# Maintaining Performance with Less Data

Dominic Sanderson  Department of Electronic and Computer Engineering
Brunel University London
London, United Kingdom
<EMAIL_ADDRESS>Tatiana Kalgonova  Department of Electronic
and Computer Engineering
Brunel University London
London, United Kingdom
0000-0003-4859-7152

###### Abstract

We propose a novel method for training a neural network for image
classification to reduce input data dynamically, in order to reduce the costs
of training a neural network model. As Deep Learning tasks become more
popular, their computational complexity increases, leading to more intricate
algorithms and models which have longer runtimes and require more input data.
The result is a greater cost on time, hardware, and environmental resources.
By using data reduction techniques, we reduce the amount of work performed,
and therefore the environmental impact of AI techniques, and with dynamic data
reduction we show that accuracy may be maintained while reducing runtime by up
to 50%, and reducing carbon emission proportionally.

###### Index Terms:

Data Reduction, Data Augmentation, Data Step, Data Increment, Data Cut

##  I Introduction

When creating a deep learning solution, there are two main factors that
determine its success: first, the model used, and second, the data used to
train with. The model, often a Neural Network, may be programmed, optimised,
and experimented with to reach an acceptable performance, but the data used to
train it is argued to be the determining factor [1]. If inappropriate data is
used to train and test a model, the model will perform poorly when deployed in
real-world situations and can be regarded as a bottleneck to the system’s
performance [2]. As such, consideration must be given to ensure not only an
appropriate quantity of data is available, but also data of suitable quality
[2]. It has been shown that increasing the data available to a model improves
average accuracy [3]. Because of this, when creating new models for different
tasks, researchers and data scientists alike will give as much data as
possible to train the model, wherever possible, without considering if it is
truly necessary. This direction is moving rapidly to a state that is
environmentally unsustainable and technically unachievable for those outside
of big tech companies [4].

In this paper, we analyze three novel methods to dynamically allocate the data
used to train a neural network model, for Image Classification tasks.

The article is organized as follows: In Section II, we review related methods
to reduce and increase the data used for training, as well as existing methods
to use dynamically changing training data. In Section III, we describe the
methods and environments used for our experimentation, in Section IV we
describe the experimentals techniques used in our investigation of dynamic
data use. In Section V we discuss the results of the experimentataion, and
conclude our work in Section VI.

##  II Previous Work

###  II-A Increasing data use

It is well established that a Deep Learning model will have increased
performance if provided with more training data. Acquiring additional data may
be achieved by manually gathering data, a process commonly performed either as
an individual or part of a group/company, by using freely available data
licenced as open-source, or via methods such as crowdfunding [2][4].
Alternatively, data augmentation may be used. Data augmentation is a powerful
technique that increases the amount of data available to a model [5]. Instead
of collecting more data samples, currently existing data samples may be
modified to give new, unique datapoints for training. This is preferable over
manual data collection, as annotated data is relatively scarce and can be
difficult to obtain, as well as requiring expert knowledge in order to label
and validate the data [2][6]. Table I shows the current State of the Art
models for image classification across six datasets. Each model uses some form
of data augmentation to artificially increase the amount and variety of data
available for training, which gives the model the improved accuracy that puts
the model at the top of their leaderboard.

TABLE I: Top performing Image Classification models and their data augmentation methods Dataset |  | No training
---
images
Model | Top Accuracy | Extra data/Augmentation used
MNIST | 60,000 |  | Homogeneous ensemble with
---
Simple CNN [7]
99.91% | Random rotation/translation
CIFAR-10 | 50,000 | ViT-H/14 [8] | 99.50% | Random horizontal flip, square crop
CIFAR-100 | 50,000 | EffNet-L2 (SAM) [9] | 96.08% | Horizontal flip, padding, random crop, AutoAugment [10]
SmallNorb | 24,300 | Efficient-CapsNet [11] | 98.77% | Downsampling, random 32x32 crop, random brightness
FlowerNet-102 | 1,020 | CCT-14/7x2 [12] | 99.72% | RandAugment [5], AutoAugment [10]
iNaturalist 2018 | 675,170 | MetaFormer [13] | 88.70% | RandAugment [5], AutoAugment [10]

###  II-B Reducing data use

While it is popular to use methods of increasing data, there are uses for data
reduction techniques. Such practises are uncommon, as basic linear random data
exclusion causes an exponential decrease in system performance [14]. Not only
is the average accuracy decreased, but the standard deviation of accuracy
increases, showing that the training outputs are inconsistent and difficult to
verify [14]. Contrary to this, some uses of data reduction techniques give an
increase to a Neural Network model’s performance. In fields where labelled
data is sparse, for example medical imaging, the data often suffers from class
imbalance. In such cases, undersampling techniques may be employed to reduce
the amount of data of the majority class, alleviating the effects of class
imbalance [15]. Less success is to be gained with random exclusion, but
methods such as Cluster Centroids [16] and Tomek Links [17] select the most
appropriate data to remove; they are often used with oversampling techniques
to give an increase in model performance [18]. Undersampling is a technique
used primarily to resolve class imbalance and is therefore unsuitable for
uniformly balanced datasets. However, for balanced datasets, methods exist for
identifying the ’usefulness’ of data, which allows less useful data to be
removed from training. Dimensionality reduction with algorithms such as
Uniform Manifold Approximation Projection (UMAP) [19] allows a dataset to be
observable in as little as 2 or 3 dimensions. By applying the UMAP algorithm
to a dataset, the centroid of each class (where all its datapoints revolve)
can be calculated, and data may then be removed based on its distance from its
centroid [20]. Surprisingly, by removing data closest to its class’s centroid,
accuracy is increased, despite fewer training evaluations being performed
[20].

###  II-C Variable data use

Varying data usage is the process of using different data throughout training.
It is a novel concept; the convention is that a model is trained with the same
data over multiple iterations, to allow the model’s weights and biases to
converge to a stable state. It would seem contradictory to add a change to a
system that would disturb this convergence; however, there are cases where its
use has improved performance, discussed below.

Real-time or online augmentation creates unique data each epoch. Compared to
offline augmentation, where the data is augmented once (before the model is
run), online augmentation performs augmentation before each epoch during
training [21]. Augmentation is commonly used to combat issues such as
overfitting and shortage of data [22], but for datasets with well-balanced
data it is also used to achieve higher levels of generalisability [23].
Transfer learning is used to improve a model from one domain by transferring
information from a related domain [24]. It may also be used to transfer
knowledge within the same domain, to reinforce the model when new data becomes
available [25].

Foremost, transfer learning is a technique used to reduce the amount of work
needed to train a model [25]; a model may be trained with one set of data, and
the outputs recorded; then, when more data is made available, training can
recommence, but with the previous outputs used as a starting point. This means
that the model need not be retrained with the original data each time more
data is added. In a way, this is varying data usage; throughout the course of
training, different data is used. The unique property of transfer learning is
that when a model is introduced to new training data, the old training data is
not also used. This can often cause catastrophic forgetting [26]. There has
been no investigation of using less data at the start of training, and
increasing data as training progresses. Unlike Transfer Learning, we have all
the data needed at the start of training, but we choose to withhold some of it
as training is performed. This is to show the effect of varying data use in a
controlled manner.

The experiments described in this paper investigate some extreme cases of
reducing data use at earlier stages, and show the effect of this on model
performance. Our contributions are as follows:

  * •

Three novel methods of dynamically introducing data to a model are described,
which each reduce the amount of evaluations performed:

    * –

Data Step

    * –

Data Increment

    * –

Data Cut

  * •

Testing on three datasets is shown, demonstrating the effect of these new
methods on network accuracy and runtime.

  * •

Evidence is given that these methods may be used reduce the resources required
for training while maintaining or improving on the performance of the training
output with a runtime reduction of over 50%.

##  III Methods

Traditionally, a deep learning task has three stages: Data Collection [2],
Data Processing [27] [28], and the training of the Network Model. These steps
are most often performed sequentially, but are sometimes somewhat interlinked;
for example, the Data Collection may be real-time image or text capture, or
the model may use Online Augmentation, where data is augmented differently
each training loop. Figure 1 shows the data flow of an Image Classification
model with Online Augmentation. After each training loop, the model returns to
the Augmentation stage. Figure 2 shows the model structure proposed in this
paper, with a data selection stage. This allows the model to select data
dynamically, at each loop of training. This in turn enables the various data
reduction techniques described further in this paper.

Figure 1: Common model structure with
Online Augmentation.
$\lambda=$ full dataset, $\lambda_{a}=$ augmented dataset.
$f_{a}()=$ augmentation function.  Figure 2: Proposed model structure with
Dynamic Data Selection.
$\lambda=$ full dataset, $\lambda_{r}=$ reduced dataset,
$\lambda_{a}=$ augmented dataset.
$f_{r}()=$ reduction function,
$f_{a}()=$ augmentation function

This paper introduces and investigates three sets of experiments where, each
epoch, data is selected for training during the data selection stage. The
result is a model that does not use all data for each epoch. These methods
are:

  * •

Data Step

  * •

Data Increment

  * •

Data Cut

###  III-A Datasets

Three datasets were used for experimentation: the handwritten digits dataset
MNIST [29], CIFAR10 [30], and the model toy dataset smallNORB [31]. Table II
shows the properties of each dataset.

Multiple variations of experiments were performed, to observe the effect of
varying amounts of data reduction on model performance. As such, datasets with
small image sizes are ideal, as they required less time to train.

