Low-Cost On-device Partial Domain Adaptation (LoCO-PDA):
Enabling efficient CNN
retraining on edge devices

Aditya Rajagopalimp
Christos-Savvas Bouganisimp

impIntelligent Digital Systems Lab, Imperial College London, UK

Aditya<EMAIL_ADDRESS>
Partial Domain Adaptation, CNN, Machine Learning, Efficient Training, Machine
Learning, ICML

With the increased deployment of Convolutional Neural Networks (CNNs) on edge devices, the
uncertainty of the observed data distribution upon deployment has led researchers to to utilise
large and extensive datasets such as ILSVRC'12 to train CNNs.
Consequently, it is likely that the observed data distribution upon deployment is a subset of
the training data distribution.
In such cases, not adapting a network to the observed data distribution can cause performance
degradation due to negative transfer and alleviating this is the focus of Partial Domain
Adaptation (PDA).
Current works targeting PDA do not focus on performing the domain adaptation on an
edge device, adapting to a changing target distribution or reducing the cost of deploying the
adapted network.
This work proposes a novel PDA methodology that targets all of these directions and opens
avenues for on-device PDA.
LoCO-PDA adapts a deployed network to the observed data distribution by enabling it to be retrained on an edge device.
Across subsets of the ILSVRC12 dataset, LoCO-PDA improves classification accuracy by 3.04pp on
average while achieving up to 15.1x reduction in retraining memory consumption and 2.07x
improvement in inference latency on the NVIDIA Jetson TX2.
The work is open-sourced at link removed for anonymity.

§ INTRODUCTION

The rapid increase in the memory and compute capability of modern edge devices has resulted in
the increased deployment of CNNs on the edge.
The deployment life cycle starts with the architecture and weights of the network being
optimised on servers using large datasets such as ILSVRC'12 [12].
Once deployed on an edge device such as the NVIDIA Jetson TX2, networks typically remain
unchanged and if adapted, require communication of data back to a server.
This raises both data-privacy concerns and is not possible when there is no edge-server

As the deployed network is trained on a large and extensive dataset such as ILSVRC'12, it is
expected that the label space of the observed data distribution after deployment (target
domain) is a subset of the label space of the training data distribution (source domain).
However utilising this deployed network to classify solely target domain data triggers negative
transfer [4].
Negative transfer refers to the loss in classification performance due to the fact that the
network has learnt to classify a much larger number of classes (including source-only classes)
than those in the target domain.
Most works in the field of Partial Domain Adaptaion (PDA) aim to prevent negative transfer, but
do not consider the possibility of a constantly changing target domain or the memory
and compute footprint of the adaptation process or the resulting adapted network - all of which
are important considerations for real-life applications.

The goal of this work is to enable the partial domain adaptation of a deployed model to be
performed directly on an edge device, within some memory and time budget, to improve the
performance of the adapted model on the observed target domain.
Additionally, the proposed methodology is sufficiently low-cost to allow for adaptation to a
changing target domain and can obtain gains in inference time memory and latency footprint by
adapting a network that is smaller than that originally deployed for inference.

The novel contribution of this work is LoCO-PDA, a methodology that uses Variational
Autoencoders (VAEs) to perform low-cost PDA on edge devices.
As demonstrated in Fig.<ref>, unlike other PDA methodologies, LoCO-PDA aims to
optimise for the trade-off between domain adapted classification accuracy, and the adaptation
process memory consumption and latency.
[Times are collected on the server grade NVIDIA 2080Ti as SOTA methodologies are too
memory intensive to be run on the Jetson TX2 edge device.]
It achieves on average a 7x and 4x improvement in adaptation time and memory respectively while
performing comparably to other SOTA PDA methodologies.

§ BACKGROUND AND RELATED WORKS

§.§ Partial Domain Adaptation (PDA)

As discussed in [5], the primary focus of unsupervised domain adaptation is
on learning domain-invariant feature representations without access to target domain
labels, so that the same feature extractor and classifier combination can be utilised
in both the source and target domains.
However, these works [17, 6] assume that both source and target
domains have the same label space, which is an unlikely scenario to be observed in
A more likely scenario in real-world deployments is PDA where the target label space is
assumed to subsume the source label space.

The challenge addressed by SOTA works targeting PDA such as ETN [5], IWAN
[21], and PADA [4] is to train a feature extractor that generates an aligned activation
distribution for classes of images that are common to both source and target domains,
while preventing distribution alignment on source only classes.
All three works utilise an adversarial framework that consists of a feature extractor
and domain classifier.
The domain classifier aims to identify in an unsupervised manner the difference between
the source and target domains, while the feature extractor is trained to reduce the
discrepancy between the domains.
The domain classifier outputs weights that identify classes as source-only or shared;
and the three works vary in their approaches to utilising these weights in training the
feature extractor.

Different to such approaches, this work proposes a novel solution to the problem of PDA
which does not change the source domain feature extractor but rather retrains just the
classifier to better discriminate between the classes observed in the target domain.

We propose to estimate the target domain label subspace using predictions of the deployed
As no ground-truth labels are used, this approach is still unsupervised.
Furthermore, compared to adversarial PDA approaches, the proposed methodology has a low
enough memory and compute requirement to be performed directly on an edge device and
benefits from adaptability to changes in the target domain.

§.§ Edge Device CNN Training

To the best of our knowledge, the only two works which also explore on-device training
of CNNs are TinyTL [3] and PersEPhonEE [13].
TinyTL proposes a variation of the MBConv [18] layer called a
Lite-Residual layer which downsamples the input activations, performs grouped and 1x1
convolutions on them, upsamples the output and adds it to the output of the original MBConv
By only retraining the lite bypass layers on the edge, the methodology reduces
memory requirements of training as only downsampled intermediate activations need to be

TinyTL is evaluated on a transfer-learning scenario where the source and target
domains do not share the same label space and there is access to ground-truth labels of the
target domain.
This is outside the scope of this work and hence is not compared to.

PersEPhonEE proposes an early-exit methodology instead of pruning to reduce the cost of
both inference and training on an edge device.
By placing an early exit after various residual blocks along ResNet50, the authors propose
to improve both training and inference time by only retraining the desired early-exit

They demonstrate between 2x-22x improvement in training latency depending on the early-exit
used and up to 20pp improvement in classification accuracy depending on the target domain.
The authors provide results for various subsets of the ILSVRC'12 dataset and propose to use
the output of the final classifier as labels to retrain earlier classifiers (unsupervised).
Thus this work falls under the PDA framework and is compared against.

§ PROBLEM DESCRIPTION

The inputs to the system are a model $\mathcal{M}^0$ and a large training data
distribution $\mathcal{D}$.
Let $\mathcal{M}^0$ be the model that has been trained on $\mathcal{D}$ and prepared for
initial deployment on an edge device.
Upon deployment, the system observes a data distribution $\mathcal{D}'$ that is assumed to be
able to change over time.
Furthermore, the label space of $\mathcal{D}'$ is assumed to be a subset of that of

Due to this assumption, it is expected that a model ($\mathcal{M}^p$), that has a lower
memory and latency footprint than $\mathcal{M}^0$ can be adapted to the observed $\mathcal{D}'$
after deployment as a smaller number of classes are being classified ($\mathcal{D}' \subset
\mathcal{D}$).
This results in an adapted network that has significant gains in inference memory and latency.
In this work we consider $\mathcal{M}^p$ to be a network that is obtained by pruning away $p$%
of the memory footprint of $\mathcal{M}^0$.
[The pruning and retraining process is performed offline on $\mathcal{D}$.
Both Taylor First Order (TFO) [14] and OFA [2]
methodologies are used for pruning.]
However, LoCO-PDA can be applied even if $\mathcal{M}^0$ and $\mathcal{M}^p$ are completely
different networks.

The goal and novel contribution of this work is to create a system that enables the retraining
of $\mathcal{M}^p$ on an edge device such that the accuracy of $\mathcal{M}^p$ on
$\mathcal{D}'$ is at least as good as that of $\mathcal{M}^0$ on $\mathcal{D}'$.

