bayesian Statistics, Regression Problem.

Abstract:

The M

uller-Lyer illusion is a well-known optical phenomenon with several competing explanations. In the

current study we reviewed the illusion in a convolutional neural network from a perspective of image-source

relationships in the process of visual functions development.

To recreate the effect of the illusion we proposed a novel method that lets us simulate the development of

visual functions in a controlled spatial environment from the state of ‘blank slate’ to effective spatial problem

solving. This process is designed to reﬂect the development of human visual system and enable us to determine

how depth perception can contribute to the appearance of the phenomenon.

We were able to successfully reproduce the effect of the classic M

uller-Lyer in 30 independent convolutional

models and also get similar results with the variants of the illusion that are thought to be unrelated to spatial

perception. For the pairs of classic stimuli we conducted additional statistical analysis using both frequentist

and Bayesian methods. The methodological and empirical insights of this study may be helpful for subsequent

investigation of visual cognition and reconsideration of the image-source relationships in optical illusions.

1 INTRODUCTION

Since the introduction of the M

uller-Lyer illusion

(Muller-Lyer, 1889) it received a number of theoreti-

cal explanations. Some of them view the phenomenon

as a result of adaptation to speciﬁc environments,

such as urban versus rural areas (Segall et al., 1963),

or the spatial perception constancy of size being mis-

applied to 2D images (Gregory, 1963) and others sug-

gest a physiological perspective and attribute the illu-

sion to the spatial summation of the postsynaptic po-

tentials in neurons in the receptive ﬁelds (Burns and

Pritchard, 1971).

Advances in the development of deep learning al-

gorithms made it possible to create complex mathe-

matical models with principles of operation similar

to natural neural networks of the human brain. This

opened up possibilities for new applications in the

ﬁeld of cognitive sciences, such as modelling seman-

tic space (Gorbunov et al., 2019) and optical illusions

(Kubota et al., 2021), (Gomez-Villa et al., 2019). This

is also valid for convolutional models of the M

uller-

https://orcid.org/0000-0002-2283-380X

https://orcid.org/0000-0002-7558-750X

Lyer illusion (Zeman et al., 2013), (Garc

ıa-Garibay

and de Lafuente, 2015). However, we believe that cur-

rent studies have not yet reached the limit of possible

approaches to optical illusion modelling.

One of the oversighted aspects is the possible us-

age of geometrically accurate computer graphics to

simulate spatial environments instead of 2D line art.

Although it was proven possible to reconstruct the

illusion without introducing image-source relation-

ships and 3D graphics, the discrepancies in human

and model performance forces one to acknowledge

the possibility of image-source relationships being

also a contributing factor (Zeman et al., 2013). We at-

tempt not only to reconstruct the effect of the M

uller-

Figure 1: The M

uller-Lyer illusion demonstration. While

the right line seems to be larger than the left one in ‘a’, it is

evident that they are equal in ‘b’.

284

Mamaev, A. and Gorbunov, I.

Spatial Simulation of the Müller-Lyer Illusion Genesis with Convolutional Neural Networks.

DOI: 10.5220/0011529100003332

In Proceedings of the 14th International Joint Conference on Computational Intelligence (IJCCI 2022), pages 284-291

ISBN: 978-989-758-611-8; ISSN: 2184-3236

Lyer illusion in a neural network, but also to simulate

the genesis of the illusion from the point of view of

depth perception explanation.

In our study we revisit the image-source explana-

tion of the illusion by pre-training a model to solve a

regression problem of object height estimation in a 3D

environment before presenting a new but similar task

of estimating the length of the lines in the M

uller-Lyer

illusion. By doing that we force the model to adapt to

a spatial environment, develop traits of depth percep-

tion, and transfer them to 2D stimuli. Moreover, the

model becomes capable of giving exact numerical es-

timations for all of the stimuli, in contrast to previous

studies that relied on the binary classiﬁcation prob-

lem. This greatly enhances the differentiation of the

results, prevents random guessing, and facilitates sta-

tistical comparison with other models or human sub-

jects.

Compared to our preliminary study, we substan-

tially increased the sample of independent models

and validation images, performed a thorough quan-

titative analysis with Bayesian and frequentist statis-

tical methods, introduced new types of stimuli, and

broadened the theoretical scope of the research (Ma-

maev and Gorbunov, 2021).