TABLE II: Used dataset properties Dataset |  | # Training
---
images
| # Test
---
images
| Image size
---
(pixels)
| # Colour
---
channels
MNIST | 60,000 | 10,000 | 28x28 | 1
CIFAR-10 | 50,000 | 10,000 | 32x32 | 3
smallNorb | 24,300 | 24,300 | 96x96 | 1

###  III-B Hardware

Multiple hardware were used to perform training, to make the most of resources
available for this research. Training for each dataset was carried out on only
one machine, to ensure metrics were consistent across a single dataset. The
average CO2 emission for each GPU is also given, which is used for emission
reduction calculations [32]. For training on MNIST:

  * •

CPU: AMD Ryzen 9 3950X 16-Core

  * •

GPU: nVidia GeForce RTX 2070 32GB

  * •

RAM: 64GB

  * •

CO2 emission per hour: 0.066kg

For training on CIFAR-10:

  * •

CPU: Intel i7 10700

  * •

GPU: RTX 4000

  * •

RAM: 64GB

  * •

CO2 emission per hour: 0.0922kg

For training on smallNorb:

  * •

CPU: AMD Ryzen 9 3950X 16-Core

  * •

GPU: nVidia GeForce RTX 2070 32GB

  * •

RAM: 64GB

  * •

CO2 emission per hour: 0.066kg

###  III-C Performance Metrics

These experiments are to observe the effect of reducing data used each epoch
of training. To quantify this, the results of the experiments show the total
number of evaluations performed; each evaluation is a single image used for
training. The runtime of each experiment is also recorded. The successfulness
of the model is quantified with the top1 accuracy achieved. The runtime and
accuracy for each experiment is directly compared with the baseline for each
dataset, to show the increase or decrease in performance in accuracy and
runtime. As the model is otherwise unchanged when run with data reduction
techniques, the CO2 reduction of each experiment is directly correlated to the
runtime. Therefore, CO2 reduction is equal to the runtime reduction.

###  III-D Network Architecture

The model used for testing is a Convolutional Neural Network with a Simple
Monolithic Architecture. It uses 9 Convolutional Layers, and a Primary and
Secondary Capsule layer. The Capsule layer uses Homogeneous Vector Capsules,
which replace the fully connected layer. The model is based on [23], which
shows relatively high performance despite its few network layers. Tests were
run for 300 epochs with a batch size of 120. Optimisation was performed using
the Adam optimiser with an initial learning rate of 0.999, with an exponential
decay rate of 0.005 per epoch. These values provide consistent initial
settings across all experiments for all datasets.

The datapoints to exclude for each of the methods were selected at random. The
experiments were performed to gather an understanding of the effect of the
novel dynamic data reduction methods with the simplest method of data
exclusion, namely random data exclusion.

##  IV Experimentation

###  IV-A Benchmark

TABLE III: Benchmark results for MNIST, CIFAR-10 and smallNorb datasets Dataset | Dataset size |  | #Training
---
epochs
| #Total Image
---
Evaluations
| Runtime
---
(hours)
| CO2 emission
---
(kg)
Test Accuracy
|  |  |  |  |  | Best | Average | Standard deviation
MNIST | 50,000 | 300 | 18,000,000 | 02:01 | 0.134 | 99.719% | 99.701% | 0.011%
CIFAR-10 | 49,920 | 300 | 14,976,000 | 02:54 | 0.268 | 88.825% | 88.689% | 0.096%
smallNorb | 24,240 | 300 | 7,272,000 | 01:27 | 0.097 | 93.152% | 92.984% | 0.135%

Table III shows the benchmark runtime and accuracy for MNIST, CIFAR-10 and
smallNorb. These results will be used to compare the data reduction
experiments, to show the reduction in runtime and effect on performance. The
number of total image evaluations is the number of training images evaluated
multiplied by the number of training epochs.

###  IV-B Data Step

The simplest method of dynamically selecting data is to ‘step up’ the data
usage at a given point during training. A fraction of the dataset is used for
a given number of training loops, after which, the full dataset is used for
training. This splits the training process into two sections; Section one
(S1), which uses less data, and Section two (S2), which uses the full dataset.
Below are definitions for the two sections:

  * •

The number of epochs run in section 1: $S_{1}^{E}$

  * •

The amount of data used in section 1: $S_{1}^{D}$

  * •

The number of epochs run in section 2: $S_{2}^{E}$

  * •

The amount of data used in section 2: $S_{2}^{D}$

With these definitions we can derive the formula for the total number of
evaluations performed:

$n_{o}\;total\;evaluations=S_{1}^{D}*S_{1}^{E}+S_{2}^{D}*S_{2}^{E}$

This data split causes a ‘step up’ in data usage, and model will only train
with a fraction of the data for some time. Figure 3, Figure 4 and Figure 5
show how this split is applied between these sections. The hypothesis is that
due to less data being processed in section S1, there will be a reduction in
runtime. After the step, the full dataset is used; this is to ensure that all
features are made available at some point in training, although not at every
epoch. This is to aid the reduction of overfitting, which is common when too
little data is used. Experimentation for the Data Step method is split into
three parts, with three experiments each.

####  IV-B1 Starting with 25% of the dataset

Three experiments were performed, and each experiment is comprised of two
sections. Each section corresponds to the amount of data used in a single
epoch. Epochs in in $S_{1}^{E}$ use $S_{1}^{D}$, and epochs in $S_{1}^{E}$ use
the full dataset. Each of the experiments switch between these Sections at a
different epoch, shown in Table IV. The values of $S^{D}$ vary between
datasets as they each have a different amount of training images.

TABLE IV: Part 1 Initial data and epoch distribution Experiment | $S_{1}^{D}$ | $S_{2}^{D}$ | $S_{1}^{E}$ | $S_{2}^{E}$
---|---|---|---|---
E1 | 25% | 100% | 25% | 75%
E2 | 25% | 100% | 50% | 50%
E3 | 25% | 100% | 75% | 25%

Figure 3 shows graphs of the data use, for the MNIST dataset with 60,000
training images. The total area under the line represents the total number of
evaluations.

Figure 3:  Data Step Experiments starting with 25% data.
(a) Step at 25% through training, (b) Step at 50% through training, (c) Step at 75% through training  TABLE V: Data Step part 1 experimental results Test type | # Image Evaluations | Runtime (hours) | Test Accuracy | Percentage difference
---|---|---|---|---
|  |  | Best | Average |  | Standard
---
deviation
Runtime |  | Best
---
Accuracy
| Average
---
Accuracy
MNIST
Benchmark | 18,000,000 | 02:01 | 99.719% | 99.701% | 1.10E-04 |  |  |
E1 | 14,678,880 | 01:43 | 99.739% | 99.711% | 2.79E-04 | -15.148% | 0.020% | 0.010%
E2 | 11,312,880 | 01:22 | 99.749% | 99.725% | 1.82E-04 | -32.627% | 0.030% | 0.024%
E3 | 7,946,880 | 00:58 | 99.749% | 99.709% | 2.35E-04 | -51.742% | 0.030% | 0.008%
CIFAR-10
Benchmark | 14,976,000 | 02:54 | 88.825% | 88.689% | 9.59E-04 |  |  |
E1 | 12,214,320 | 02:25 | 88.675% | 88.239% | 2.78E-03 | -16.467% | -0.170% | -0.507%
E2 | 9,415,320 | 01:54 | 87.801% | 87.677% | 1.49E-03 | -34.293% | -1.153% | -1.141%
E3 | 6,616,320 | 01:22 | 86.878% | 86.608% | 2.55E-03 | -52.555% | -2.193% | -2.346%
smallNORB
Benchmark | 7,272,000 | 01:27 | 93.152% | 92.984% | 1.35E-03 |  |  |
E1 | 5,931,120 | 01:15 | 93.172% | 92.944% | 2.22E-03 | -13.922% | 0.022% | -0.043%
E2 | 4,572,120 | 01:04 | 93.003% | 92.588% | 3.70E-03 | -26.674% | -0.159% | -0.425%
E3 | 3,213,120 | 00:51 | 92.628% | 92.089% | 4.59E-03 | -41.191% | -0.562% | -0.962%

This first phase of experiments sees the largest decrease in evaluations
performed compared to each benchmark. In particular, experiment E3 has the
largest decrease out of all the Data Step experiments. As such, if it is
assumed that modifying number of evaluations performed would affect the
outputs of the program, then these experiments would differ the most from the
benchmarks. The results of this first phase of experiments have shown that
when fewer evaluations are performed, runtime is reduced. This was expected.
More importantly is the effect this has the accuracy:

  * •

With the MNIST dataset, there has been a slight increase in average and best
accuracy: E1, E2 and E3 show an average accuracy increase of 0.01%, 0.024% and
0.008% respectively.

  * •

For the CIFAR-10 dataset, there is a much more noticeable decrease in
accuracies. E1, E2 and E3 show average accuracy decrease of 0.507%, 1.141% and
2.346% respectively.

  * •

For the smallNorb dataset, there is also a decrease in accuracy: E1, E2 and E3
show average accuracy decrease of 0.043%, 0.425% and 0.962% respectively.

Based on previous research on reducing data used for training, the results on
CIFAR-10 and smallNorb datasets is perhaps to be expected. It is curious that
the MNIST shows an increase in performance, albeit a slight one. What is
interesting still is that for both MNIST and smallNorb datasets, the standard
deviation of their accuracies is greater than their benchmarks, but for
CIFAR-10 the standard deviation is lower. There seems to be no pattern, and
all three datasets show unique characteristics.

####  IV-B2 Starting with 50% of the dataset

The next three experiments under the Data Step method follow the same form as
in part one, but the initial data $S_{1}^{D}$ is 50% of the full dataset. This
is shown in Table VI, and Figure 4 shows graphs that demonstrate this with the
MNIST dataset. This set of experiments is to observe the effect of a different
amount of data reduction Experimental results are shown in Table VII.