§ LOW-COST TRAINING

[node distance = 1.8cm,thick,font=]
minimum width = 1cm,
minimum height = 1cm
] () at (0,0) ;
minimum width = 1cm,
minimum height = 1cm
] (images) at (0.1,-0.1) ;
minimum width = 1cm,
minimum height = 1cm,
] () at (0.2,-0.2) ;
[below = 0.2cm of images] (X_label) $\mathcal{X}_t$;

right of = images,
rotate = -90,
minimum width = 1cm,
minimum height = 1.5cm,
trapezium stretches body,
] (m0) ;
[] at (m0.center) $\mathcal{M}^0$;

[-stealth] (images.east) – (m0.south);

[right of = m0] (m0_preds)

$\forall y_t \in \mathcal{Y}_t$

$P(y_t | \mathcal{X}_t)$
[-stealth] (m0.north) – (m0_preds.west);

below right of = m0_preds,
minimum width = 1cm,
minimum height = 1cm,
trapezium stretches body,
] (vae_decoder) ;
[] at (vae_decoder.center)

[-stealth] (m0_preds.east) -| (vae_decoder.north);

[left of = vae_decoder] (z) $z \in \mathcal{N}(0,I)$;
[-stealth] (z.east) – (vae_decoder.west);

[below of = vae_decoder] (acts) $\overline{\mathcal{A}_s}$;
[-stealth] (vae_decoder.south) – (acts.north);

right of = acts,
rotate = -90,
minimum width = 1cm,
minimum height = 1cm,
trapezium stretches body
] (student_fc) ;
[] at (student_fc.center) $\mathcal{M}^p_{FC}$;

[-stealth] (acts.east) – (student_fc.south);

minimum width = 2cm,
minimum height = 1cm,
left of = acts,
] (student_fe) $\mathcal{M}^p_{FE}$;

[-stealth, gray!40] (X_label.south) |- (student_fe.west);
[-stealth, gray!40] (student_fe.east) – (acts.west);

Proposed methdology for low-cost adaptation of the pruned network's classifier
$\mathcal{M}^0$ produces a set of predicted labels ($\mathcal{Y}_t$), for target domain images
The predicted classes along with their frequency of occurrence ($P(y_t | \mathcal{X}_t)$) is  provided to the trained decoder.
The decoder then generates a set of estimated activations
$\overline{\mathcal{A}_s}$ which can be used to train the pruned network's classifier.
Consequently, this process bypasses the feature extractor of the pruned network ($\mathcal{M}^p_{FE}$).

This section motivates the proposed training methodology by identifying methods to reduce the
cost of retraining.

The greyed out path in Fig.<ref> is an abstraction of a CNN network
The feature extractor ($\mathcal{M}^p_{FE}$) takes as input an image and outputs activations
that are then passed to the classifier ($\mathcal{M}^p_{FC}$) to predict the class of the
The cost of training the feature extractor is significantly larger than that of the
As stated in [3], the limiting factor of training CNNs on edge devices is the
memory required to store intermediate activations used by backpropagation.
Consequently, the primary cost of retraining a CNN lies in the feature extractor stage and
bypassing this would significantly reduce the cost of retraining a network.

One way to bypass the training of the feature extractor stage is to only train the classifier.
This only requires inference, which is a highly optimised process on edge devices.
However, doing so also requires the storage of images with which retraining can be performed.
This can be infeasible on edge devices due to the limited available memory.
Consequently, LoCO-PDA aims to retrain just the classifier but also bypass the execution of the
feature extractor, thereby significantly reducing the memory requirements of the retraining

§ METHODOLOGY

Most modern networks have linear classifier layers as they have highly expressive feature
extractors that create representations of images (activations) that can be linearly separated.
LoCO-PDA exploits the simplicity of these activations and uses light-weight VAEs to generate
them instead of performing inference through the feature extractor of $\mathcal{M}^{p}$.
These generated activations can then be used to train the classifier.
This process is described by the black path in Fig.<ref>.
Utilising a pruned model achieves gains in inference memory consumption and latency, while
retraining the classifier allows the pruned network to regain the accuracy lost by pruning
while also optimising the network's weights to the observed data distribution.
There is no added overhead to performing inference through $\mathcal{M}^0$ as this is the
originally deployed model which is being used for inference and thus statistics of
$P(y_t|\mathcal{X}_t)$ can be collected over time before initiating retraining.

§.§ VAE Training Process

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minimum width = 1cm,
minimum height = 1cm
] (images) at (0.1,-0.1) ;
minimum width = 1cm,
minimum height = 1cm,
] () at (0.2,-0.2) ;
[below = 0.2cm of images] (X_label) $\mathcal{X}_s$;

minimum width = 2cm,
minimum height = 1cm,
right of = images,
] (student_fe) $\mathcal{M}^p_{FE}$;

[-stealth] (images.east) – (student_fe.west);

[right of = student_fe] (acts) $\mathcal{A}_s$;
[-stealth] (student_fe.east) – (acts.west);

below right = 0.8 and 1.4 of acts,
rotate = -180,
minimum width = 1cm,
minimum height = 1cm,
trapezium stretches body,
] (vae_encoder) ;
[] at (vae_encoder.center)

[-stealth] (acts.east) – ++(0.3,0) -| (vae_encoder.south);

[left = 0.4 of vae_encoder.20] (src_labels) $y_s \in \mathcal{Y}_s$;
[-stealth] (src_labels.east) – (vae_encoder.20);

[below = 0.25 of vae_encoder.110] (var) $\Sigma$;
[below = 0.3 of vae_encoder.65] (mean) $\mu$;
[-stealth] (vae_encoder.65) – (mean.north);
[-stealth] (vae_encoder.110) – (var.north);

scale = 0.5,
below = 0.2 of var] (mul) $\times$;
scale = 0.5,
left = 0.2 of mul] (add) $+$;

[right = 0.2 of mul] (n) $n \in$ ${\mathcal{N}(0,\mathcal{I})}$;
[left = 0.2 of add] (z) $z$;

[-stealth] (n.west) – (mul.east);
[-stealth] (mul.west) – (add.east);
[-stealth] (mean.south) – (add.north);
[-stealth] (var.south) – (mul.north);

below of = student_fe,
rotate = -90,
minimum width = 1cm,
minimum height = 1cm,
trapezium stretches body,
] (vae_decoder) ;
[] at (vae_decoder.center)

[-stealth] (add.west) – (z.east);
[-stealth] (z.west) – ++(-0.3,0) |- (vae_decoder.70);
[-stealth] (src_labels.south) |- (vae_decoder.110);

[left of = vae_decoder] (recon_x) $\overline{\mathcal{A}_s}$;
[-stealth] (vae_decoder.south) – (recon_x.east);

[thin, blue!40]
below = 0.9 of vae_decoder] (recon_loss)
$|| \overline{\mathcal{A}_s} - \mathcal{A}_s ||^2_2$;
right = 0.9 of recon_loss] (kld_loss)

VAE training process used to train the decoder that is used in
The whole process if performed offline using source domain images ($\mathcal{X}_s$) and
available ground truth source domain labels ($\mathcal{Y}_s$).
The CVAEs are trained to reproduce source domain activations ($\mathcal{A}_s$) when provided
with the desired class $y_s$ and a random vector $z$ sampled from a multivariate standard
normal distribution.
The blue boxes show the loss functions used to train the CVAEs, where the left box is the
reconstruction mean-squared-error (MSE) between the estimated activations
$\overline{\mathcal{A}_s}$ and the true activations $\mathcal{A}_s$ and the right box is the
KL-divergence between the learnt distribution of the latent space ($\mathcal{N}(\mu,\Sigma)$)
and $\mathcal{N}(0,\mathcal{I})$.

LoCO-PDA uses an extension of VAEs known as Conditional-VAEs (CVAEs).
VAEs [10] are generative models that learn to model some distribution $P(t)$
where $t$ are high-dimensional data points which can represent a variety of inputs.
They do this by learning a latent representation that can then be sampled to generate data
points close $P(t)$.
Fig.<ref> shows the process of training the CVAEs to generate the
activations of $\mathcal{M}^p$ at the output of the feature extractor.
CVAEs [20] allow for the sampling of the latent space to be conditioned on a
desired class of $\mathcal{A}_s$.
One CVAE is required per source domain dataset $\mathcal{D}$ and model $\mathcal{M}^{p}$
and they are trained on $\mathcal{M}^{p}$'s activations for all images ($\mathcal{X}_s$) in
$\mathcal{D}$ before deployment.

