11institutetext:  Dalian University of Technology, China
{yrpiao<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Dynamic Fusion Network For Light Field Depth Estimation

Yongri Piao  Yukun Zhang  Miao Zhang Corresponding Author Xinxin Ji

###### Abstract

Focus-based methods have shown promising results for the task of depth
estimation. However, most existing focus-based depth estimation approaches
depend on maximal sharpness of the focal stack. Out-of-focus information in
the focal stack poses challenges for this task. In this paper, we propose a
dynamically multi-modal learning strategy which incorporates RGB data and the
focal stack in our framework. Our goal is to deeply excavate the spatial
correlation in the focal stack by designing the spatial-correlation perception
module and dynamically fuse multi-modal information between RGB data and the
focal stack in a adaptive way by designing the multi-modal dynamic fusion
module. The success of our method is demonstrated by achieving the state-of-
the-art performance on two datasets. Furthermore, we test our network on a set
of different focused images generated by a smart-phone camera to prove that
the proposed method not only broke the limitation of only using light field
data, but also open a path toward practical applications of depth estimation
on common consumer level cameras data. The code is available:
https://github.com/OIPLab-DUT/Light-Field-for-Depth-Estimation.

##  1 Introduction

Depth estimation is a crucial step for understanding geometric relations
within a scene. Accurate and reliable depth information plays an important
role in computer vision including object tracking [9, 18], scene understanding
[10], virtual reality (VR) [37, 4], autonomous driving [3] and pose estimation
[26, 28].

Figure 1: From left to right: we show focal stack, RGB image, depth map, the
result of DDFF, the result of our method in a scene.

Depth estimation can be broadly classified into active and passive
acquisition. In contrast to active techniques that involve sending a
controlled energy beam and detecting the reflected energy [20, 2], passive
techniques are image-based methods and are more in accord with the human
visual perception of the depth, i.e., humans use a great variety of vision-
based passive depth cues such as texture, edges, size perspective, binocular
disparity, motion parallax, occlusion effects and variations in shading.
Monocular depth estimation [31, 6, 23], as low-cost, convenient and efficient
passive techniques, has attracted lots of interest lately. However, depth
estimation from a single image of a generic scene is an ill-posed problem, due
to the inherent ambiguity of mapping an intensity or color measurement into a
depth value. On the other hand, inspired by the analogy to human depth
perception, multiview depth estimation has achieved great success, including
binocular and multi-view stereo [29, 17, 25]. The similarities and
correspondences produced between each pixel in the images produce far superior
results. However, these approachs are sensitive to imaging systems and require
careful alignment and calibration in the setup.

The light field enables a unique capability of post-capture refocusing. Fig.1
shows an example of the light field. A stack of focal slices are generated as
they are taken at different depths, which contain abundant spatial parallax
information. Furthermore, focusness information caters to human’s visual
fixation that allows our eyes to maximize the focus we can give to the object
in a scene. In spite of this promising characteristics for focusness
information, there are a few studies documenting their efficacy for depth
estimation. Early works mainly aim at determining the depth of a pixel by
measuring its sharpness or focus at different images of the focal stack [27,
33, 22]. Later on, several approaches based on deep learning have been
proposed [11]. These methods use convolutional neural networks (CNNs) to
extract effective focusness information for facilitating depth estimation
instead of hand-crafted features.

While these methods demonstrate that focusness information is useful for depth
estimation, there is still large room for further improvement in terms of two
key aspects: (1) How we deeply excavate the spatial correlation between focal
slices for obtaining useful focusness information is critical for depth
estimation. Since the different focal slices are focused at different depths,
the spatial correlation between the focal slices is closely related to depth
variation of the objects in a scene. Most of previous focal-based depth
estimation networks used a standard 2D CNN to learns filters that extend
across the entire focal stack. However, spatial correlation between the focal
slices is likely to be ignored. As a result, the focusness information has not
been well-captured. (2) How do we effectively fuse focusness features and RGB
information to reduce information loss in the depth map? While focusness
information in the focal slices provides implicit depth cues, leading to
better depth estimation, out-of-focus areas with unknown sharpness could be
prone to information loss error, leading to inaccurate depth estimation. As
shown in Fig. 1, the result of DDFF which based on focal slices [11] losses
some detail and contains a lot of noise. Considering that the RGB image
contains high quality sharpness and can be used to compensate for missing data
in the out-of-focus area of the focal slices, we believe combining multi-modal
information is beneficial to improve the accuracy of depth maps. Albeit most
of recent methods performed data fusion by employing some manually set, such
as, sum fusion, weighted fusion, concatenate fusion. Those methods are unable
to take full advantages of multi-modal information between RGB images and
focal slices.