TABLE VI: Part 2 Initial data and epoch distribution Experiment | $S_{1}^{D}$ | $S_{2}^{D}$ | $S_{1}^{E}$ | $S_{2}^{E}$
---|---|---|---|---
E1 | 50% | 100% | 25% | 75%
E2 | 50% | 100% | 50% | 50%
E3 | 50% | 100% | 75% | 25%
TABLE VII: Data Step part 2 experimentation results Test type | # Image Evaluations | Runtime (hours) | Test Accuracy | Percentage difference
---|---|---|---|---
|  |  | Best | Average |  | Standard
---
deviation
Runtime |  | Best
---
Accuracy
| Average
---
Accuracy
MNIST
Benchmark | 18,000,000 | 02:01 | 99.719% | 99.701% | 1.10E-04 |  |  |
E4 | 15,788,880 | 01:53 | 99.759% | 99.715% | 2.52E-04 | -6.638% | 0.040% | 0.014%
E5 | 13,547,880 | 01:39 | 99.719% | 99.699% | 1.59E-04 | -18.038% | 0.000% | -0.002%
E6 | 11,306,880 | 01:24 | 99.729% | 99.695% | 2.98E-04 | -30.597% | 0.010% | -0.006%
CIFAR-10
Benchmark | 14,976,000 | 02:54 | 88.825% | 88.689% | 9.59E-04 |  |  |
E4 | 13,137,840 | 02:35 | 88.464% | 88.283% | 1.78E-03 | -10.458% | -0.407% | -0.457%
E5 | 11,274,840 | 02:14 | 88.434% | 87.998% | 3.01E-03 | -22.581% | -0.441% | -0.779%
E6 | 9,411,840 | 01:55 | 88.243% | 87.711% | 3.00E-03 | -33.831% | -0.656% | -1.103%
smallNORB
Benchmark | 7,272,000 | 01:27 | 93.152% | 92.984% | 1.35E-03 |  |  |
E4 | 6,384,000 | 01:20 | 93.564% | 92.978% | 4.45E-03 | -8.475% | 0.443% | -0.006%
E5 | 5,484,000 | 01:11 | 93.263% | 92.500% | 6.19E-03 | -18.317% | 0.120% | -0.520%
E6 | 4,584,000 | 01:04 | 93.544% | 92.905% | 4.39E-03 | -26.675% | 0.421% | -0.084%

The results of this second phase of experiments mirrors those of the previous
phase of experiments. We observe that with fewer evaluations performed,
runtime is reduced; though as more evaluations are performed compared to the
previous phase, the runtime is decreased less. The accuracies show the
following patterns:

  * •

With the MNIST dataset, there is a slight increase in average and best
accuracy for all but one experiments: E4 shows an average accuracy increase of
0.014%, while E5 and E6 show average accuracy decrease of 0.002% and 0.006%
respectively.

  * •

For the CIFAR-10 dataset, there is still a decrease in accuracies across all
experiments, but not as steep a decline as in the previous experiments. E4, E5
and E6 show average accuracy decrease of 0.457%, 0.779% and 1.103%
respectively.

  * •

For the smallNorb dataset, there is also an accuracy decrease, but less
extreme than in the previous phase. E4, E5 and E6 show average accuracy
decrease of 0.006%, 0.520% and 0.084% respectively. The average accuracy of
experiment E5 is also much higher than the others, and also has a higher
standard deviation, which shows that the metrics are less stable.

Figure 4:  Data Step Experiments starting with 50% data.
(a) Step at 25% through training, (b) Step at 50% through training, (c) Step
at 75% through training

The expectation is that as these experiments use more data than the last set,
the accuracies should be improved. However, experiments E5 and E6 for MNIST
and E5 for smallNorb show worse accuracies. This may show that dynamic data
reduction in this particular way hinders the accuracy of the model, for these
particular datasets.

####  IV-B3 Starting with 75% of the dataset

The last lot of three experiments were performed, also with two sections each.
Epochs in $S_{1}^{E}$ use $S_{1}^{D}$, and epochs in $S_{1}^{E}$ use the full
dataset $S_{2}^{D}$. In the same manner as the previous two lots of
experiments, each of the experiments switch between these sections at a
different epoch, shown in Table VIII. Figure 5 shows graphs that demonstrate
this. The aim of these experiments is to observe the effect of a smaller data
reduction than in previous tests. Table IX shows the results of the
experimentation.

TABLE VIII: Part 3 Initial data and epoch distribution Experiment | $S_{1}^{D}$ | $S_{2}^{D}$ | $S_{1}^{E}$ | $S_{2}^{E}$
---|---|---|---|---
E1 | 75% | 100% | 25% | 75%
E2 | 75% | 100% | 50% | 50%
E3 | 75% | 100% | 75% | 25%

Figure 5:  Data Step Experiments starting with 75% data.
(a) Step at 25% through training, (b) Step at 50% through training, (c) Step at 75% through training  TABLE IX: Data Step part 3 experimentation results Test type | # Image Evaluations | Runtime (hours) | Test Accuracy | Percentage difference
---|---|---|---|---
|  |  | Best | Average |  | Standard
---
deviation
Runtime |  | Best
---
Accuracy
| Average
---
Accuracy
MNIST
Benchmark | 18,000,000 | 02:01 | 99.719% | 99.701% | 1.10E-04 |  |  |
E7 | 16,898,880 | 01:55 | 99.729% | 99.719% | 1.00E-04 | -5.412% | 0.010% | 0.018%
E8 | 15,782,880 | 01:54 | 99.729% | 99.717% | 1.31E-04 | -5.713% | 0.010% | 0.016%
E9 | 14,666,880 | 01:47 | 99.729% | 99.713% | 1.52E-04 | -11.948% | 0.010% | 0.012%
CIFAR-10
Benchmark | 14,976,000 | 02:54 | 88.825% | 88.689% | 9.59E-04 |  |  |
E7 | 14,061,360 | 02:47 | 88.675% | 88.625% | 5.59E-04 | -4.027% | -0.170% | -0.072%
E8 | 13,134,360 | 02:36 | 88.655% | 88.402% | 2.98E-03 | -9.888% | -0.192% | -0.324%
E9 | 12,207,360 | 02:25 | 88.353% | 88.239% | 1.14E-03 | -16.565% | -0.531% | -0.507%
smallNORB
Benchmark | 7,272,000 | 01:27 | 93.152% | 92.984% | 1.35E-03 |  |  |
E7 | 6,828,000 | 01:23 | 93.705% | 93.059% | 5.45E-03 | -5.036% | 0.593% | 0.081%
E8 | 6,378,000 | 01:19 | 93.581% | 92.926% | 5.19E-03 | -9.528% | 0.461% | -0.062%
E9 | 5,928,000 | 01:16 | 93.300% | 92.751% | 3.85E-03 | -13.423% | 0.159% | -0.250%

This final phase of testing uses the least data reduction, and as such, the
decrease in both runtime and average accuracy is the smallest. The following
patterns are observed:

  * •

With the MNIST dataset there is an increase in accuracy for all experiments:
E7, E8 and E9 show an average accuracy increase of 0.018%, 0.016% and 0.012%
respectively. Experiment E7 shows a decrease in standard deviation of
accuracies, compared to the benchmark.

  * •

With the CIFAR-10 dataset, the results follow the same patterns as with the
results of the last groups of experiments - the accuracy decrease directly
correlates to the reduction in data used, although this correlation is non-
linear. E7, E8 and E9 show an average accuracy decrease of 0.072%, 0.324% and
0.507% respectively.

  * •

With the smallNorb dataset, E7 shows an average accuracy increase of 0.081%,
and E8 and E9 show an average accuracy decrease of 0.062% and 0.250%
respectively. There is an increase in performance with experiment E7; this is
the only experiment on the smallNorb dataset to have an increase in
performance. The average accuracies of experiments E8 and E9 are decreased,
which shows how much of an effect such a slight difference in number of
evaluations performed has on the model.

###  IV-C Data Increment

Like the methods described in the Step section, the incrementing method uses a
fraction of the data at the start of training. At a previously defined
interval, more of the data is added incrementally, causing multiple smaller
steps in data usage throughout the training. As such, the number of sections
present is equal to the number of increments.

Adding the data this way will decrease the time spent training for the earlier
epochs, while ensuring that all training data is eventually used. Figure 6
shows example data usage graphs with varying increment intervals. Table X
shows the results of the experimentation.

Figure 6:  Data Increment experiments, with (a) 33% data increments. (b) 10% data increments.  TABLE X: Data Increment experimentation results Test type | # Image Evaluations | Runtime (hours) | Test Accuracy | Percentage difference
---|---|---|---|---
|  |  | Best | Average |  | Standard
---
deviation
Runtime |  | Best
---
Accuracy
| Average
---
Accuracy
MNIST
Benchmark | 18,000,000 | 02:01 | 99.719% | 99.701% | 1.10E-04 |  |  |
33% | 11,992,080 | 01:23 | 99.739% | 99.703% | 2.89E-04 | -31.790% | 0.020% | 0.000%
25% | 11,241,120 | 01:20 | 99.719% | 99.699% | 1.42E-04 | -34.210% | 0.000% | 0.000%
10% | 9,903,720 | 01:09 | 99.709% | 99.687% | 2.50E-04 | -42.630% | -0.010% | -0.010%
5% | 9,456,720 | 01:07 | 99.729% | 99.701% | 1.93E-04 | -44.820% | 0.010% | 0.000%
1% | 9,099,120 | 01:08 | 99.709% | 99.677% | 2.18E-04 | -44.090% | -0.010% | -0.020%
0.33% | 9,041,880 | 01:04 | 99.719% | 99.695% | 1.96E-04 | -47.310% | 0.000% | -0.010%
CIFAR-10
Benchmark | 14,976,000 | 02:54 | 88.825% | 88.689% | 9.59E-04 |  |  |
33% | 9,981,360 | 02:00 | 88.153% | 87.723% | 2.69E-03 | -30.920% | -0.760% | -1.090%
25% | 9,361,560 | 01:52 | 87.771% | 87.582% | 2.18E-03 | -35.130% | -1.190% | -1.250%
10% | 8,253,000 | 01:42 | 87.550% | 87.317% | 1.52E-03 | -41.380% | -1.440% | -1.550%
5% | 7,877,520 | 01:36 | 87.490% | 87.275% | 1.92E-03 | -44.530% | -1.500% | -1.590%
1% | 7,581,360 | 01:33 | 87.299% | 87.106% | 1.25E-03 | -46.370% | -1.720% | -1.780%
0.33% | 7,531,560 | 01:33 | 87.560% | 87.225% | 2.06E-03 | -46.560% | -1.420% | -1.650%
smallNORB
Benchmark | 7,272,000 | 01:27 | 93.152% | 92.984% | 1.35E-03 |  |  |
33% | 4,840,200 | 01:07 | 92.954% | 92.482% | 3.61E-03 | -23.040% | -0.210% | -0.540%
25% | 4,536,240 | 01:03 | 92.628% | 92.463% | 2.21E-03 | -28.180% | -0.560% | -0.560%
10% | 3,999,840 | 01:02 | 92.768% | 92.373% | 3.51E-03 | -28.850% | -0.410% | -0.660%
5% | 3,821,040 | 00:59 | 92.735% | 92.291% | 3.45E-03 | -32.180% | -0.450% | -0.740%
1% | 3,676,560 | 00:57 | 92.562% | 92.115% | 4.91E-03 | -34.520% | -0.630% | -0.930%
0.33% | 3,651,600 | 00:56 | 92.525% | 92.248% | 2.39E-03 | -35.700% | -0.670% | -0.790%