The encoder in Fig.<ref> learns the function $q_{\phi}(z |
\mathcal{A}_s, y_s)$ which maps activations to latent space variables.
The decoder then learns the function $p_{\theta}(\mathcal{A}_s | z, y_s)$ which
generates an activation corresponding to provided class.
It learns to generate activations for all source domain classes ($\mathcal{Y}_s$) in order
to be able to adapt to varying target domains.

The loss function used is that proposed in [8] for conditional VAEs and is
given by:

\begin{equation}
\begin{aligned}
\mathcal{L}(\theta, \phi; \mathcal{A}_s, z, y_s, \beta) = &
\mathbb{E}_{q_{\phi}(z | \mathcal{A}_s, y_s)}
[\textrm{log}\, p_\theta(\mathcal{A}_s | z, y_s)] \\
&- \beta\ \mathcal{KL}(q_\phi(z | \mathcal{A}_s, y_s) || p(\bm z))
\end{aligned}
\end{equation}

The weights of the encoder and decoder that are learnt are given by $\phi$ and $\theta$
$\beta$ is a parameter that balances the trade-off between reconstruction error and latent
space regularisation.
$p(\bm z)$ is the prior distribution and is set to $\mathcal{N}(\bm 0, \bm I)$ in order to
obtain a closed form solution to the regularisation term.
The conditional decoding process samples $\bm z \sim \mathcal{N}(\bm 0, \bm I)$, augments
the variable with $y_s$ corresponding to the desired class and provides this as input to
the trained decoder as shown in Fig.<ref>.
This process is facilitated by learning a latent space that closely follows
$\mathcal{N}(\bm 0, \bm I)$ (KL-divergence loss term), thus training a decoder that can map
a randomly sampled variable in this distribution to the desired network activation based on
the conditioning information.[Refer to Appendix
<ref> for the training hyperparameters.]

§.§ Identifying $\mathcal{D}'$

Once the VAE is trained, it is deployed along with the initial deployment model
$\mathcal{M}^0$ and the pruned model $\mathcal{M}^p$ that will be adapted to the observed
data distribution.
Before retraining $\mathcal{M}^p$ on $\mathcal{D}'$, it is necessary to identify the labels
We propose to identify these labels from the distribution predicted by the currently
deployed network $\mathcal{M}^0$ (Fig.<ref>).
The assumption here is that $\mathcal{M}^0$ has sufficient accuracy to identify the classes
observed upon deployment.

§.§ Retraining $\mathcal{M}^p$

The retraining process gets as input a distribution $P(y_t|\mathcal{X}_t)$
which assigns a probability per observed class based on the frequency with which that class
was observed.
The tunable parameter $R$ decides how many total activations to use when performing the
classifier retraining.
The $R$ activations that are generated follow the class distribution
$P(y_t|\mathcal{X}_t)$, thus a sufficiently small probability could result in no activation
being generated for that class.
Under the assumption that $\mathcal{M}^0$ predicts the correct class with a sufficiently
high probability, this approach can be robust to noisy class distributions generated by
$\mathcal{M}^0$ that may contain incorrect class predictions.
These activations are then used to train $\mathcal{M}^{p}$'s classifier.
Compared to the feature extractor, the decoder can be up to 20x smaller in terms of memory
footprint which reduces the cost of retraining such that it can be performed on a CPU
within a reasonable time budget.
[Refer to Appendix <ref> for the retraining

§ EVALUATION

This section first evaluates LoCO-PDA under the constraints of a practical deployment scenario.
Sec.<ref> then compares LoCO-PDA to other SOTA PDA methodologies.

Consider the following scenario, as proposed by [15], of an autonomous
vehicle running an image classification CNN on an NVIDIA Jetson TX2 edge device.
As the environment the car observes varies over time, the weights of a pruned version of the
deployed CNN can be adapted to specialise this network to the data observed.
The Jetson TX2 has only 8GB of memory that is shared between the CPU and GPU.
Apart from the image classification network, there are other safety critical applications
running on the device.

This scenario introduces the following considerations.
As the device has a unified memory system, memory intensive processes such as retraining
started on either processor can affect processes from executing on the other.
In safety critical scenarios, this could have catastrophic consequences.
On-device retraining of CNNs is primarily limited by its memory consumption and
for many CNNs even batch sizes as low as 32 requires too much memory.
Additionally, the robustness of the methodology to the lack of ground-truth labels of the
observed data is an important consideration.

Consequently all approaches in this section are evaluated on the following metrics.
Static (storage) and runtime memory consumption.
Classification accuracy of the adapted network on $\mathcal{D}'$.
Performance under noisy $\mathcal{D}'$ estimation.
Wall-clock time to perform on-device adaptation on both the GPU and CPU.
Inference latency and total floating-point operations (FLOPs) of the adapted network.

§.§ Evaluation Datasets and Methodologies

This section details the methodologies that will be evaluated and the datasets that they
will be evaluated on.
For all experiments in this section, the training hyperparameters provided in Appendix
<ref> and <ref>
were used.

§.§.§ Datasets

For all experiments in this section, the source domain dataset ($\mathcal{D}$) on which
model training and pruning are performed is the entire ILSVRC'12 training set.

For the practical deployment scenario, adaptation to four example target domain
datasets ($\mathcal{D}'$) will be explored.
These are constructed as subsets of the ImageNet dataset that emulate different
environments that an autonomous vehicle could encounter [15].
They are City (185 classes), Motorway (26 classes), Country-side (204 classes),
Off-road (26 classes).
Furthermore, for this evaluation, an edge-retraining dataset is created by randomly
sampling 20% of the ImageNet training dataset.
For each $\mathcal{D}'$, images from this dataset are utilised to emulate images that a
deployed network would classify and are used to retrain the network on the edge.
Although the deployed network has already observed these images during training on
ImageNet, a different random seed is used for image preprocessing and augmentation to
emulate  variation from the training distribution.
Doing so allows for all accuracy results to be reported on the full ImageNet validation
dataset (unseen by models) to allow for fair comparison in the future.

When evaluating the methodology against other resource unconstrained PDA approaches,
the Caltech-84 and Office-31 datasets are used.
The Caltech-84 dataset uses the 84 classes that are shared between the
ILSVRC'12 and Caltech-256 [7] datasets.
The Office-31 dataset consists of 3 categories corresponding to images from
Amazon, a DSLR and a Webcam and each category has images
corresponding to 31 classes.

These target domains are common transfer learning benchmarks and details of the classes
present in all target domains are provided in Appendix

§.§.§ Methodologies

The methodologies that LoCO-PDA is compared against under the practical deployment
scenario setting store a set of $N$ possible pruned models on-device, each of which can
be retrained to the observed data distribution.
The choice of model that is retrained depends on the target inference memory and
latency at the time of deployment.
The methodologies vary in either pruning strategy or the manner in which multiple
networks with different budgets are exposed.

On-device Retraining A set of $N$ pruned models are obtained using both Taylor
First Order [14] and OFA pruning [2].
All pruning is performed using the tool open sourced in [16].
For both pruning strategies, on device retraining on $\mathcal{D}'$ of all-layers and
classifier-only are considered as baselines.

PersEPhonEE This methodology appends early-exit modules which contain two
convolution layers and a classifier at various points along the backbone network.
After training of all exits and the backbone network offline on $\mathcal{D}$,
on-device retraining cost is minimised by only finetuning the early exit that falls
within the available memory and compute budget.

To the best of our knowledge, PersEPhonEE is the only other work that explores edge
device retraining for PDA.
The TFO and OFA approaches are baselines to evaluate the robustness of LoCO-PDA to
different pruning strategies.

For all methodologies, both memory bounded and unbounded training approaches are
This refers to placing a memory limit on the number of images that can be stored in
order to perform retraining versus utilising the entire retraining dataset which as
discussed in Sec.<ref> corresponds
to 20% of the ILSVRC'12 training dataset.
In the memory bounded scenario, a limit of 400MB is set as this is deemed to be a
reasonable allocation of the available 8GB on a Jetson TX2 for this purpose.