Apart from investigating the causes of the M

uller-

Lyer illusion and introducing unique methodology

that can be important for the advancement of com-

putational cognitive sciences, the study also can be

valuable from an industrial standpoint. Data set gen-

eration software can be used to pre-train computer vi-

sion models, and convolutional models such as the

one presented in this paper can be used as a measure-

ment tool. Finally, the susceptibility of convolutional

neural networks to optical illusions should be noted in

the development of models aimed for precision.

In the paper we will ﬁrst review the speciﬁcations

of the neural model and the data sets, then present the

results of the statistical analysis and ﬁnally general-

ize the ﬁndings, provide our interpretation and sug-

gest the prospects for further studies.

2 MATERIALS AND METHODS

2.1 Data Sets

The model input data was divided into three separate

data sets:

1. Training data set (n = 500)

2. Validation data set (n = 100)

3. Testing data set (n = 12)

Images of the training data set were used to ﬁt the

model. The validation data set has been excluded

from ﬁtting and was used later to assess the model’s

accuracy on the new data. Both data sets were ren-

dered from a virtual 3D environment and contained

equivalent images. Each image depicted an internal

or external view on a cuboid mesh with a resemblance

to the geometry of a building or a room.

The training and validation data sets had target

values designated for every image in the array, as

shown in Figure 2. As the target values were the ob-

ject height values extracted directly from the image

source, the testing data set consisted only of input im-

ages.

Figure 2: Training and validation images with target values.

The testing images were conventional and uncon-

ventional variants of the M

uller-Lyer illusion stimuli.

Original M

uller-Lyer stimuli have arrows on the ends

of the lines that point either inside, towards the line

or outside, from the line. According to the illusion,

the ‘in’ lines are expected to be seen bigger than ‘out’

lines. Similarly, we also classiﬁed the variant stim-

uli with arrows replaced with other shapes as ‘in’ or

‘out’ depending on whether they are seen as larger or

smaller. All test stimuli are presented in Figure 3.

Figure 3: Testing data set stimuli pairs: a. narrow original;

b. wide original; c-e. variants from literature: d. (Hatwell,

1960), e. (Parker and Newbigging, 1963), c. Hatwell,

halved; f. our variant.

The virtual 3D environment was created with the

open source Godot 3.3.3 game engine. The scene

shown in Figure 4 included two cuboid meshes with

a mutual edge and a camera positioned inside one of

the meshes and pointed towards the edge.

Spatial Simulation of the Müller-Lyer Illusion Genesis with Convolutional Neural Networks

285

Figure 4: 3D environment overview: a1–2. object meshes;

b. mutual edge, length = target value; c: camera.

Several spatial parameters were randomly altered

for every picture taken by the camera in given ranges:

• Mesh heights (2.1,4.2),

• Camera distance (0,5),

• Camera y-axis rotation (−25,25),

• Mesh y-axis rotation (35,55).

The ranges were limited to selected values to ensure

that the angle ﬁts into the ﬁeld of view of the camera.

The heights of the meshes, being also the lengths

of their mutual edge, were exported in the image

metadata and used as target values to ﬁt the model.

Other parameters such as mesh rotation, camera dis-

tance and camera rotation were randomized to aug-

ment the training data set so the model is trained in a

dynamic environment. This is done to enhance model

validity, as the model is forced to educe the laws of

spatial geometry to solve the problem from different

positions. It is also necessary to prevent overﬁtting, as

using highly similar data can render the model inca-

pable of generalizing on new data, making it impossi-

ble to estimate the illusion stimuli. The target values

are also necessary to be randomized, as the model is

expected to give a precise estimation for a picture with

an object of any given height.