Using the data increment method causes much less data to be used throughout
training. The most extreme case of this is with steps of 0.33%, where roughly
half of the evaluations are performed. This makes the data increment method a
more extreme data reduction method than the data step method. For each
dataset, the observations are as follows:

  * •

For the MNIST dataset, average accuracy is maintained with 33% and 25%
increments, but is never improved upon the baseline. For both of these cases,
runtime is shortened by at least 30%.

  * •

For the CIFAR-10 dataset, all average accuracies are worse by at least 1%.
Like MNIST, runtime is shortened by a minimum of 30%.

  * •

For the smallNorb dataset, as the dataset has much fewer training images, both
the runtime and accuracy reduction are not as large as with other datasets.
For all experiments, average accuracy is reduced by less than 1%, while
runtime is reduced by between 20% to 40%.

Based on these results, the data increment method seems to have a more
detrimental effect on network performance. It is also worth noting that the
standard deviation of average accuracies are worse than the benchmark across
all experiments, meaning the networks are less reliable at getting consistent
results.

###  IV-D Data Cut

The data cut method splits a dataset into multiple cuts of equal size. Each
cut is then used for an equal number of training epochs. The more splits a
deck is cut into, the less data is present in each split. As such, the more
splits there are, the fewer evaluations will be performed. Figure 7 shows how
a dataset may be divided into 6 cuts, for training over 300 epochs.

This approach is similar to Transfer Learning techniques, in which a network
that has already been trained on some data is introduced to a new set of
training data. The new input data contains the same classes as it had already
been trained with, so the network does not need to adapt to new classes.
However, while previous methods in this paper introduced new training data to
accompany previously used data, this method does not include the previously
used data when new data is introduced - it continues training on only new
data. Normally, when introducing new data via transfer learning, a fresh
network is deployed, and given the weights and biases of previously trained
networks. The Data Cut method aims to give the network different data at
different times, while preserving the state of the network throughout a single
training cycle. Table XI shows the results of the experimentation.

Figure 7:  A dataset is split into 6 sections (cuts).
A single split is used for training every 50 epochs out of 300.  TABLE XI: Data Cut experimentation results Test type | # Image Evaluations | Runtime (hours) | Test Accuracy | Percentage difference
---|---|---|---|---
|  |  | Best | Average |  | Standard
---
deviation
Runtime |  | Best
---
Accuracy
| Average
---
Accuracy
MNIST
Benchmark | 18,000,000 | 02:01 | 99.719% | 99.701% | 1.10E-04 |  |  |
2 Splits | 9,000,000 | 01:07 | 99.719% | 99.699% | 1.23E-04 | -44.272% | 0.000% | -0.002%
3 Splits | 5,976,000 | 00:44 | 99.679% | 99.655% | 1.68E-04 | -63.331% | -0.040% | -0.046%
6 Splits | 2,988,000 | 00:25 | 99.659% | 99.596% | 4.10E-04 | -78.800% | -0.060% | -0.105%
9 Splits | 1,980,000 | 00:19 | 99.598% | 99.526% | 6.82E-04 | -84.016% | -0.121% | -0.175%
12 Splits | 1,476,000 | 00:16 | 99.508% | 99.337% | 1.41E-03 | -86.604% | -0.211% | -0.365%
15 Splits | 1,188,000 | 00:14 | 99.367% | 99.102% | 2.85E-03 | -88.074% | -0.352% | -0.600%
18 Splits | 972,000 | 00:13 | 99.297% | 98.855% | 3.86E-03 | -89.124% | -0.423% | -0.848%
21 Splits | 828,000 | 00:12 | 99.237% | 97.476% | 1.83E-02 | -89.909% | -0.483% | -2.232%
24 Splits | 720,000 | 00:11 | 96.365% | 95.645% | 7.33E-03 | -90.501% | -3.363% | -4.068%
CIFAR-10
Benchmark | 14,976,000 | 02:54 | 88.825% | 88.689% | 9.59E-04 |  |  |
2 Piles | 7,488,000 | 01:33 | 86.777% | 86.528% | 1.54E-03 | -46.518% | -2.306% | -2.436%
3 Splits | 4,968,000 | 01:04 | 85.492% | 85.143% | 3.20E-03 | -62.692% | -3.753% | -3.998%
6 Splits | 2,484,000 | 00:37 | 82.269% | 81.984% | 3.27E-03 | -78.491% | -7.381% | -7.560%
9 Splits | 1,656,000 | 00:28 | 80.110% | 79.673% | 2.65E-03 | -83.690% | -9.811% | -10.166%
12 Splits | 1,224,000 | 00:23 | 77.791% | 77.072% | 6.68E-03 | -86.431% | -12.422% | -13.098%
15 Splits | 972,000 | 00:20 | 75.552% | 72.888% | 2.02E-02 | -88.013% | -14.943% | -17.816%
18 Splits | 828,000 | 00:19 | 72.440% | 69.066% | 2.70E-02 | -88.971% | -18.447% | -22.125%
21 Splits | 684,000 | 00:17 | 67.731% | 66.030% | 1.94E-02 | -89.880% | -23.748% | -25.548%
24 Splits | 612,000 | 00:16 | 64.518% | 60.600% | 4.62E-02 | -90.370% | -27.365% | -31.671%
smallNORB
Benchmark | 7,272,000 | 01:27 | 93.152% | 92.984% | 1.35E-03 |  |  |
2 Splits | 3,636,000 | 00:59 | 93.106% | 92.744% | 3.37E-03 | -32.490% | -0.049% | -0.257%
3 Splits | 2,412,000 | 00:51 | 92.880% | 92.616% | 2.36E-03 | -41.598% | -0.292% | -0.396%
6 Splits | 1,188,000 | 00:42 | 92.946% | 92.104% | 5.30E-03 | -52.148% | -0.221% | -0.946%
9 Splits | 792,000 | 00:38 | 91.840% | 91.408% | 3.93E-03 | -55.659% | -1.408% | -1.694%
12 Splits | 576,000 | 00:36 | 90.924% | 90.012% | 7.59E-03 | -58.625% | -2.392% | -3.195%
15 Splits | 468,000 | 00:36 | 89.975% | 87.791% | 2.33E-02 | -58.774% | -3.410% | -5.584%
18 Splits | 396,000 | 00:36 | 88.461% | 87.139% | 1.05E-02 | -58.445% | -5.035% | -6.286%
21 Splits | 324,000 | 00:36 | 86.865% | 80.511% | 8.85E-02 | -58.097% | -6.749% | -13.414%
24 Splits | 288,000 | 00:37 | 86.894% | 80.776% | 7.48E-02 | -57.390% | -6.718% | -13.128%
Figure 8:  Runtime reduction and accuracy loss with Data Cut method, on
CIFAR-10 dataset

Of the three experimental methods described in this paper, the Data Cut method
cuts the number of evaluations performed down the most. It becomes apparent
quickly that the more splits are used, the more the accuracy is degraded.
Results on the MNIST dataset show that by cutting the dataset into two splits,
we almost cut the runtime in half while maintaining the benchmark accuracy.
With 24 splits, there is a large drop in performance. This shows a ‘cut-off
point’, where too few evaluations are performed, and the model doesn’t have
enough iterations to allow the weights and biases to converge, causing more
classifications to fail. Figure 8 shows the percentage difference in runtime
and accuracy of the CIFAR-10 dataset, compared to its benchmark. The results
of this dataset show the biggest percentage differences. The graph shows that
by introducing more data splits, the runtime decreases non-linearly, while the
accuracy decreases more steadily. Therefore, the most appropriate number of
data splits must be chosen to benefit the most from this dataset reduction
technique.

##  V Discussion

The runtime of the experiment correlates directly with the total number of
evaluations performed. This is as expected, as the fewer evaluations are
performed, the fewer calculations are performed by the neural network.

For the Data Step method, the most notable results are those of the MNIST
dataset, and E7, E8 and E9 of smallNorb, as they show an increase in accuracy
despite the fewer evaluations performed. An future research direction would be
to investigate why there is a performance increase in these cases, and how the
detrimental effect on stability may be rectified. Also, further research into
why CIFAR-10 does not show any increase could be explored.

As with the Data Step method, the Data Increment method shows that reducing
the number of evaluations performed decreases the runtime. However, despite
both methods performing the same fundamental task of data reduction, they
produce different results for roughly the same number of evaluations
performed. For example, comparing experiments E2 of the data step method and
data increments of 25%, we can observe the accuracies when roughly the same
number of evaluations are performed.

The number of evaluations for the two experiments are:

  * •

E2 for MNIST dataset uses 11,312,880 evaluations, data increments of 25% use
11,241,120 evaluations, a difference of 0.634%.

  * •

E2 for CIFAR-10 dataset uses 9,415,320 evaluations, data increments of 25% use
9,361,560 evaluations, a difference of 0.571%.