In the memory unbounded scenario, as each $\mathcal{D}'$ has a different number of
classes, the storage memory utilised varies and is 9.40 GB, 1.36 GB, 10.3 GB and 1.36
GB for the City, Motorway, Country-side, and Off-road $\mathcal{D}'$ respectively.
The memory unbounded scenario evaluates the best case performance of each methodology
when not subjected to any memory constraints.
All retraining based approaches are evaluated on the NVIDIA Jetson TX2 using PyTorch
v1.6, CUDA 10.2, cuDNN 8.0 and batch-size 32.
Under the scenario where ground-truth labels in the target domain are used, these
methodologies act as an "oracle" for retraining a network directly on the target

For resource unconstrained PDA, LoCO-PDA is compared against ETN [5], IWAN
[21] and PADA [4].

§.§.§ Networks

The $\mathcal{M}^0$ and $\mathcal{M}^p$ networks utilised in this evaluation are shown
in Table.<ref>.
As discussed in Sec.<ref>, $N$ $\mathcal{M}^{p}$ networks are stored
on device, however for simplicity this section only evaluates the case of $N=1$.

$\mathcal{M}^0_{TFO}$ is an unpruned ResNet50 network and $\mathcal{M}^{80}_{TFO}$ is
the same network TFO pruned by 80%.
$\mathcal{M}^0_{OFA}$ is the largest sub-network that can be sampled from the OFA super
network and $\mathcal{M}^{80}_{OFA}$ is an OFA sub-network that has a
similar FLOPs budget to $\mathcal{M}^{80}_{TFO}$.
$\mathcal{M}^0_{PE}$ is an unpruned ResNet50 network augmented with 6 early exits
located at FLOPs normalised locations along the network.
Consequently, apart from model memory, all other metrics are the same as
$\mathcal{M}^0_{TFO}$ in Table.<ref>.
$\mathcal{M}^{80}_{PE}$ is the second early exit (after layer2.1) as this has a similar
FLOPs budget to $\mathcal{M}^{80}_{TFO}$.
[Refer to Appendix <ref> for results on other

§.§ Static and Runtime Memory Footprint

Static memory footprint of a methodology corresponds to the storage memory consumed by the
various parts of the methodology required to perform network adaptation.
Runtime memory footprint corresponds to the memory consumed during network adaptation.
Note that this is the footprint of the network adaptation process and not that of performing
inference on the network.
The total static and runtime memory consumption of the various methodologies on a Jetson
TX2 is displayed in Table.<ref>.

TFO, OFA: Image based retraining For each methodology (TFO, OFA), the static
network storage component consists of the corresponding $\mathcal{M}^0$ to estimate labels
and $\mathcal{M}^{80}$ which is adapted through retraining.
Note depending on the number ($N$) of pruned networks ($\mathcal{M}^{p}$) stored, the
static network storage component will increase.
Here, the case of $N=1$ is evaluated.
The static image storage component consists of 400MB of images for memory bounded
The runtime memory consumption is affected by whether classifier-only or all-layer
retraining is performed and is quoted for retraining $\mathcal{M}^{80}$ on the GPU of
the Jetson TX2.

PersEPhonEE: Early-exit retraining The network component of the static memory
consumption is from $\mathcal{M}^{80}_{PE}$.
The images component of the static memory consumption is from the 400 MB of images
stored to perform memory bounded early-exit retraining.
The runtime component is from performing on-device retraining of the second early exit
(Sec.<ref>) on the GPU.

LoCO-PDA For LoCO-PDA, the additional static network component is storage of
the CVAE model used to generate activations.
This component is 9.65 MB for $\mathcal{M}^{80}_{TFO}$ and 16.83 MB for
As PersEPhonEE uses $\mathcal{M}^{0}_{TFO}$ as the backbone network, the CVAE trained
on activations generated by $\mathcal{M}^{80}_{TFO}$ is used for comparison.
No images are used for retraining with this approach, hence the static image component
is 0 MB.
The memory consumed by the generated activations is incorporated into the runtime
memory consumption which is evaluated both on the CPU and GPU of the Jetson TX2.
The significantly extra memory consumption of the GPU is due to CUDA initialisation
overheads which can be close to 1 GB depending on the device and framework
LoCO-PDA executed on a CPU consumes 13.3x, 8.76x and 15.1x less memory compared to TFO
classifier-only retraining, OFA classifier-only retraining and second exit training of
PersEPhonEE respectively executed on a GPU.

§.§ Accuracy of $\mathcal{M}^p$ on $\mathcal{D}'$

This section compares the best achieved Top1-Test accuracy after adapting the network to
the four autonomous driving $\mathcal{D}'$s for all the methodologies discussed in
In order to evaluate the methodologies under a no-uncertainty situation, all results here
assume knowledge of the ground-truth labels for the images in $\mathcal{D}'$.
Later sections evaluate the effect of uncertain labels when using $\mathcal{M}^0$ to
identify $\mathcal{D}'$.

First, memory bounded and unbounded; and all-layers and classifier-only training approaches
are evaluated.

§.§.§ Memory bounded vs. unbounded

As seen in Table <ref>, the memory unbounded approach outperforms the memory bounded approach by 2.64pp on
average across $\mathcal{D}'$ and methods.
Given the small performance difference (2.64pp) and the large memory requirements (3.4x) of the memory unbounded approach vs. the memory bounded approach, the rest of the evaluation focuses on the memory bounded case.

§.§.§ All layers vs. Classifier-only

Retraining all layers performs on average across $\mathcal{D}'$ 0.2pp worse than
classifier-only training.
Furthermore, retraining all layers has a runtime memory footprint that is 1.8x and
training latency that is 3.4x larger than classifier-only training.
Hence the following sections use classifier-only retraining as a baseline.

§.§.§ LoCO-PDA

The No Label Space Uncertainty columns of Table.<ref> display
the achieved Test Top1 accuracy on each autonomous driving $\mathcal{D}'$ for
$\mathcal{M}^{80}_{TFO}$ and $\mathcal{M}^{80}_{OFA}$ when
no retraining, memory bounded classifier-only retraining and
LoCO-PDA are performed on-device.
PersEPhonEE was not compared against here due to its poor performance
This poor performance is due to utilising predictions from an early exit placed only
after the second Bottleneck block of ResNet50 to allow for a FLOPs normalised
comparison between methodologies (<ref>).

The following results are averages across both the OFA and TFO networks.
LoCO-PDA consistently outperforms memory bounded retraining of $\mathcal{M}^{80}$ by
1.46pp on average across all $\mathcal{D}'$.
Furthermore, on the smaller $\mathcal{D}'$ (Motorway, Off-road), on which domain
adaptation is more beneficial, LoCO-PDA also outperforms its corresponding unpruned
network ($\mathcal{M}^0$) by 3.04pp on average.
Intuitively, larger subsets do not benefit significantly from on-device adaptation as
the initial network optimised on many classes learns a classifier that best separates a
space with many classes.

§.§ Storage Memory Limitations

The results from the previous sections utilise a storage memory limit of 400MB for image
based retraining approaches.
However, as discussed in Sec.<ref> placing this restriction did
have a reduction in accuracy when compared to the memory unbounded scenario.
This section expands on this to demonstrate the effect of various storage limits on the
achieved accuracy when performing image based classifier-only retraining.
The red line in Fig.<ref> displays the achieved classifier-only retraining
accuracy for $\mathcal{M}^{80}_{TFO}$ on the Off-road $\mathcal{D}'$ as the available
memory to store images is varied (x-axis).
These values are the average of five retraining runs.
The solid green line displays the no retraining accuracy of $\mathcal{M}^{80}_{TFO}$ and the
solid blue line displays the accuracy achieved by LoCO-PDA.
The dashed blue line is the static memory consumed by LoCO-PDA.
The graphs show that as the available memory budget is increased, the achieved retraining
accuracy improves and depending on $\mathcal{D}'$ can outperform LoCO-PDA under no label
uncertainty while utilising 6.8x times the memory.
For the same memory budget as LoCO-PDA, the memory-bounded retraining approach
performs on average 4.88pp worse than LoCO-PDA.