The script used to generate batches of images with

randomized spatial parameters written in GDScript

is given below.

for i in range(n): #Generate n images

height = rand_range(2.1,4.2) #Mesh height

$Interior.mesh.size.y = height

$Exterior.mesh.size.y = height

if rand_rot == true: #Mesh rotation

rot = rand_range(35,55)

$Interior.rotation_degrees.y = rot

$Exterior.rotation_degrees.y = rot

if rand_dst == true: #Camera distance

Python-inspired scripting language of Godot

$Camera.translation.z =

rand_range(0,5)

if rand_angle == true: #Camera rotation

$Camera.rotation_degrees.y =

rand_range(-25,25)

if armed == true: #Toggle screenshots

image = get_viewport().get_texture()

.get_data()

image.flip_y()

image.save_png("../screenshots/"

+str(height)+".png")

print(str(i+1),". ",str(height)) #Log

2.2 Neural Network Architecture

The convolutional neural network was initially de-

veloped and tested on the limited datasets as part

of the previous stage of our research (Mamaev and

Gorbunov, 2021). It was designed and ﬁtted with

Keras and Tensorﬂow frameworks for Python 3 in the

Google Colaboratory environment and followed a tra-

ditional architecture for its class, as shown in Figure

5. The shape of the input layer matched the grayscale

input images that were converted to 200×200×1 ma-

trices. The convolutional and max-pooling layers al-

ternated until the third max-pooling layer, after which

the data got ﬂattened and transferred to the fully con-

nected layers. Except the output, all of the layers were

activated with the ReLU function.

Input/Output

Convolutional

Max Pooling

Fully connected

Conv1

32 3×3

Conv2

64 3×3

Conv3

128 3×3

MP3

2×2

MP2

2×2

MP1

2×2

FC1

n=1024

FC2

n=512

Input

200×200

Output

n=1

Figure 5: Convolutional neural network architecture

overview.

In contrast to typical convolutional neural net-

work architectures used for classiﬁcation, our model

was initially designed to solve the regression prob-

lem and was converging on a single linear-activated

output neuron. Accordingly, the loss between the es-

timate and the real target value was calculated using

the mean squared error (eq. 1) function and the adap-

tive moment estimation function (Adam) was used as

a loss optimiser.

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286

MSE =

∑

i=1

− ˆy

)

(1)

The image datasets, Godot project ﬁle of the-

dataset generator, the Jupyter notebook with Keras

source code and the ﬁnal ﬁtted model mentioned at

this section are available at Open Science Framework

(Mamaev, 2022).

2.3 Statistical Analysis

To determine the factors and the extent of the illu-

sion’s effect, we ﬁtted a total of 30 independent mod-

els with the same training data set and made estima-

tions for the classic M

uller-Lyer stimuli, classic stim-

uli with wide arrowheads, and four types of illusion

variants. Then we conducted both Bayesian and fre-

quentist statistical analyses represented by Markov

chain Monte Carlo (MCMC), Bayes Factor, Leave-

One-Out Cross-Validation (LOO), Watanabe-Akaike

Information Criterion (WAIC) and repeated measures

Analysis of Variance (ANOVA) respectively for the

estimations of narrow and wide classic stimuli.

As the results were obtained from a non-linear

model, the normality of the distribution could have

been substantially disrupted. This calls for statisti-

cal methods that would be more suitable for the as-

sessment of the deep neural networks. One of such

methods is Bayesian inference. Using MCMC algo-

rithms, we ﬁtted four linear models that explain the

estimated values with the parameters of endpoint ar-

rows, the mean estimated line length and an indepen-

dent variable. The models corresponded to the hy-

potheses about the contribution of the arrowheads’

width and direction factors to the estimation results:

Model 1. Null model, arrowheads do not affect the

estimation:

Y = M

+ B

+ Err (2)

Model 2. In-Out model, the direction of the arrow-

heads affects the estimation:

Y = M

+ B

· X

in/out

+ Err (3)

Model 3. Narrow-Wide model, the width of the ar-

rowheads affects the estimation:

Y = M

+ B

· X

nrw/wd

+ Err (4)

Model 4. Full model, both the width and the direc-

tion of the arrowheads affect the estimation:

Y = M

+ B

· X

in/out

+ B

· X

nrw/wd

+ Err (5)

where:

Y : is the estimation made by the model

: is the mean of all estimations made by model i

: is the independent variable

: is the regression coefﬁcient of In-Out factor (eq.