  * •

E2 for smallNorb dataset uses 4,572,120 evaluations, data increments of 25%
use 4,536,240 evaluations, a difference of 0.785%.

However, despite the marginal difference in number of evaluations performed,
the difference in accuracy between these two experiments are:

  * •

E2 for MNIST dataset gives an accuracy increase of 0.024%, data increments of
25% gives an accuracy difference of 0%.

  * •

E2 for CIFAR-10 dataset gives an accuracy decrease of 1.141%, data increments
of 25% gives an accuracy decrease of 1.250%.

  * •

E2 for smallNorb dataset gives an accuracy decrease of 0.425%, data increments
of 25% gives an accuracy difference of 0.560%.

For all cases, the data increment method shows worse accuracy. While it is
true that the data increment method has fewer evaluations performed, the
difference in data (less than 1% for all datasets) amounts to less than 3
epochs of training. This amount is negligable as the top accuracy of a model
settles at much earlier epochs. The results of the experimentation show that
decreasing the number of evaluations with the Increment method has a more
detrimental effect on the accuracy of the model.

Finally, the data cut method shows the largest reduction in evaluations
performed. Below is a comparison between experiments data increment of 0.33%
and data cut with 2 splits:

  * •

Data Increment for MNIST dataset uses 9,041,880 evaluations, data cut with 2
splits use 9,000,000 evaluations, a difference of 0.463%.

  * •

Data Increment for CIFAR-10 dataset uses 7,531,560 evaluations, data cut with
2 splits use 7,488,000 evaluations, a difference of 0.578%.

  * •

Data Increment for smallNorb dataset uses 3,651,600 evaluations, data cut with
2 splits use 3,636,000 evaluations, a difference of 0.427%.

Comparing the accuracies between these two experiments, we observe:

  * •

Data Increment for MNIST gives an accuracy decrease of 0.01%, data cut with 2
splits gives an accuracy decrease of 0.02%.

  * •

Data Increment for CIFAR-10 gives an accuracy decrease of 1.65%, data cut with
2 splits gives an accuracy decrease of 2.436%.

  * •

Data Increment for smallNorb gives an accuracy decrease of 0.79%, data cut
with 2 splits gives an accuracy decrease of 0.257%.

Accuracy for all observed data cut experiments are worse than those of the
data increment method. We can conclude that, as the data increment method is
inferior to the data step method, the data cut method is least effective at
maintaining accuracy. However, if the need is to reduce the runtime by as much
as possible without hindering accuracy too greatly, a data cut of 9 splits
appears to be the best compromise between runtime and accuracy.

##  VI Conclusion

The results of this paper have shown that, contrary to the norm, reducing the
data used for training has in some cases improved the performance of the
model. It proves that not all data may be necessary for training, and in fact
some data may hinder it.

The approaches used are somewhat brutish, excluding data randomly without
consideration of the value of the datapoints removed. Other works cited have
shown algorithmic approaches to select which data to remove to enhance
performance, which is a next step for varying data usage. Still, having shown
that even random exclusion has improved results, it would seem that varying
data use is to be explored in further detail.

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$\rho^{2}\leq\hat{\epsilon}/|S|$, we further have:

$\displaystyle\mathrm{Pr}\left(\theta^{k}\in
N_{\hat{\epsilon}}(\bar{\theta}),~{}q^{k}\in
N_{\delta}\left(\mathrm{EQ}(\bar{\theta})\right),~{}\forall
k\right)\geq\mathrm{Pr}\left(\begin{array}[]{l}|\theta^{k}(s)-\bar{\theta}(s)|<\frac{\hat{\epsilon}}{|S|},~{}\forall
s\in[\bar{\theta}],~{}\forall k\text{ and}\\\ \theta^{k}<\rho^{2},~{}\forall
s\in S\setminus[\bar{\theta}],~{}q^{k}\in
N_{\delta}\left(\mathrm{EQ}(\bar{\theta})\right),~{}\forall
k\end{array}\right)$
$\displaystyle=\mathrm{Pr}\left(\theta^{k}(s)<\rho^{2},\forall s\in
S\setminus[\bar{\theta}],\forall
k\right)\cdot\mathrm{Pr}\left(\left.\begin{array}[]{l}|\theta^{k}(s)-\bar{\theta}(s)|<\frac{\hat{\epsilon}}{|S|},\forall
s\in[\bar{\theta}],\forall k\\\ \text{and }q^{k}\in
N_{\delta}\left(\mathrm{EQ}(\bar{\theta})\right),~{}\forall
k\end{array}\right|\begin{array}[]{l}\theta^{k}(s)<\rho^{2}.\\\ \forall s\in
S\setminus[\bar{\theta}],\forall k\end{array}\right)$

For any $\theta^{1}\in N_{\epsilon^{1}}(\bar{\theta})$ and any $q^{1}\in
N_{\delta^{1}}\left(\mathrm{EQ}(\bar{\theta})\right)$, we know from Lemmas 5 –
6 that:

$\displaystyle\mathrm{Pr}\left(\theta^{k}(s)<\rho^{2},~{}\forall s\in
S\setminus[\bar{\theta}],~{}\forall k\right)>\gamma,\text{ and }$
$\displaystyle\mathrm{Pr}\left(\left.\begin{array}[]{l}|\theta^{k}(s)-\bar{\theta}(s)|<\frac{\hat{\epsilon}}{|S|},~{}\forall
s\in[\bar{\theta}],~{}\forall k\\\ \text{and }q^{k}\in
N_{\delta}\left(\mathrm{EQ}(\bar{\theta})\right),~{}\forall
k\end{array}\right|\theta^{k}(s)<\rho^{2},~{}\forall s\in
S\setminus[\bar{\theta}],~{}\forall k\right)=1$

Therefore, for any $\theta^{1}\in N_{\epsilon^{1}}(\bar{\theta})$ and any
$q^{1}\in N_{\delta^{1}}\left(\mathrm{EQ}(\bar{\theta})\right)$, the states of
learning dynamics satisfy $\lim_{k\to\infty}\mathrm{Pr}\left(\theta^{k}\in
N_{\bar{\epsilon}}(\theta),~{}q^{k}\in
N_{\bar{\delta}}\left(\mathrm{EQ}(\bar{\theta})\right)\right)>\gamma$. Thus,
$\left(\bar{\theta},\bar{q}\right)$ is locally stable. $\square$


Paper already accepted by the American Journal of Physics

# Practical study of optical stellar interferometry

P. Rodríguez-Ovalle, A. Mendi-Martos, A .Angulo-Manzanas, I. Reyes-Rodríguez,
M.Pérez-Arrieta
Aula Espazio-Gela, Escuela de Ingeniería de Bilbao
UPV/EHU
Plaza Ingeniero Torres Quevedo, 1, Bilbao, 48013, Bizkaia, Spain
&M. A. Illarramendi, A. Sánchez-Lavega
Departamento de Física Aplicada, Escuela de Ingeniería de Bilbaon
UPV/EHU
Plaza Ingeniero Torres Quevedo, 1, Bilbao, 48013, Bizkaia, Spain
<EMAIL_ADDRESS>

###### Abstract

We present a technique to observe stellar inteferograms, as well as the
detailed analysis of those created by three bright stars: Betelgeuse, Rigel,
and Sirius. It is shown that the atmospheric turbulence is responsible for the
reduction of the long-exposure fringe visibility of the obtained interference
patterns. By using different baselines in our interferometer, we are able to
distinguish the decay of the visibility with the baseline, observe how
different parameters such as the diameter or the distribution of the holes in
our interferometer affect the pattern, and measure the atmospheric turbulence.
This experiment can be performed by postgraduate students to gain practical
experience in optical interferometry in astronomy.

_K_ eywords Interferometry, stars, seeing effect, interference patterns,
atmospherical turbulence

##  1 Introduction

In optical astronomy, interferometric techniques have been used ever since
their discovery over a hundred years ago, for they enable increased angular
spatial resolution [1]. Interferometric techniques are used in both ground-
based and space-based observatories. Of particular interest is their use in
the study of the properties of stars, for example to measure their sizes, to
characterize multiple stellar systems, and to precisely determine their
positions in the sky and their motions (astrometry) [2, 3, 4, 5, 6]. A list of
optical and infrared astronomical interferometers can be found in ref. [7].
The technique is not simple to work with and is limited by the sky’s
atmospheric conditions at the time of observation, called ’seeing conditions’.

Teaching astronomical interferometry to undergraduate or graduate physics
students is therefore not an easy task. The topic is usually presented briefly
in optics courses [8, 9, 10] and, when dealing with astronomy, most textbooks
mainly focus on the classic Michelson interferometer [11]. Few practical
activities are carried out for training in these techniques at university
level and this is done mostly in relation to astronomy, astrophysics and space
science courses. At laboratory level, Michelson-type radio interferometer [12]
or interferometric experiments [13] have been introduced. In the Master in
Space Science and Technology of the Basque Country University UPV/EHU [14], we
have incorporated practical work into the educational program with optical
telescopes coupled to laser sources with polymeric optical fibers simulating
stars [15, 16].

In this paper we describe a new interferometry experiment. By coupling
different plates using several diaphragms, with variable apertures and
separations, at the entrance of a 28 cm telescope, we have built a simple
interferometer and, with it, we have observed three bright stars (Betelgeuse,
Rigel and Sirius). We have analyzed the stellar interferograms using optical
interferometry theory. It is shown that, very much like what happens in
research-grade telescopes or in stellar interferometers [17], atmospheric
turbulence causes a reduction in the long-exposure fringe visibility by a
factor that depends on the atmospheric coherence length, or Fried parameter
($r_{0}$). By studying the decay of the visibility with the baseline, we have
estimated the Fried parameter for each case. The star sizes could not be
estimated due to the small values of the baselines provided by the experiment
[1, 2, 3, 4, 5], but our work describes a simple technique to obtain
interference patterns produced by bright stars, the analysis of which
introduces important physical concepts, such as spatial and temporal
coherence.