§.§ Noisy $\mathcal{D}'$ Estimation

All results presented thus far assume knowledge of the label space of
$\mathcal{D}'$ which is not representative of real deployment scenarios.
In this work we propose to estimate the label space of $\mathcal{D}'$ using the predictions
of the initially deployed model $\mathcal{M}^{0}$.
As the accuracy of this model is not 100%, some of the labels obtained from this model are
This section evaluates the effect on achieved adaptation accuracy of noisy $\mathcal{D}'$
The following results are averages across both the OFA and TFO networks.

The Label Space Uncertainty columns in Table.<ref> show the
achieved test top1 accuracy on $\mathcal{D}'$ for both the memory bounded classifier-only
retraining strategy (Retrained) and the LoCO-PDA approach.
For the image based retraining approaches, memory bounded retraining with uncertain labels
causes a 7.27pp degradation in achieved accuracy compared to retraining with certain labels
across $\mathcal{D}'$.
Furthermore in most cases, retraining with noisy labels actually degrades the network's
accuracy compared to $\mathcal{M}^{80}$ without any retraining.
On the other hand, LoCO-PDA only incurs a 0.67pp reduction in achieved accuracy across
$\mathcal{D}'$ and never a degradation compared to $\mathcal{M}^{80}$ without retraining.

With image based retraining, predicting an incorrect label and utilising this combination
to retrain the network causes increased confusion as the network is now being trained to
associate incorrect labels and images.
As LoCO-PDA generates the activations corresponding to the estimated class, only an
incorrect ratio of the labels is generated and the retraining stage still associates labels
with their correct corresponding activations.
Thus, LoCO-PDA is significantly more robust to noisy estimations of $\mathcal{D}'$ than
image based retraining approaches.

§.§ On-device Adaptation Wall-clock Time

This section evaluates the wall-clock time to perform on-device adaptation using each
methodology on the Jetson TX2.
The values in Table.<ref> are the time taken to retrain the
pruned network until the accuracy under the No Label Space Uncertainty case in
Table.<ref> is achieved.
As different $\mathcal{D}'$ behave differently under retraining, the quoted values are
averaged across the four $\mathcal{D}'$.

The time taken to perform on-device adaptation for image based retraining approaches is
infeasible on a CPU, hence the GPU times are utilised here for comparison.

The results show that LoCO-PDA executed on a CPU and GPU outperform classifier-only
retraining on a GPU by 2.9x and 12.9x respectively.

By bringing domain adaptation latencies down to less than a minute, while consuming less
than 300 MB of memory and being significantly robust to noisy estimations in
$\mathcal{D}'$, LoCO-PDA opens up avenues for on-device partial domain adaptation of
networks that have not been explored before.

§.§ Partial Domain Adaptation

This section compares LoCO-PDA against SOTA PDA methodologies ETN [5], IWAN
[21], and PADA [4].
As these works do not consider the case of pruning a network, the reported results in this
section are for an unpruned ResNet50.

For the ILSVRC'12 to Caltech-84 transfer setting, activations corresponding to the 84
classes shared between the ILSVRC'12 and Caltech-256 datasets are generated using a CVAE
trained on ILSVRC'12 data that has no knowledge of the target Caltech-84 activation
Furthermore, the target distribution is estimated using the output of classifying the
entire Caltech-84 dataset on the unpruned ResNet50 network, hence no ground-truth labels
were used.
For the Office-31 setting, ResNet50 is first finetuned to each of the three categories
using all 31 classes and 3 CVAEs are trained to generate activations for each of these
source domain networks.
The target domain images are then classified on the source domain finetuned network to
obtain a distribution and the corresponding CVAE is used to generate activations to retrain
the network.

On average across all PDA settings, Table.<ref> shows that LoCO-PDA improves
performance compared to no domain adaptation and PADA by 5.42pp and
1.20pp respectively; and performs comparably to IWAN (only 0.95pp worse).
Although performing 3.43pp worse compared to ETN, LoCO-PDA has the significant advantage
over all three methodologies that it can rapidly adapt to changes in the target
PADA, IWAN and ETN cannot be performed on an edge device to re-adapt a network to new
domains due to their large memory and latency footprints (Fig.<ref>).

§ CONCLUSION

This work proposes LoCO-PDA, a methodology to perform low cost on-device partial domain
adaptation of CNNs along with an open-sourced implementation.

In the case where a pruned version of a network is adapted to the observed target domain,
the proposed methodology achieves up to 3.27x reduction in model memory footprint and 2.07x
improvement in inference latency while achieving on average 3.04pp improvement in
classification accuracy compared to an unpruned version of the network that is deployed without
any domain adaptation.
Compared to other image based on-device retraining methodologies, LoCO-PDA achieves up to 15.1x
runtime memory consumption improvements while also being significantly more robust to noisy
estimations of the target domain class
The entire retraining process can be performed on a CPU with sub-minute domain adaptation times
on a Jetson TX2 device, opening avenues for on-device PDA which have not been explored before.
LoCO-PDA also performs comparably to other SOTA PDA methodologies while benefiting from the
ability to adapt to varying target domains, which the other methodologies cannot.

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# Mueller matrix based characterization of Cervical tissue sections: A
quantitative comparison of Polar and Differential decomposition methods

Nishkarsh Kumar  Department of Physics, Indian Institue of Technology Kanpur,
Kalyanpur, India- 208016.  Jeeban Kumar Nayak<EMAIL_ADDRESS>Department of Physical Sciences, Indian Institute of Science Education and
Research Kolkata, Mohanpur, India, 741246  Asima Pradhan<EMAIL_ADDRESS>Department of Physics, Indian Institue of Technology Kanpur, Kalyanpur, India-
208016.  Center for Lasers and Photonics (CELP), Indian Institue of Technology
Kanpur, Kalyanpur, India- 208016.  Nirmalya Ghosh  Department of Physical
Sciences, Indian Institute of Science Education and Research Kolkata,
Mohanpur, India, 741246

###### Abstract

Detection of cervical intraepithelial Neoplasia (CIN) at the early stage
enables prevention of cervical cancer, which is one of the leading cause of
cancer deaths among women. Recently there is a great interest to use the
optical polarimetry as a non-invasive diagnosis tool to characterize the
cervical tissues. In this context, it is crucial to validate the performance
of various Mueller matrix decomposition techniques, that are utilized to
extract the intrinsic polarization properties of complex turbid media, such as
biological tissues. To quantitatively compare the performance of polar and
differential MM decomposition methods for probing the polarization properties
in various complex optical media. Also to utilize the derived individual
polarization parameters through differential decomposition method as useful
metrics to distinguish between the cervical tissue sections with different
grades of CIN. Complete polarization responses of the cervical tissue
sections, and other media are recorded by preparing a home-built Mueller
matrix imaging set up with a spatial resolution of $\approx 400nm$. The
Mueller matrices are then processed with the polar and differential
decomposition methods to separate, and quantify the individual polarization
parameters. Pronounced differences in the extracted polarization properties
are observed for different CIN grades with both the decomposition methods.
While a significant increase in the depolarization parameter $(\Delta)$ is
observed with progression of CIN stages ($\Delta=0.35$ for CIN-I, and
$\Delta=0.56$ for CIN-II), a trend reversal is seen for the linear
diattenuation parameter $(d_{L})$, where the magnitude of the $d_{L}$
decreases from $0.26$ to $0.19$ with the growth of CIN, in case of
differential decomposition method.
Our results indicate that the differential decomposition of MM have certain
advantages over the polar decomposition method to extract the intrinsic
polarization properties of a complex tissue medium. The quantified
polarization parameters obtained through the decomposition methods can be used
as useful metrics to distinguish between the different grades of CIN, and to
describe the healing efficiency of a self-healing organic crystal. Thus the
Mueller matrix polarimetry shows great potential as an label-free, non-
invasive diagnostic and imaging tool with potential applications in biomedical
clinical research, and in various other disciplines.