Our core insight is that we can leverage RGB data and the focal stack to learn
an estimation model of depth by deeply excavating the spatial correlation in
the focal stack and fusing multi-modality cues between RGB image and focal
slices. Concretely, our contributions are mainly three-fold:

  * •

We proposed a spatial-correlation perception module for correlating the focus
and depth. Based on the observe that different focal slices possess focus area
of multiple scales and focused at different depth, a pyramid ConvGRU is
designed to excavate the spatial correlation between different focal slices
and sequently pass multi-scale focusness information along the depth
direction.

  * •

We propose a multi-modal dynamic fusion module (MDFM) in which multi-
modalities features are fused in an adaptive manner. This fusion strategy
allows the filter parameters to dynamically change with the input focusness
features in the process of convolution with RGB features, thereby avoiding
information loss in the depth map.

  * •

We demonstrate the effectiveness of the proposed model on two light field
datasets [19]. The results show that our approach achieves superior
performance over the state-of-the-art approaches. Moreover, we also validate
our model on Mobile Phone dataset [32] which contains focal slices captured by
a smart-phone camera. This further shows a positive step towards practical
application of our model with common consumer level cameras.

##  2 Related Work

The related works can be divided into two categories: light field depth
estimation using traditional approaches and learning-based approaches.

###  2.1 Traditional Approaches

There is a wide range of methods for light field depth estimation. The field
can be roughly divided into methods based on EPI analysis, multi-view stereo
matching based approaches and focus-based approaches. [40] proposed a 4D
light-field depth estimation method via epipolar plane image analysis and
locally linear embedding. [15] employed a sparse decomposition design to
leverage the depth-orientation relationship on its epipolar plane images.
Another popular approach is using total variation regularization. [21]
proposed a nonlinear Total Variation (TV) based method for recovering 3D shape
of an object by diffusing several initial depth maps obtained through
different focus measures. [24] proposed an non-convex minimization scheme to
determine depth maps based on prior knowledge. [13] applied a preconditioned
alternating direction method of multipliers (PADMM) with a new cost function
to generate a noise-free depth map. Jeon, et al. proposed an algorithm to
estimate the multi-view stereo correspondences with sub-pixel accuracy using
the cost volume [14]. [35] developed an occlusion-aware depth estimation
algorithm to deal with occlusion edges.

Although the above traditional methods can produce good depth maps, these
methods require presetting for the depth estimation with careful parameter
tuning. Furthermore, they tend to be difficult to generalize into all scenes.

###  2.2 Learning-based Approaches

Recently, deep learning, in particular Convolutional Neural Networks (CNNs),
has successfully broken the bottleneck of traditional methods in a wide range
of fields. Such as, super-resolution [39], novel view generation [16] and
material recognition [36]. For the depth estimation, [30] introduced a fully-
convolutional neural network for highly accurate depth estimation. [1]
predicted depth from a single focal slice using deep neural networks by
exploiting dense overlapping patches. [11] suggested the first end-to-end
learning method to compute depth maps from the focal stack. [31] estimated
scene depths from a single image, using the information provided by a camera’s
aperture as supervision. They introduced two differentiable aperture rendering
functions that use the input image and predicted depths to simulate the depth-
of-field effects caused by real camera apertures. Then they trained a
monocular depth estimation network end-to-end to conduct depth estimation from
RGB images.

These methods demonstrate that focusness cues can greatly contribute to depth
estimation, achieving superior results. However, there is still large room for
further improvement in terms of spatial correlation excavation in the focal
stack and multi-modal fusion between the RGB image and the focal stack. In
this paper, we propose a deep learning based method that dynamically
incorporates a RGB and a focal stack to confront these challenges.

##  3 Method

###  3.1 The Overall Architecture

Figure 2: The whole pipeline.

In this paper, our goal is to excavate the spatial correlation among focal
slices and dynamically fuse RGB information and focusness features for focus-
based depth estimation. The overall framework is described in Figure 1. First,
the RGB image is fed into the RGB stream which consists of a SeNet-154 network
[12] pretrained on ImageNet [7], a decoder employs four upsampling layer to
gradually up-scale the final feature from the encoder and a refinement module
concatenate the feature from decoder and encoder in color channels with three
convolutional layers. Second, all focal slices are fed into another focal
stream to generate focusness information from focal slices, relying on the
proposed spatial-correlation perception modelling (SCPM) designed to excavate
the spatial correlation in focal slices. Last, we propose a multi-modal
dynamic fusion module (MDFM) to dynamically fuse multi-modal information
between RGB data and the focal stack in an adaptive way for providing good
depth map. The details of our two modules will be discussed in the following
two sections.

###  3.2 Spatial-Correlation Perception Module (SCPM)

Considering the stack of focal slices in a scene possess focus-area of
multiple scales and focused at different depth, we aim to correlate the depth
information with multi-scale focusness features. To do this, we propose a
spatial-correlation perception module (SCPM), in which the spatial correlation
between different focal slices is excavated by a proposed pyramid ConvGRU. In
this way, multi-scale focusness information in different slices can be
transfered along the depth direction.