##  2 Theoretical background

A simplified version of a stellar interferometer can be obtained by using a
single telescope whose aperture is covered by a lid with two circular pinholes
separated by a variable distance, called the baseline $B$. The working
principle of this simple interferometer is the same as Young’s double-slit
experiment, where the beams emerging from each pinhole of diameter $D$ form
interference fringes in the focal plane of the telescope or plane of
observation. A schematic illustration of this interferometer is shown in Fig.
1. The quality of the interference fringes detected at the observation plane
is measured by the fringe visibility or contrast $V$:

$V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}\text{ },$ (1)

where $I_{max}$ and $I_{min}$ are the maximum and minimum intensities of the
interference fringes, respectively. The visibility is scaled from 0 to 1,
where 0 means no fringes and 1 denotes fringes with perfect contrast.

Figure 1: Simple scheme of a double pinhole stellar interferometer. The two
pinholes ($Q_{1}$ and $Q_{2}$) of diameter $D$ are separated by a distance
$B$, and are placed far away from the source.

Taking the simplest model to describe the light emission of a star, i.e. a
circular, uniform, and spatially incoherent source emitting quasi-
monochromatic light of wavelength $\lambda$, the diffraction-limited intensity
at the plane of observation can be expressed as follows [9]:

$I(\alpha)=I_{0}\left(\frac{J_{1}\left(\frac{\pi}{\lambda}\alpha
D\right)}{\frac{\pi}{\lambda}\alpha
D}\right)^{2}\left(1+V_{s}\cos\left(\frac{2\pi}{\lambda}\alpha
B\right)\right),$ (2)

with:

$V_{s}=2\left\lvert{\frac{J_{1}\left(\frac{\pi\alpha^{\prime}B}{\lambda}\right)}{\frac{\pi\alpha^{\prime}B}{\lambda}}}\right\rvert,$
(3)

where $J_{1}$ is the first-order Bessel function of the first kind,
$\alpha^{{}^{\prime}}$ is the angular size of the source and $\alpha$ is the
observation angle. The product $\alpha B$ is the optical-path difference for
small values of $\alpha$, and $I_{0}$ is a constant related to the incident
intensity. $V_{s}$ is the spatial fringe visibility for this simple model.
$V_{s}$ does not depend on $\alpha$, and is inversely proportional to the
source angular size and to the baseline distance. In fact, the steady decrease
of $V_{s}$ from a value of 1 when $(\pi\alpha^{{}^{\prime}}B)/\lambda=0$ to a
value of 0 when $(\pi\alpha^{{}^{\prime}}B)/\lambda=1.22\pi$ allows the
determination of the source size if $V_{s}$ is measured as a function of the
baseline distance $B$. The easiest procedure to estimate the source diameter
is to determine the lowest value of B for which the interference fringes
disappear. The reduction of $V_{s}$ as the source size or the baseline
increases is a result of the spatial coherence of the light. The function
$\frac{J_{1}(\frac{\pi\alpha D}{\lambda})}{\frac{\pi\alpha D}{\lambda}}$
represents the irradiance distribution of the diffraction-limited response to
a point source illuminating one circular pinhole, which is the Airy pattern.
The total number of visible fringes is limited by the diffraction effect, as
well as by the value of the visibility function $V_{s}$. If the baselines used
in the measurements are short enough to provide very small values of
$(\pi\alpha^{{}^{\prime}}B)/\lambda$ , the fringe visibility is very close to
1 and, therefore, Eq. 2 could be simplified to:

$I(\alpha)=I_{0}\left(\frac{J_{1}\left(\frac{\pi}{\lambda}\alpha
D\right)}{\frac{\pi}{\lambda}\alpha
D}\right)^{2}\left(1+\cos\left(\frac{2\pi}{\lambda}\alpha B\right)\right),$
(4)

which is the diffraction-limited irradiance distribution produced by a point
source emitting a quasi-monochromatic light. If a finite spectral bandwidth
$\Delta\lambda$ is taken into account for the source, the resulting fringe
pattern can be inferred by adding up the interference patterns given by Eq. 4
at all wavelengths. This effect reduces the fringe visibility as the
observation angle $\alpha$ or, equivalently, the time delay between
interfering beams $\tau$ increases. The time delay for the interference
fringes to vanish is called the coherence time $\tau_{c}$, which is defined as
$\tau_{c}$=$\lambda^{2}$/c$\Delta\lambda$. Taking this definition into
account, an approximate diffraction-limited irradiance distribution for a
polychromatic light at the plane of observation can be written as:

$I(\alpha)=I_{0}\left(\frac{J_{1}\left(\frac{\pi}{\lambda}\alpha
D\right)}{\frac{\pi}{\lambda}\alpha
D}\right)^{2}\left(1+V_{t}\cos\left(\frac{2\pi}{\lambda}\alpha
B\right)\right),$ (5)

with:

$V_{t}=1-\alpha
B\frac{\Delta\lambda}{\lambda^{2}}=1-m\frac{\Delta\lambda}{\lambda}$ (6)

The exact expression for the temporal fringe visibility $V_{t}$ depends on the
form of the spectral bandwidth [9, 10]. The parameter $m$ in Eq.6 is an
integer number that indicates the order of interference. One of the
consequences of observing a source with a significant bandwidth is the
dependence of $V_{t}$ on the position on the observation plane. In particular,
its value decreases down to 0 as $\alpha$ increases up to
$\alpha_{e}=\lambda^{2}/B\Delta\lambda$, or as the interference order becomes
$m=\lambda/\Delta\lambda$. This decrease in the visibility is therefore more
pronounced when $\Delta\lambda$ is larger. The behavior of $V_{t}$ as a
function of the wavelength distribution is a result of the temporal coherence
of the light. If the values of $B$ and $\Delta\lambda$ used in the
measurements result in very high values of $\alpha_{e}$, $V_{t}$ at the
observation positions close to the optical axis would be nearly 1. In that
case, Eq. 5 could also be simplified to Eq. 4.

The preceding equations are only valid for an atmosphere without turbulence.
Turbulence causes the fringes to undergo random changes due to inhomogeneities
encountered by light along its path to the interferometer. For long-exposure
times and under the assumption $V_{s}$=$V_{t}$=1, the diffraction-limited
irradiance distribution at the plane of observation can be expressed as
follows:

$I(\alpha)=I_{0}\left(\frac{J_{1}\left(\frac{\pi}{\lambda}\alpha
D\right)}{\frac{\pi}{\lambda}\alpha
D}\right)^{2}\bigg{(}1+V_{a}\cos\left(\frac{2\pi}{\lambda}\alpha
B\right)\bigg{)},$ (7)

with:

$V_{a}=exp\left[-3.44\left(\frac{B}{r_{0}}\right)^{5/3}\right]$ (8)

$V_{a}$ is the atmospheric fringe visibility of the detector’s time-averaged
interference pattern produced by the light that has passed through the Earth’s
turbulent atmosphere [18]. The Fried parameter, $r_{0}$, is an atmospheric
coherence length that indicates the length over which the wavefront remains
unperturbed and it is a measure of the seeing quality, i.e., the quality of
the atmospheric conditions at the time of observation [17]. The smaller
$r_{0}$ is, the larger the effects of turbulence on the propagating wave.
$r_{0}$ varies as $\lambda^{6/5}$, so that it becomes smaller at shorter
wavelengths which implies a more severe turbulence effect on the wavefront.
Typical values for $r_{0}$ under good seeing conditions are 15-20 cm for
visible wavelengths [19]. In our case, the turbulence-induced random phase
fluctuations of the fields drive visibility rapidly to zero as the baseline
$B$ is increased (see expression of $V_{a}$ in Eq.8); that is, atmospheric
turbulence causes the light field to become spatially incoherent for baselines
$B$ $\geq$ $r_{0}$. By analyzing the dependence of $V_{a}$ on baseline, we can
estimate $r_{0}$, which allows us to characterize the atmospheric turbulence
[18].

##  3 Experimental set-up

Interference fringes were obtained for the bright stars Betelgeuse, Rigel and
Sirius. Betelgeuse has the largest angular size of any star in our night sky.
It was the first star that was resolved by Michelson and Pease using stellar
interferometry [1]. As a red giant, its large size comes along with a
relatively low temperature. Rigel, a blue super-giant star ($\beta$ Orion) has
a surface temperature higher than 10000 K. Despite having an angular size much
smaller than Betelgeuse’s, its high temperature makes it a bright object in
the sky. Sirius has a much smaller radius compared with Betelgeuse or Rigel.
As an A-type star, its temperature is close to 10000 K. However, this star is
closer to Earth than the other two, at only 8.6 light years. Table 1 shows the
stars’ relevant properties. Sirius is a binary star, consisting of Sirius A
(the brighter star) and Sirius B. Since the ratio between the emitted
brightness of both stars, known as the contrast factor $f$ of the binary star,
is very small, Sirius can be approximated as a single star for this study.

Table 1: Properties of the studied stars: their radius (in units of the Sun’s radius $R_{\odot}$), their distance from Earth (in light-years), their angular aperture (mas = miliarcseconds), $f$ represents the brightness ratio between the two stars in the case of double stars, $m_{v}$ is their apparent magnitude (measure of the brightness), $T$ their surface temperature. Star | Radius ($\boldsymbol{\rm{R_{\odot}}}$) | Distance ($\boldsymbol{\rm{Ly}}$) | Size (mas) | f | $\text{{m}}_{\text{{V}}}$ | T(K) |
---|---|---|---|---|---|---|---
Betelgeuse | 887 | 497.95 | 54.04 | - | 0.52 | 3600 |
Rigel | 79 | 863 | 2.6 | - | 0.15 | 12100 |
Sirius A | 2 | 8.6 | 6.04 | 0.0001 | -1.46 | 9940 |

The observations were carried out with a Celestron C11 XLT telescope with a
280 mm aperture and focal distance of 2800 mm from the Aula Espazio Gela
Observatory in Bilbao (Spain) [14]. Once the telescope was well focused, its
aperture was blocked by a lid with two or four circular pinholes. Only the
case of two pinholes will be discussed in the main text. The restults for the
four pinholes set-up are given in the Supplementary Material (See online
supplemental material at url to be inserted by AIPP). Fig. 2 illustrates the
arrangement of the pinholes on the lids. Several two-pinhole lids were made
with different combinations of pinhole diameters $D$ and baselines $B$. The
lids were made of a robust cardboard and the pinholes were carefully shaped in
order to minimize errors. All the chosen values for the pinhole diameters $D$
allowed us to observe the interference fringes clearly. Table 2 summarizes the
lid parameters used for the three stars.