††preprint: APS/123-QED

##  I Introduction

Many of the natural objects with biological or non-biological origin possess
some intrinsic polarization anisotropy such as birefringence, dichroism, and
depolarization. Quantification of these polarization anisotropic parameters
with desirable spatial resolution, can provide essential morphological
structural, and functional information regarding a specimen, and hence
quantitative polarimetry has emerged as an powerful tool for the
characterization of a wide range of optical media in various disciplines[1, 2,
3, 4]. In the realm of quantitative polarimetry, polarization resolved Mueller
matrix imaging is a widely adopted technique as it enables probing and
quantifying complete polarization response of a sample in a single
experimental embodiment[1, 2, 5, 6, 7]. Owing to the recent developments in
the field of Mueller matrix polarimetry such as, increasing the measurement
precision by optimising the calibration methods[1], adoption of snapshot
techniques to extract the polarization information of dynamic objects[8],
establishment of various Mueller matrix decomposition and analysis techniques
to extract the polarization anisotropy parameters, and etc;[9] the MM imaging
has gained a special place in biomedical and clinical applications.[1, 10]
However, carrying out quantitative polarimetry in optically thick turbid media
like biological tissues is still a challenge[11, 12, 2]. Although such complex
media possess intrinsic polarization anisotropy, the inevitable depolarization
through multiple scattering and simultaneous exhibition of several
polarization effects obstruct the quantification of these polarization
parameters. Therefore, in recent years a number of Mueller matrix
decomposition methods were proposed aiming to separately quantify the
polarization anisotropy parameters in a lumped system [13, 1, 14, 15] . These
extracted polarization parameters contain a wealth of information regarding
the medium properties. The proposed techniques come with different
assumptions, and hence one has to be judicious while choosing an appropriate
decomposition techniques according to the targeted real-life applications.
Some of the recent studies have shown that among the various decomposition
methods, the polar and differential matrix decomposition are the efficient
techniques to extract the polarization parameters in a complex media, where
alongside several polarization anisotropic effects, the medium possess
significant depolarization effect. In polar decomposition, a given Mueller
matrix is decomposed into sequential products of three basis matrices
corresponding to the three general polarization effects (depolarization,
diattenuation (linear and circular), retardance (linear and circular)). Yet
owing to the non-comuting nature of the matrix multiplication, there is an
ambiguity in the derived anisotropic parameters depending on the order of
multiplication[7]. On the other hand differential matrix formalism is a more
general kind of decomposition methods, which consider simultaneous exhibition
of multiple polarization effects, hence is more appropriate for complex turbid
media such as biological tissues. Several researchers have utilized the MM
polarimetry for the early stage detection of the cervical cancer.[16, 17, 18]

In this work, we have used MM polarimetry to characterize cervical tissue
sections with different grades of CIN. Detection of cervical intraepithelial
Neoplasia (CIN) at the early stage enables prevention of cervical cancer,
which is a leading cause of cancer death among women[19, 20]. Polarization
resolved MM imaging of the cervical tissues are recorded, and processed with
both polar and differential decomposition methods to extract the associated
polarization properties. Individual polarization parameters are obtained by
plotting the histogram of spatially quantified polarization images of cervical
tissues, and its subsequent fitting with the Gaussian distribution. The
obtained parameters are then used as useful metrics to distinguish between the
different grades of CIN. Our results not-only enables realization of a non-
invasive, label free polarimetric diagnostic tool for the early detection of
cervical cancer, but also facilitates quantitative comparison of the
performance of various MM decomposition methods in complex turbid media. In
addition we have also presented the quantified polarization properties of a
self-healing organic crystal to further validate the utilization of different
MM decomposition techniques in complex optical media.

##  II Materials and Methods

###  II.1 Materials

Sections of $20\mu m$ thickness from the stromal region of the cervical
tissues are used in this study. The cervical tissues are obtained from
G.S.V.M. medical college, Kanpur and the sectioning of these cervical tissues
are done by Dr. Asha agarwal.

###  II.2 Mueller matrix polarimetry set-up

We have utilized a custom-designed polarization microscopic arrangement, where
one can determine the complete polarization response of a sample by recording
the $4\times 4$ Mueller matrix [21, 22](Figure 1a). The experimental system
employs broadband white light excitation and the subsequent recording of the
polarization resolved images of the sample at any selected wavelengths between
($\lambda=400-725nm$). Alongside imaging, the spectral Mueller matrix of a
given sample can also be recorded simultaneously with the Mueller matrix
polarimetry system utilized in our experimental configuration. The thirty six
projective polarization measurements required for the construction of Mueller
matrix is recorded by sequentially generating and analyzing six different
linear and circular polarization states[23, 24]. A collimated white light from
the microscope’s (Olympus, IX-71) inbuilt illumination source (halogen lamp,
JC 12V 100W) is passed through a polarization state generator unit which
comprises of a rotatable (i) linear polarizer and (ii) achromatic quarter wave
plate (QWP, Thorlabs AQWP05M-600 ). Polarization of the scattered light is
subsequently analyzed by the polarization state analyzer unit that consists of
a rotatable achromatic QWP and a linear polarizer. All the optical polarizing
elements are mounted on computer controlled motorized rotational mounts
(PRM1/M-Z7E, Thorlabs, USA) for precision control. For recording the spectral
Mueller matrices of the WPCs, the scattered light is relayed to a spectrometer
(HR 4000, Ocean Optics) for spectrally resolved signal detection (resolution
$.2nm$). The Mueller matrix images at selected wavelengths were recorded by
using an EM CCD camera (Andor, IXON 3). The polarization resolved intensity
images of the cervical tissue sections (dimension = $650\mu m\times 650\mu m$)
are recorded using a $650nm$ band-pass filter $(\Delta\lambda\approx 25nm)$.
The imaging Mueller matrices are constructed using 36 polarization-resolved
projective measurements (given in the table below)[23].

Table 1: Scheme for construction of Mueller matrix using 36 polarization-resolved projective measurements. Here, the first letter represents the input polarization state, the second letter stands for the analyzer or the projected polarization state. The states are defined as $I_{H}(horizontal)$, $I_{V}(vertical)$, $I_{P}(+45\deg)$, $I_{M}(-45\deg)$, $I_{L}$ left circular polarized $(LCP)$, $I_{R}$ right circular polarized $(RCP)$ HH+HV+VH+VV | HH+HV$-$VH$-$VV | PH+PV$-$MH$-$MV | RH+RV$-$LH$-$LV
---|---|---|---
HH$-$HV+VH$-$VV | HH$-$HV$-$VH+VV | PH$-$PV$-$MH+MV | RH$-$RV$-$LH+LV
HP+VP$-$HM$-$VM | HP$-$VP$-$HM+VM | PP$-$PM$-$MP+MM | RP$-$RM$-$LP+LM
HR+VR$-$HL$-$VL | HR$-$VR$-$HL+VL | PR$-$PR$-$MR+ML | RR$-$RL$-$LR+LL

##  III Theory

Here we will briefly describe the theoretical treatment for the extraction of
polarization anisotropy parameters using both polar and differential MM
decomposition methods[7, 11, 13]. In the polar decomposition method, a given
Mueller matrix is decomposed into sequential product of three basis matrices.

$M=M_{\Delta}.M_{R}.M_{D}$ (1)

Where $M_{\Delta}$ corresponds to the depolarization effect associated with
the medium, and $M_{R}\&M_{D}$ describe the retardance (linear and circular),
and diattenuation (linear and circular) effects respectively. The magnitudes
of the anisotropy parameters, diattenuation $(D)$, retardance $(\delta)$, and
depolarization $(\Delta)$ are calculated using the respective basis matrices.
On the other hand, in the differential matrix formalism, the anisotropic
polarization and depolarization effects are stored simultaneously in various
elements of a differential matrix $m$ [14, 15]. The differential matrix is
related to the Mueller matrix $M$ and its spatial derivative along the
direction $(\vec{z})$ of propagation of light as[25]:

$\frac{dM}{dz}=mM$ (2)

The above equation assumes that the sample is laterally homogeneous and both
polarization and depolarization effects are occurring simultaneously. In
uniformly distributed polarization properties along the propagation direction
the integration of differential matrix equation yields:

$L=\ln M=m.l$ (3)

Here $L$ is the matrix logarithm of recorded Mueller matrix $M$ and $l$
represents the optical pathlength in the medium. The polarization properties
of the underlying system can be constructed using the Lorentz symmetric
$(L_{u})$ and Lorentz antisymmetric $(L_{m})$ components of logarithmic
Mueller matrix $L=L_{m}+L_{u}$ as:

$\begin{split}L_{m}=1/2(L-GL^{T}G)\\\ L_{u}=1/2(L+GL^{T}G)\end{split}$ (4)

$G$ is the Minkowski metric tensor represented as $G=diag(1,-1,-1,-1)$. The
corresponding anisotropic parameters can be calculated directly from the
respective matrix elements of the $L_{m}$ matrix.