3 and 5)

: is the regression coefﬁcient of the width factor

(eq. 4 and 5)

in/out

(

0, Inward arrows

1, Outward arrows

(eq. 3 and 5)

nrw/wd

(

0, Narrow arrows

1, Wide arrows

(eq. 4 and 5)

Err: is the model error

The models were ﬁtted using PyMC3 and visual-

ized using Matplotlib and Arviz libraries for Python

3. An example of the source code is given below:

dev=numpy.std(Dependent)

with pymc3.Model() as Regr2Model:

#Define variables

b0=pymc3.Normal(’B0’,0,sd=dev*2)

b1=pymc3.Normal(’B1’,0,sd=dev*2

b2=pymc3.Normal(’B2’,0,sd=dev*2)

RegrError=pm.HalfCauchy(’Err’,

beta=10, testval=1.)

#Define likelihood

likelihood = pymc3.Normal(’Y’,

mu=MeanNet + b0

+ b1*Predictor1 + b2*Predictor2,

sd=RegrError, observed=Dependent)

#Trace posterior probability

Regr2FTrace = pymc3.sample(6000,

cores=3, tune=200)

Predictor 1 and 2 arrays represent the factors that

affect the values of the Dependent array.

3 RESULTS

3.1 Model Validation

After around 25-30 ﬁtting epochs the neural network

loss was dropping lower than < 0.001. To evaluate

model’s generalization and accuracy on a new data

set we made estimations on the hold-out validation

data set of 100 images similar to the training data set.

Spatial Simulation of the Müller-Lyer Illusion Genesis with Convolutional Neural Networks

287

Figure 6 shows the bias between the model estima-

tions given for each image and the ground truth. The

values are close to zero, with the biggest peaks be-

ing ≤ ±0.5. The probable cause for the larger bias

in some images of the data set is the random nature

of the 3D scene. It is possible that some of the im-

ages lacked the visual features necessary for a precise

estimation because of an unfortunate camera-object

positioning.

Figure 6: Bias between the estimations and the ground truth

in the validation data set.

X = −0.181,σ = 0.083.

X: mean,

σ: std. deviation.

3.2 ANOVA

For an initial analysis, we used ANOVA with repeated

measures. The repeated measures were the estima-

tions given by 30 models for the same four M

uller-

Lyer illusion stimuli. Details of the results are given

in Table 1. We have discovered a signiﬁcant effect of

both the direction and the width of the arrowheads on

the variance of the estimates given by the model. The

directional factor η

= 0.822 inﬂuences the estima-

tions stronger than the width η

= 0.728. Together,

they also produce a smaller effect η

= 0.633.

Table 1: Detailed ANOVA results. SS: sum of squares, df:

degrees of freedom, MS: mean squares, F: F ratio.

SS df MS F

Intercept 1206.74 1 1206.74 32151.95

Error 1.088 29 0.038

N-W 0.04 1 0.040 77.74

Error 0.015 29 0.001

In-Out 0.191 1 0.191 135.06

Error 0.041 29 0.001

N-W×In-Out 0.010 1 0.010 49.96

Error 0.006 29 0.000

As shown in Figure 7, the lines with the arrow-

heads pointing inside are estimated to be longer than

those with the arrowheads pointing outside, accord-

ing to the principles of the M

uller-Lyer illusion in hu-

mans. In addition, stimuli with wider arrows show a

greater divergence.

Figure 7: Marginal means for two factors: arrowheads di-

rection and angle. Vertical bars denote 95% conﬁdence in-

tervals.

3.3 Bayesian Models

Figure 8 shows model traces and posterior plots made

with MCMC (Metropolis) algorithm. The similarities

of posterior distributions in all of the MCMC chains

is a favourable result in terms of the model ﬁtness.

Figure 8: MCMC full model traces and posterior distribu-

tions.

As shown in Figure 9, the only parameter that is

not signiﬁcantly different from zero is the indepen-

dent variable B

of the Null hypothesis model. Other

parameters are signiﬁcantly divergent from zero with

94% highest density intervals.

The model error is at maximum in the Null hy-

pothesis model and at minimum in the Full model.

The In-Out model also has low error. This, again,

speaks in favour of the Full and In-Out models.

For further model comparison we used Bayes fac-

tor, WAIC and LOO:

BF(M

Alt

) =

exp(

∑

log(P

(B|A))

exp(

∑

log(P

Alt

(B|A))

(6)

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288

Table 2: Speciﬁcations of the MCMC models.