Figure 2: Diagram of the lid with two holes. The diameters $D$ of all the
holes are equal for a given lid. Table 2: Values of the experimental
parameters used for each observation night. The absolute error for the values
of D and B is 1 mm. For each filter, the bandwidth is indicated.

BETELGEUSE
---
D (mm) | B (mm) | Filter | Date (yy-mm-dd)
30 | 133 | H$\alpha$ 35 nm | 17-03-15
30 | 238 | H$\alpha$ 35 nm | 17-03-15
51 | 147 | H$\alpha$ 35 nm | 17-03-15
RIGEL
D (mm) | B (mm) | Filter | Date (yy-mm-dd)
51 | 147 | H$\alpha$ 35 nm | 17-03-15
51 | 222 | H$\alpha$ 35 nm | 17-03-15

SIRIUS
---
D (mm) | B (mm) | Filter | Date (yy-mm-dd)
22 | 220 | H$\alpha$ 7 nm | 17-03-10
22 | 158 | H$\alpha$ 7 nm | 17-03-10
61 | 178 | H$\alpha$ 7 nm | 17-03-10
30 | 238 | H$\alpha$ 35 nm | 17-03-23
30 | 133 | H$\alpha$ 35 nm | 17-03-23
50 | 222 | H$\alpha$ 35 nm | 17-03-23
65 | 162 | H$\alpha$ 35 nm | 17-03-23
66 | 162 | H$\alpha$ 35 nm | 17-03-29
65 | 205 | H$\alpha$ 35 nm | 17-03-29
90 | 187 | H$\alpha$ 35 nm | 17-03-29
66 | 162 | H$\beta$ 8.5 nm | 17-03-29
90 | 187 | H$\beta$ 8.5 nm | 17-03-29

Images of the fringe pattern were acquired with a camera attached to the focal
plane of the telescope. We used a DMK41AU02 canera equipped with the Sony
ICX205AL CCD chipset with a pixel size of 4.65 $\mu m$. To improve the quality
of the images, we processed them by stacking video sequences for every
observation with the same methodology described in previous papers [20, 21].
Hence, we can obtain images with a better quality and remove the random noise
that could appear in individual frames due to the effects of atmospheric
turbulence and telescope vibrations. To obtain resolved fringes at different
wavelengths, we used two $H_{\alpha}$ filters centered on a wavelength of
656.3 nm with bandwidths of 7 and 35 nm respectively, and a $H_{\beta}$ filter
with a 8.5 nm bandwidth centered at 486.1 nm [22].

We used very long exposure times, ranging from 0.3 s to 2.4 s so that suitable
fringe images were detected. This fact adversely affected the measurements due
to the additional effect of different turbulence scales in the atmosphere,
which usually change on timescales longer than a few miliseconds. The obtained
long-exposure images were converted to digital values, according to the camera
range. The brightness levels from the interference patterns were digitized
into 512 levels. Those levels were treated with the free software ImageJ,
which allowed us to obtain the profile of the interference patterns [23]. The
software also allowed us to determine the values of the irradiance $I_{max}$
and $I_{min}$ to calculate the visibility of the fringe pattern. In order to
show the photometric cuts as a function of the angular size in the focal
plane, the distances in pixels were transformed to radians as seen through the
2800 mm focal length of our telescope, by taking into account the magnifying
effect of the telescope’s Barlow lens.

##  4 Results and discussion

In this Section, we will study the interference patterns obtained for the
different values of baselines ($B$) and spectral bandwidths ($\Delta\lambda$)
displayed in Table 2. Using such small values of $B$ and such narrowband
filters leads to the atmospheric turbulence being the main cause of the
reduction in the visibility of the observed fringe patterns. Fig. 3 shows
examples of such interference patterns and the corresponding relative
irradiance profiles obtained for different combinations of $D$ and $B$ with
the 35 nm bandwidth-$H_{\alpha}$ filter and the two pinholes lid. Figs. 3(a)
and (b) correspond to the Betelgeuse star, Figs. 3(c) and (d) to Rigel, and
Figs. 3(e) and (f) to Sirius. As can be seen, the fringe patterns obtained
with the double-pinhole lid deviate little from the diffraction-limited
interference patterns. We can notice both that the interference fringes are
modulated by the larger Airy pattern generated by the circular apertures and
that the spatial frequency of the fringes increases with $B$. We also notice
that the fringe visibility is poor in all cases ($V$ $<$ 0.88). Fig. 4
displays the interference fringes from Sirius obtained with the narrowest
$H_{\alpha}$ filter and the $H_{\beta}$ filter, which also have a low-
visibility. The interference patterns obtained with other experimental
parameters (see Table 2) are very similar to those shown in Figs. 3 and 4.
Since the conditions to approximate $V_{s}$ and $V_{t}$ to 1 are satisfied in
our experiment, the obtained visibilities are mainly due to the effect of the
atmospheric turbulence. On the one hand, the very short baselines are such
that $\frac{\pi\alpha^{{}^{\prime}}}{\lambda}$ $<$ 0.4 rad, thus providing
that $V_{s}$ is very close to 1. Note that the fact that the visibility does
not decrease with B implies that we cannot estimate the source size. For
Betelgeuse, which is the largest star analyzed in this work, interference
fringes would disappear entirely with a baseline of about 3 meters [1], which
is unrealistic in most practical setups. On the other hand, the order of
interference for which the fringe visibility goes to zero using the broadest
filter ($H_{\alpha}$ filter with $\Delta\lambda$ = 35 nm) is 18, which is far
away from the maximum of interference. This implies that $V_{t}$ at the
observation positions near the optical axis can also be approximated by 1.
Therefore, we can conclude that the cause of the reduction in the visibility
of our patterns is due to atmospheric turbulence and use Eq. 7 to describe the
irradiance distribution of the interference fringes. From the analysis of the
dependence of the visibility values on the baseline, we will be able to derive
the Fried parameter for each of our experimental conditions.

Fig. 5 shows the visibility $V$ of the stars analyzed as a function of the
two-pinhole distance. The data plotted in each figure have been obtained on
the same night. The values of the fringe visibility are quite low and they
tend to diminish as the baseline increases, as expected from Eq. 8. In the
calculation of $V$, $I_{max}$ has been determined from the irradiance of the
central maximum and $I_{min}$ from the average of the two adjacent minima. All
irradiance measurements have been corrected for the background irradiance and
the errors in the visibility values were estimated from the standard
deviations calculated from five independent measurements taken for each
interference patterns. It can be shown that the error in the visibility
arising from the uncertainties of the baselines and pinhole diameters ($\pm$
$1$ $mm$) is negligible in comparison with the standard-deviation error.

Figure 3: Two-pinhole interference pattern produced with the $H_{\alpha}$
filter, $\Delta\lambda$ = 35nm. On the left and in the center, the
interference patterns obtained experimentally and the corresponding
photometric cuts; on the right, the theoretical curves calculated from Eq. 7
with $\lambda$ = 656.3 nm, and the corresponding values of $D$, $B$, and
$r_{0}$. (a) Betelgeuse star, $D$ = 51 mm, $B$ = 147 mm; (b) Betelgeuse star,
$D$ = 30 mm, $B$ = 238 mm; (c) Rigel star, $D$ = 51 mm, $B$ = 147 mm; (d)
Rigel star, $D$ = 51 mm, $B$ = 222 mm; (e) Sirius star, $D$ = 90 mm, $B$ = 187
mm; (f) Sirius star, $D$ = 66 mm, $B$ = 205 mm. Figure 4: Two-pinhole
interference patterns produced by Sirius with the narrow $H_{\alpha}$ filter
($\lambda$ = 656.3 nm, $\Delta\lambda$ = 7 nm, $D$=22 mm, $B$=220 mm, top),
and the $H_{\beta}$ filter ($\lambda$ = 486.1 nm, $\Delta\lambda$ = 8.5 nm,
$D$=66 mm, $B$=162 mm, bottom). On the left and in the center, the
interference patterns obtained experimentally and the corresponding
photometric cuts. On the right the theoretical curve calculated from Eq. 7.
Figure 5: Visibility measurements as a function of the two-pinhole baseline
distance obtained for the three stars. The dashed lines are the fittings of
the data to the expression of $V_{a}$. Data obtained from (a) Betelgeuse and
Rigel on March 15, 2017; (b) Sirius on March 10, 2017; (c) Sirius on March 23,
2017; (d) Sirius with $H_{\alpha}$ and $H_{\beta}$ on March 29, 2017.