##  IV Results and Discussion

The cervix is composed of squamous epithelium, connective tissues, and other
components. Collagen is a primary components of the cervical connective
tissue, and cross linking between the individual collagen molecules lead to
the formation of microfibril and collagen fibers. Anisotropic organization of
these collagen fibers makes the cervix polarization anisotropy. Thus, through
quantitative MM polarimetric imaging, essential morphological information
regarding the cervical tissue sections can be captured. More importantly in
case of most cancers, the structural organization of the collagen fibers gets
distorted. Consequently it leads to the alteration of the polarization
properties of the cervical tissue sections. This enables quantitative MM
imaging to be a non-invasive diagnostic tool for pre cancer detection, where
quantitative polarimetric analysis can distinguish between a healthy and
defected cervical tissue section.
Here we have quantitatively analysed the polarization properties of an
essential cervical pathologies, the cervical intraepithelial Neoplasia (CIN),
which is of particular interest for the early detection of cervical cancer.
Polarization MM imaging of cervical tissue sections with CIN grade one (CIN-I)
and grade two (CIN-II) are recorded using the MM imaging set up (Fig. 1a).
Polarization resolved images of both these tissue sections are captured with a
spatial resolution of $\approx 400nm$. Consequently the full $4\times 4$ MM is
constructed from the projective polarization measurements as described in the
table 1. The experimentally observed MM for both CIN-I (Fig. 1b) and CIN-II
(Fig. 1c) tissue sections demonstrate simultaneous presence of multiple
polarization effects such as linear diattenuation $(d_{L})$, linear retardance
$(\delta_{L})$, and depolarization $(\Delta)$. This is evident from the non-
zero magnitude of the respective MM elements. Besides the linear diattenuation
$(d_{L})$ descriptor elements $(M_{12/21},M_{13/31})$, pronounced magnitude of
the elements $(M_{23/32},M_{24/42})$ describing the linear birefringence
$(\delta_{L})$ properties are also observed. Significant depolarization of the
incident polarized light is also observed, which is expected owing to the
multiple scattering in the turbid tissue media.

---
Figure 1:  (a) Schematic of the polarization Mueller matrix microscopy system
to capture the spectral and imaging polarization response of a sample. PSG and
PSA: Polarization state generator and analyzer units, P: Polarizer, Q: quarter
wave-plate, BS: beam splitter. The $4\times 4$ polarization Mueller matrix
images of the cervical tissue section with CIN grade one (CIN-I) (b), and CIN
grade two (CIN-II) (c) are presented. Besides significant depolarization,
simultaneous occurrence of multiple polarization effects are manifested in the
different Mueller matrix elements. The drawn scale bar is $130\mu m$.

It is pertinent to note that although the intrinsic polarization properties of
the cervical tissue sections are reflected in the respective MM elements, the
obtained Mueller matrices (fig. 1b, 1c) does not possess a proper symmetry as
generally observed for the conventional polarizing optical elements[24]. This
deviation originates due to the presence of significant depolarization effect
and simultaneous occurrence of multiple polarization effects, which are
manifested in a complex inter-related way in the MM. Thus to extract and
quantify the polarization properties, Processing of the observed MM with
decomposition methods is indispensable. The derived polarization parameters
then can be utilized as useful diagnostic metrics to distinguish between the
different grades of CIN, This highlights the significance of MM decomposition
technique to establish the quantitative MM polarimetry as a diagnostic tool.
The associated decomposed matrices are discussed in the supplementary
information.
The experimental recorded Mueller matrices (Fig. 1b, 1c) are processed with
both polar and differential decomposition methods to obtain the spatial
distribution of the individual polarization parameters throughout the tissue
sections. Utilizing the corresponding algorithms spatial various polarization
anisotropic properties are derived, and the corresponding results are
presented in Figure 2. All the obtained polarization parameters; linear
diattenuation $(d_{L})$ (Fig. 2a), linear retardance $(\delta_{L})$ (Fig. 2b),
and depolarization $(\Delta)$ (Fig. 2c) show significant strength to be
considered as potential metrics to describe the morphological properties of
the cervical tissue sections. However, there is an spatial inhomogenity for
all the extracted polarization parameters (Fig. 2a(I),(II), b(I)(II),
c(I)(II)). Hence we go on to plot histograms for the respective polarization
images, which enables a better quantitative presentation of the polarization
properties. Histograms shown in Fig. 2a(III), 2b(III) and 2c(III) are plotted
by taking the magnitude of the polarization parameters with no. of pixels
attaining that particular magnitude. Furthermore, the histograms are fitted
with the Gaussian distribution, and the statistical moments of the
distributions (mean $\mu$, standard deviation $(\sigma)$) are calculated.
These can serve as more precise and convenient metrics to distinguish between
the different grades of CIN.Comparing the magnitude of the mean value obtained
with polar and differential decomposition methods, a significant difference in
the magnitude of the linear diattenuation parameter $(d_{L})$ is clearly
visible in the gaussian fit of the histogram (Fig. 2a(III)). Magnitude of the
mean values of $d_{L}$ correspond to the differential and polar decomposition
are found out to be 0.26 and 0.10 respectively. On the other hand, the phase
anisotropy parameter $(\delta_{l})$ (mean values: $0.13$ and $0.13$) and
depolarization (mean values: $0.35$ and $0.32$) does not show much of the
variation in the magnitudes for the polar and differential decomposition
methods.

---
Figure 2:  Intrinsic polarization parameters, such as linear diattenuation
$(d_{L})$ (a), linear retardance $(\delta_{L})$ (b), and depolarization
$(\Delta)$ (c) are extracted and quantified from the recorded Mueller matrix
of the cervical tissue section of grade CIN-I. While the results obtained for
the differential matrix decomposition method is presented in the first
column(a(I), b(I), c(I)), the spatial variation of the polarization parameters
obtained with the polar decomposition method are given in the second column
(a(II), b(II), c(II)). The histogram plots corresponding to the polarization
images are presented (a(III),b(III), c(III)), and further the histograms are
fitted with Gaussian distribution to calculate the statistical averages (mean
$(\mu)$, and standard deviation $(\sigma)$), which are used as metrics to
distinguish between the cervical tissue sections. The depicted scale bar is
$130\mu m$