X: mean, σ: standard deviation, HDI: highest density intervals, ESS: effective

sample size,

R: Gelman-Rubin convergence statistic.

Model Parameter

X σ HDI 3% HDI 97% ESS (bulk) ESS (trial)

Null B0 -0.000 0.005 -0.009 0.009 9201 7616 1

Null Err 0.051 0.003 0.044 0.067 7753 8702 1

In-Out B0 0.040 0.004 0.032 0.047 6863 8892 1

In-Out B1 -0.080 0.006 -0.090 -0.069 5021 6496 1

In-Out Err 0.031 0.002 0.027 0.035 9991 9487 1

N-W B0 -0.018 0.006 -0.030 -0.007 2981 6009 1

N-W B1 0.037 0.009 0.021 0.054 5822 8254 1

N-W Err 0.048 0.003 0.042 0.054 12445 11408 1

Full B0 0.021 0.004 0.014 0.029 6015 7011 1

Full B1 -0.080 0.005 -0.088 -0.071 10192 10726 1

Full B2 0.037 0.005 0.028 0.045 5608 5301 1

Full Err 0.025 0.002 0.022 0.028 11296 10509 1

Figure 9: Overall comparison of the calculated model pa-

rameters.

WAIC = −2

∑

i=1

log

∑

s=1

p(y

|θ

)

∑

i=1

∑

s=1

log p(y

|θ

)

(7)

el pd

LOO

∑

i=1

log(p(y

−i

)) (8)

Table 3 illustrates the comparison between mod-

els with Bayes factor. The two-factor Full model

(3 · 10

) was chosen as the most probable, while the

In-Out (6.8 · 10

) and the Narrow-Wide (6.5 · 10

)

models were the second and the third respectively.

The scores were given relative to the least probable

Null model.

Comparison with the LOO method conﬁrms the

Full model as the most probable. Small deviations in-

dicated in Figure 10 signify valid differences in the

models’ probabilities. Detailed results are shown in

Table 4. The WAIC results were similar but were not

Table 3: Bayes Factor model comparison.

Full In-Out Nrw-Wd Null

Full 1 2.3· 10

−12

2.2 ·10

−34

3.3 ·10

−38

I-O 4.4 ·10

1 9.7 · 10

−23

1.5 ·10

−26

N-W 4.6 · 10

1 ·10

1 1.5 ·10

−4

Null 3 ·10

6.8 ·10

6.5 ·10

plotted due to the warnings raised for two of the mod-

els.

Figure 10: Leave-One-Out Cross-Validation.

3.4 Illusion Variants

We used all of the 30 models used in the statistical

analysis to test them on the unconventional M

uller-

Lyer illusion stimuli. Figure 11 illustrates the differ-

ences between the length estimations of ‘in’ and ‘out’

categories in each pair. The letters in the legend cor-

respond to images in Figure 3 for quick reference. As

there are no negative values the lines from the ‘in’ cat-

egory were estimated to be larger than from ‘out’ cat-

egory every single time across 30 models. The most

dramatic differences are found in pairs ‘d’ and ‘e’: in

those pairs, the line ended with either a square or a

circle. The smallest differences are in ‘c’ category:

the lines with square halves at the ends. Classic stim-

Spatial Simulation of the Müller-Lyer Illusion Genesis with Convolutional Neural Networks

289

Table 4: Detailed LOO and WAIC model comparison report. pLOO, pWAIC: effective number of parameters, ∆LOO, ∆WAIC:

difference from the best-ﬁtting model, SE: standard error, ∆SE: standard error of criterion’s difference from the best model.

Model Rank LOO pLOO ∆LOO Weight SE ∆SE

Full 0 270.70 4.67 0 9.89

−1

10.5 0

In-Out 1 245.21 3.31 25.49 1.14

−2

9.33 5.85

N-W 2 194.34 2.72 76.36 2.01

−12

6.86 6.4

Null 3 186.66 1.98 84.04 0 7.69 7.016029

Model Rank WAIC pWAIC ∆WAIC Weight SE ∆SE Warning

Full 0 270.73 4.64 0 9.89

−1

10.48 0 True

In-Out 1 245.22 3.3 25.51 1.11

−2

9.32 5.85 True

N-W 2 194.35 2.71 76.39 0 6.86 6.38 False

Null 3 186.66 1.98 84.07 1.23

−11

7.69 7.01 False

uli ‘a’ and our version ‘f’ showed comparable results

even though the new stimuli is highly different from

other M

uller-Lyer illusion variations.