By fitting the expression of $V_{a}$ in Eq. 8 to the measured visibility data
plotted in Fig. 5, we have estimated the Fried parameter $r_{0}$ for each
observation night at the wavelengths used (Table 3), i.e. at 656.3 nm for the
fringes obtained with the $H_{\alpha}$ filters and at 486.1 nm for those
captured with the $H_{\beta}$. The highest values of the Fried parameter were
obtained for the night of March 15, 2017. It can be noticed that the $r_{0}$
values at 656.3 nm obtained from the interference fringes of Sirius are lower
than those obtained from Betelgeuse and Rigel. This could be due to stronger
atmospheric turbulence on the corresponding observation nights. The longer
exposure times used to obtain the interference fringes from Sirius due to its
lower emission in the spectral range of the $H_{\alpha}$ filter, could also
affect the values obtained. On the other hand, the spectral dependence of the
two $r_{0}$ values calculated from Fig. 5 (d), at 656.3 nm and 486.1 nm agrees
very well with the theoretical spectral dependence of the Fried parameter,
that is, $r_{0}\propto\lambda^{6/5}$. By using the proportionality constant
($0.0133\pm 0.0008\;cm/nm^{6/5}$) calculated from the value of $r_{0}$ at
$\lambda$ = 656.3 nm, we get $r_{0}$ = 22 $\pm$ 2 cm at $\lambda$ = 486.1 nm,
which is consitent with the experimental value displayed in Table 3. Using the
same proportionality constant, we estimate that $r_{0}$ at 500 nm is 23 $\pm$
2 cm. This value is slightly higher than the typical Fried parameter at a good
observation site [19]. This result, together with the larger values obtained
for $r_{0}$ on the other observation nights, indicates that the atmospheric
effect on the experimental images was small in all cases. In all of them, the
obtained values for $r_{0}$ are of the order of the size of the telescope and,
of course, larger than the diameter of the holes. Hence, these values of the
atmospheric coherence lengths $r_{0}$, are in agreement with the fact that the
shape of the interference fringes are dominated by the diffraction effect, as
was shown in Figs. 3 and 4. Nevertheless, more accurate results for Fried
parameters could be obtained with a systematic study of the dependence of the
visibility with $B$.

Table 3: Fried parameters at the corresponding wavelength for each observation night obtained from the fittings of the experimental points of Fig 5. Errors for the Fried parameters have been estimated from the fittings using Eq. 7. Date (yy-mm-dd) | Star |  $r_{0}$ (cm) |  $\lambda$ (nm) |  $\Delta\lambda$ (nm)
---|---|---|---|---
17-03-15 | Betelgeuse |  36 $\pm$ 10 | 656.3 | 35
17-03-15 | Rigel | 33 $\pm$ 2 | 656.3 | 35
17-03-10 | Sirius | 24 $\pm$ 5 | 656.3 | 7
17-03-23 | Sirius | 25 $\pm$ 2 | 656.3 | 35
17-03-29 | Sirius | 32 $\pm$ 2 | 656.3 | 35
17-03-29 | Sirius | 22 $\pm$ 4 | 486.1 | 8.5

In the third column of Fig. 3 and Fig. 4, we have included the theoretical
photometric curves obtained from Eq. 7. As can be seen, the experimental and
theoretical photometric curves are in good agreement regarding the variation
of the number of visible fringes with different combinations of $D$ and $B$.
The fringe visibilities are also well described by Eq. 8. If the fringe
visibility is high enough, the number of visible fringes can be easily
calculated from the angular radius of the Airy disk ($1.22\lambda/D$) and the
angular fringe spacing ($\lambda/B$), in the same way as in the
diffraction–limited interference fringes. For instance, one can predict that
the change in $D$ and $B$ for the two fringe patterns produced by Betelgeuse
(Figs. 3(a) and (b)) results in a strong increase in the total number of
visible fringes from 7 to 19, which is clearly seen in the experimental
curves. In contrast, if the fringe visibility is low, it is more difficult to
analyze the effect of changing the values of $D$ and $B$ on the interference
patterns, as in the case of the Sirius interferograms (Figs. 3(e) and (f)).
Likewise, if the fringes were only diffraction-limited, we should observe 25
fringes in the interference pattern displayed in Fig. 4 (a) and 5 fringes in
that displayed in Fig. 4 (b). In spite of the low visibility values of the two
interference patterns displayed in Fig. 4, the theoretical predictions indeed
agree quite well with the experimental results.

##  5 Conclusion

In this work, we show a simple experiment in which long-exposure interference
patterns produced by three stars, Betelgeuse, Rigel and Sirius, can be
detected using a bandpass filter and a digital camera coupled to a small
telescope obscured by lids with two (or more) pinholes. It has been
demonstrated that the obtained long-exposure fringe patterns produced by the
stars can be very well described by diffraction-limited interference patterns
produced by a quasi-monochromatic point source, but with a fringe visibility
reduced due to the atmospheric turbulence. Through the analysis of the fringe
visibility as a function of the two-pinhole baseline, we have been able to
characterize the affectation of the atmospheric turbulence to astronomical
observations by calculating the Fried parameter for each observation night.
The experiment can be useful for students and/or teachers in universities. Its
simplicity and the interest of the obtained results make this experiment ideal
for postgraduate subjects, such as: Astrophysics, Astronomy, Optics or
Astronomical and Space Signal Processing. In addition to showing the principle
of operation of the Michelson stellar interferometer, the experiment
underlines important concepts related to spatial interferometry, such as
spatial and temporal coherence and astronomical seeing.

## acknowledgments

This work has been supported by a research grant from Diputación Foral de
Bizkaia-Bizkaiko Foru Aldundia to Aula EspaZio Gela at UPV/EHU, with
additional support in the frame of Spanish project PID2019-109467GB-I00
(MINECO/FEDER, UE), with additional support in the frame of the Grant
PID2019-109467GB-I00 funded by MCIN/AEI/10.1303 /501100011033/ and by Grupos
Gobierno Vasco IT1742-22.

## Competing interests

The authors declare no competing interests.

## Availability of data and materials

The datasets used and/or analyzed during the current study available from the
corresponding author upon reasonable request.

##  Appendix A Suplementary materials: 4-pinhole study case

Another type of interferometer can be made by covering the telescope with a
lid having more than two pinholes. This arrangement could provide more
baselines, with different modules and orientations, thus allowing to conduct
additional measurements and therefore to obtain more information [2, 3, 4, 5].
In astronomical interferometers, as many different baselines as possible are
required in order to get a good quality image.

The interference pattern produced by four pinholes can be generated by the
superposition of all light beams coming from each hole. The general expression
for the irradiance distribution can be given in terms of six different
visibilities describing the correlation of the optical fields for each pair of
holes. We have worked out the diffraction-limited interference pattern
produced by four pinholes placed as shown in Fig. A.1. Since two of the
baselines are equal in this case, only four visibilities are necessary.
Following Eq. 7, the intensity is given by:

$\displaystyle I(\alpha)=I_{0}\left(\frac{J_{1}\left(\frac{\pi}{\lambda}\alpha
D\right)}{\frac{\pi}{\lambda}\alpha
D}\right)^{2}\bigg{(}4+4V_{1}\cos\bigg{(}\frac{\pi}{\lambda}\alpha\left(B_{2}-B_{1}\right)\bigg{)}\,\,+$
(9) $\displaystyle
4V_{2}\cos\bigg{(}\frac{\pi}{\lambda}\alpha\left(B_{2}+B_{1}\right)\bigg{)}+2V_{3}\cos\left(\frac{2\pi}{\lambda}\alpha
B_{1}\right)+2V_{4}\cos\left(\frac{2\pi}{\lambda}\alpha B_{2}\right)\bigg{)}.$

For very short baseline distance ($B$ $<<$$\lambda$/$\pi$ $\alpha^{\prime}$)
and a quasi-monochromatic light, the visibilities can be determined by the
Earth’s atmosphere turbulent effects by using the expression of $V_{a}$ shown
in Eq. 8. In an atmosphere without turbulence, Eq. 9 could be simplified to:

$I(\alpha)=I_{0}\left(\frac{J_{1}\left(\frac{\pi}{\lambda}\alpha
D\right)}{\frac{\pi}{\lambda}\alpha
D}\right)^{2}\bigg{(}\cos\left(\frac{\pi}{\lambda}\alpha
B_{1}\right)+\cos\left(\frac{\pi}{\lambda}\alpha B_{2}\right)\bigg{)}^{2}$
(10)

If, in addition, the holes are equally spaced, $B_{1}=B_{2}/3=B$, by applying
trigonometric relations, Eq. 10 could be simplified to [8, 9, 10]:

$I(\alpha)=I_{0}\left(\frac{J_{1}\left(\frac{\pi}{\lambda}\alpha
D\right)}{\frac{\pi}{\lambda}\alpha
D}\right)^{2}\frac{\sin^{2}\left(4\left(\frac{\pi}{\lambda}\alpha\mathrm{B}\right)\right)}{\sin^{2}\left(\frac{\pi}{\lambda}\alpha\mathrm{B}\right)}.$
(11)

Figure A.1: Diagram of the lid with four holes: $D$ = 30 mm, $B_{1}$ = 133 mm,
$B_{2}$ = 238 mm. The diameters $D$ of all the holes are equal.

Fig. A.2 shows the interference pattern produced by Betelgeuse on the night of
17-03-15, using the lid with the four holes and the $H_{\alpha}$ filter of 35
nm in bandwidth. We can clearly appreciate the effect of having more than two
holes on the interference pattern: the emergence of secondary maxima of
irradiance and the narrowing and brightening of the fringes (principal maxima
of irradiance) [8, 9, 10]. Since the baselines are very short and the light
can be assumed to be quasi-monochromatic, the obtained interference pattern
can be described from Eq. 9 using the parameters of the measurement and the
four visibilities calculated from the expression of $V_{a}$ of Eq. 8. The
experimental photometric cut and the theoretical prediction have been plotted
in the center and right panels of Fig. A.2, respectively. The four
visibilities have been calculated by using the equation of $V_{a}$ with the
four different baseline distances of the lid and with the value of the Fried
parameter estimated previously from Betelgeuse measurements carried out in the
same night. As can be seen, a very good agreement is obtained between the
theoretical and experimental curves, confirming the value of the Fried
parameter.

Figure A.2: Four-pinhole interference pattern produced by Betelgeuse with the
$H_{\alpha}$ filter of 35 nm (17-03-15). On the left and in the center, the
interference pattern obtained experimentally and its photometric cut. On the
right, the theoretical curve calculated from Eq. 9 with $\lambda$ = 656.3 nm,
$D$= 30 mm, $B_{1}$ = 133 mm $B_{2}$= 238 mm, $V_{1}$ = 0.87, $V_{2}$ = 0.32,
$V_{3}$ = 0.18, $V_{4}$ = 0.52.

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