The mean values obtained from the Gaussian fitting of the histograms are used
to distinguish between the cervical tissue sections with CIN grade one (CIN-I)
and two (CIN-II). The results are presented for both polar (Fig. 3a) and
differential decomposition method (Fig. 3b), so that besides demonstrating a
polarimetric diagnosis tool, we can quantitatively compare the performance of
the polar and differential decomposition of MM in case of the cervical tissue
sections or in more general for optical interaction in complex turbid media.
While the histograms with grey colour corresponds to the CIN-I cervical tissue
sections, the magenta colour histograms describe the polarization properties
of CIN-II cervical tissue sections. With polar decomposition method, the mean
value of the linear diattenuation parameter $(d_{L})$ obtained for both CIN-I
$(\mu=0.10)$ and CIN-II $(\mu=0.09)$ are nearly equal (Fig. 3a (I)). However,
a pronounced difference in the depolarization $(\Delta)$ (Fig. 3a (II))and
linear retaradance $(\delta_{L})$ (Fig. 3a (III)) parameters between the CIN-I
and CIN-II is observed. The mean value of the $\Delta$ significantly increases
for CIN-II $(\Delta=0.53)$ as compared to the CIN-I $(\Delta=0.32)$ indicating
the change in the morphology of the cervical tissue sections. The disruption
or randomization of the collagen fibers can lead to the rise of depolarization
effect in the cervical tissue. The mean value of the $\delta_{L}$ also exhibit
dissimilarities, where the CIN-II $(\delta_{L}=0.21)$ has a relatively higher
magnitudes with respect to the CIN-I $(\delta_{L}=0.13)$ tissue. Now coming to
the MM processing with differential decomposition method, unlike to the case
of polar decomposition, there is a pronounced change in the linear
diattenuation parameter $(d_{L})$ between the CIN-I and CIN-II tissues (Fig.
3b (I)). The obtained mean value of $d_{L}$ decreases with the progress of CIN
stages, and the obtained mean values for CIN-I and CIN-II are $0.26$, and
$0.19$ respectively. This describes the degradation of the structural
orientation of the anisotropic collagen fibers. The results obtained for
depolarization also validates this, where a significant growth in the
depolarization magnitude is observed in the CIN-II $(\Delta_{L}=0.56)$ as
compared to the CIN-I $(\Delta_{L}=0.35)$ tissue sections (Fig. 3b (II)). A
deviation in the trend is also observed for the case of linear retardance
$(\delta_{L})$, (Fig. 3b (III)) which is consistent with the results obtained
with the polar decomposition method. Altogether, these exclusively
demonstrates that, the quantitative MM imaging with incorporation of
appropriate decomposition techniques can serve as an efficient label free
diagnostic tool for early detection of cervical cancer. We also want to note
that the other parameters of the Gaussian distributions such as skewness,
kurtosis can also be used as useful metrics to distinguish between the
cervical tissues with different grades of CIN.[26]

---
Figure 3:  Histograms corresponding to the spatial variation of the
polarization parameters quantified with the polar (a) and differential
decomposition method (b) are presented. Comparative evaluation of both the
polar and differential decomposition methods are performed by considering the
mean values obtained from the Gaussian fitting. Although with polar
decomposition, the linear diattenuation $(d_{L})$ does not shows much
variation for CIN-I and CIN-II, a pronounced change is observed with
differential decomposition method (a(I), b(I)). A significant increase in the
depolarization parameter with the growth of CIN is observed in the extracted
polarization parameter for both polar and differential decomposition
techniques (a(II), b(II)). Also increment in the magnitude of the linear
retardance $(\delta_{L})$ parameter are observed with the progression of CIN
stages (a(III), b(III))

Looking at the results demonstrated above, it is evident that the differential
decomposition of MM have certain advantages over the polar decomposition
method to extract relevant information regarding the polarization properties
of a complex tissue medium. To further compare the performance of both these
decomposition techniques, we now consider an anisotropic organic crystal and
discuss it’s polarization properties. In our recent work, we have utilized MM
polarimetry to probe the healing efficiency of a self-healing bipyrazole
piezoelectric crystal. Such extraordinary crystals heal themselves anonymously
when subjected to the mechanical fracture through a three-point bending test.
Highly ordered crystalline structure of crystals make them a polarization rich
entity which shows the strong anisotropic effects. We are examining the
different stages of crystals in a similar way as it was shown for the cervical
tissues. Three different crystal stages; pristine, neatly healed and
imperfectly healed are taken for carrying out the MM imaging, and subsequent
quantification of the polarization parameters ($d_{L}$, $\delta_{L}$ and
$\Delta$) through both polar and differential decomposition methods. In all
the scenarios, the magnitude of linear diattenuation $(d_{L})$ is found to be
very low, showing a very little presence of amplitude anisotropy due to lack
of the imaginary part of the refractive index. Hence the other two
polarization parameters (linear retardance $(\delta_{L}),$, and depolarization
$(\Delta_{L})$) are considered for the comparative study of the crystals in
different stages. The corresponding results are shown in figure 4. A region of
$26\mu m\times 104\mu m$ dimension (shown with a rectangle in images of the
Figure 4) is selected in each crystal through the microscope eyepiece and
verified by calculating the higher correlation values for the specific
dimensions between the pristine and healed crystals. Linear retardance mean
values with their standard deviation for the pristine and neatly healed
crystal were $0.42\pm 0.04$ and $0.34\pm 0.05$ respectively resulting in the
retrieval $(80-85\%)$ of the phase anisotropy of the crystal in healing
process (Fig. 4a(II),(IV), 4b(II)(IV)). Depolarization $\Delta$ shows the
comparable values for the pristine $(0.46\pm 0.03)$ and healed $(0.44\pm
0.03)$ (Fig. 4a(I)(III), 4b(I)(III)) . In case of the imperfectly healed,
crack junction is clearly observed in the crystal (Fig. 4c), $\delta_{L}$ in
left and right part of the crystal is remain intact with respect to the
pristine while depolarization magnitude has increased. While being under
mechanical fractures, the order of the crystalline structure gets scrambled
leading to the decrease in anisotropic parameters and store their order as the
crystal heals itself. Orientation angle of linear retarder axis provide a
better understanding for the efficient repairing of the fractured crystal
(Fig. 4a(V), 4b(V), 4c(V)). While the long-range crystalline order remains
intact for both pristine (Fig. 4a(V)) and the healed crystal (Fig. 4b(V)), a
clear disorder in the orientation of the retarder axis is observed for the
imperfectly healed crystal (Fig. 4c(V)). Important thing to note here is that,
both the polar and differential decomposition of the MM gives rise to the
near-equal magnitude of the extracted polarization parameters which was not
the case for the cervical tissue sections. A table containing all the mean and
standard deviation values of the respective polarization parameters is given
in the supplementary information.

---
Figure 4:  Quantified polarization parameters of the pristine (a), healed (b)
and imperfectly healed crystal (c) are provided for both polar and
differential decomposition of the MM. Spatial variation of the depolarization
$(\Delta)$ (I.)(II.) and linear retardance $(\delta_{L})$ (III.)(IV.)
parameters are extracted from the inverse analysis of the Mueller matrix.
Variation in such homogeneous organic crystals is negligible while processing
through different analysis methods. An area with dimensions $26\mu m\times
104\mu m$ (black colored box present in each image) has been analyzed for the
quantitative comparison of the derived polarization parameters between the
pristine, healed, and imperfectly-healed crystal. (V) Spatial variation of the
orientation axis of the linear retarder is also plotted for the considered
region of interest (black solid box), where a clear discontinuity is visible
along the crack region (c(V.)) demonstrating the decrease in the linear
retardance parameter. Scale bar is $50\mu m$.

##  V Conclusion

In summary, we have demonstrated that the differential decomposition of MM
have certain advantages over the polar decomposition method to extract the
intrinsic polarization properties of a complex turbid media. For this purpose,
cervical tissue sections with different grades of CIN are considered, which
alongside significant depolarization, exhibits multiple polarization effect
simultaneously. The polarization properties of these cervical tissue sections
are quantified through the incorporation of both polar and differential MM
decomposition methods. We have shown that, the individual derived polarization
parameters through differential decomposition can be treated as useful metrics
to distinguish between different stages of CIN, which enables realization of a
non-invasive, label-free diagnosis tool for early detection of the cervical
cancer. In addition we have also investigated the performance of both polar
and differential decomposition MM methods to quantify the healing efficiency
of a self-healing organic crystal. This highlights the ability of MM
polarimetry with an appropriate decomposition technique to be an efficient
diagnosis and imaging tool with potential applications in various disciplines.

## Disclosures

The authors declare no conflict of interests.

## Code, Data, and Materials Availability

The data, utilized algorithms for the decomposition of Mueller matrix
supporting the results that have been reported in this study are available
from the corresponding authors upon reasonable request.

## Acknowledgments

Jeeban Kumar Nayak and Nirmalya Ghosh acknowledge the support of the Indian
Institute of Science Education and Research (IISER) Kolkata, an autonomous
institute under the Ministry of Human Resource Development (MHRD), Government
of India. Nishkarsh Kumar and Asima Pradhan acknowledge the support from
Indian Institue of Technology (IIT) Kanpur. The authors acknowledge Dr. Kiran
Pandey from GSVM medical college for providing the cervical tissues and Dr.
Asha Agarwal for the histopathology sectioning. We acknowledge Dr. Chilla
Malla Reddy and Dr. Surojit Bhunia for providing the self-healing crystals.

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