Figure 11: Estimated difference between ‘in’ and ‘out’

stimuli from the categories of the testing data set.

4 DISCUSSION

First of all, the model achieved high accuracy both

in the training and the validation data sets. With the

loss of < 0.001 and the estimate-truth divergence in

the validation set of 0.181 ± 0.083, the neural net-

work have been proven capable of solving spatial

tasks despite the randomized parameters of the envi-

ronment. This acknowledges the versatility of convo-

lutional neural networks and their capabilities in vi-

sual pattern recognition.

Secondly, the successful implementation of the

spatial simulation in the study of visual perception

opens a new opportunity for following studies. Opti-

cal illusions or other visual phenomena can be investi-

gated with geometrically accurate computer graphics

where maximum control over the environment is re-

quired.

Both Bayesian and frequentist statistical methods

that we used detected a signiﬁcant effect of the direc-

tion and width of the arrows at the ends of the M

uller-

Lyer illusion stimuli on the estimations made by the

model. The results of Bayesian model comparison

were unambiguous: all of the methods ranked the Full

model to be the most probable, followed by the In-

Out and Narrow-Wide models. The Null model was

ranked as the most unlikely. The consistency of the

ranking speaks in favour of its integrity. Such results

were not unexpected, as the original illusion is pri-

marily based on the direction of ending elements but

width also affects the estimations, perhaps depending

on the similarity of the wider arrows and the training

data set images.

The surprising discovery of the unconventional

uller-Lyer stimuli being able to cause similar ef-

fects without depth cues not only in humans, but also

in a convolutional model ﬁtted almost exclusively for

detection of depth cues requires thorough theoretical

review. However, in general, it is possible to pre-

sume that depth perception either coexists with an ad-

ditional independent cause of the illusion or both the

classic M

uller-Lyer illusion and its non-perspective

variants take roots in the same phenomenon. Recep-

tive ﬁelds are good candidates, especially if we note

the similarities between them and the convolutional

ﬁlters. Convolutional neural networks are inspired by

the visual system’s anatomy and physiology, so they

are expected to have much in common. The weighted

spatial summation of retinal cells’ activation is re-

ﬂected by the weights of ﬁlter matrix applied to the

values of the input image in an artiﬁcial neural net-

work, this way it is entirely possible to transfer the

neurophysiological explanation of the line ends being

subjectively shifted due to the context of additional

receptive ﬁeld activation from the arrowheads to the

computational models. Yet the models are different

from nature as it is possible not only to directly study

single-cell activations or small groups of neurons but

also to study fully functional small-scale neural net-

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290

works (Lindsay, 2021). Those two approaches com-

plement the limitations of each other and broaden the

understanding of the phenomenon. We suppose both

the structure of visual system and the function of spa-

tial perception contribute to the appearance of the op-

tical illusions, however we are yet to understand the

interaction of those components while working with

various stimuli. A possible route we can take is to

study the inner states of the model: for example, the

ﬁlter weights and the feature maps of a neural model

are much easier to access than the internal states of

living neurons.

5 CONCLUSIONS

We have successfully recreated the M

uller-Lyer illu-

sion in a convolutional neural network that was pre-

trained to estimate heights of the 3D object in a spatial

simulation. Transfer learning was successful and the

model was substantially biased when estimating the

illusion stimuli. We used Bayesian statistics to calcu-

late the impact of the image properties on the estima-

tions of the neural network and tested the model on

unconventional versions of the illusion.

Still, it is necessary to cover additional aspects of

the illusion in the next studies, such as the compar-

ison between the estimations of the neural network

and the mean estimations provided by humans for the

same images. Moreover, convolutional models may

be successfully used with other optical illusions, and

the study of the models’ inner states, as mentioned in

the previous section, can also be fruitful.

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