Specifically, we first feed 12 focal slices $I_{1}(x,y,s,t)$,
$I_{2}(x,y,s,t)$…$I_{n}(x,y,s,t)$ to four 5$\times$5 convolutional layers to
encode focusness features $x_{i}$. This procedure can be defined as:

$\begin{array}[]{l}f_{i}\left({I_{i}(x,y,s,t);\theta_{i}}\right)\to
x_{i},\end{array}$ (1)

where $s\times t$ indicates the angular resolution, $x\times y$ indicates the
spatial resolution and $i$ represents the i-th focus slice. $\theta$ denotes
the parameters of the encoder layer and $f_{i}$ is a learning mapping
function. Then, in order to excavate the spatial correlation between different
focal slices, we draw ideas from recent works in [8, 38, 43]. Concretely, they
use a typical ConvGRU to capture short and long term temporal dependencies. In
our work, we consider the focusness features as a feature map sequence and
design a pyramid ConvGRU. In order to pass multi-scale focusness features
corresponding to focus area of multiple scales, the proposed pyramid ConvGRU
use an atrous spatial pyramid pooling (ASPP) module [5] for each of the gates
instead of convolution, which can encode multi-scale foucsness information by
applying atrous convolution at multiple parallel filters with different rates
and field-of-views. The dilation rates are 1, 3 and 5, respectively. The
formulation of the proposed pyramid ConvGRU is:

$r_{i}=\sigma\left[\begin{array}[]{l}W_{r1}*[x_{i},h_{i-1}],\\\
W_{r3}*[x_{i},h_{i-1}],\\\ W_{r5}*[x_{i},h_{i-1}]\\\ \end{array}\right],$ (2)
$z_{i}=\sigma\left[\begin{array}[]{l}W_{z1}*[x_{i},h_{i-1}],\\\
W_{z3}*[x_{i},h_{i-1}],\\\ W_{z5}*[x_{i},h_{i-1}]\\\ \end{array}\right],$ (3)
$\begin{array}[]{l}n_{i}=\tanh(x_{i}*W_{xn}+r_{i}\odot
h_{i-1}*W_{hn}+b_{n}),\end{array}$ (4)
$\begin{array}[]{l}h_{i}=\left({1-z_{i}}\right)\odot h_{i-1}+z_{i}\odot
n_{i},\end{array}$ (5)

Figure 3:  The architectures of ConvGRU and our pyramid ConvGRU. (a) ConvGRU;
(b) pyramid ConvGRU. In ConvGRU, both update and reset gates are standard
convolutional. In our pyramid ConvGRU, we use an atrous spatial pyramid
pooling (ASPP) module instead of convolution for each of the gates.

where all $W_{*}$ and $b_{*}$ are model parameters to be learned. $\sigma$ is
sigmoid function. $\odot$ and $\ast$ are element-wise multiplication and
convolution, respectively. The pyramid ConvGRU takes the i-th focusness
feature $x_{i}$ and a previous output $h_{i-1}$ as input, outputs a new
feature $h_{i}$ by combining $h_{i-1}$ and a candidate state $n_{i}$ weighted
by an output of an update gate $z_{i}$. The update and reset gates, $z_{i}$
and $r_{i}$, selectively update multi-scale focusness information from the
input focusness feature $x_{i}$ and the previous output $h_{i-1}$,
automatically learn the spatial correlation from neighboring slices. Note that
the updated focusness feature $h_{i}$ is not disappeared with information
transfering, but passed to the next slice.

###  3.3  Multi-Modal Dynamic Fusion Module (MDFM)

Figure 4:  The architectures of static fusion and our fuse fusion. (a) Three
different static fusion methods: sum fusion, weighted fusion, concatenate
fusion. (b) our MDFM.

As RGB images and focus slices imply different depth information, we consider
that fusing different multi-modal information is important for depth
estimation task. However, existing fusion schemes usually employ some manually
set, including sum fusion, weighted fusion, concatenate fusion. These static
fusion schemes not suitable for our task because the RGB and focusness feature
are not equivalent quantities, is prone to information loss. As shown in the
Fig.4(a), these static fusion methods processes the entire image. When the
network parameters are fixed, the convolution kernel does not change with the
input features, ignoring the relationship between multi-modal features.
Therefore, a more proper and effective strategy should be considered. To this
end, we introduce a multi-modal dynamic fusion module (MDFM) to dynamically
fuse the RGB features and focusness features in an adaptive manner. In this
module, the filter varies with focusness features is used to convolve with RGB
information , thereby avoiding information loss.

Figure 5: Visual results of the effect of MDFM. $Concatenate$ represents
Concatenate Fusion. $SUM$ represents Sum Fusion. $Weight$ represents Weighted
Fusion.

Specifically, the process of our module consists of two steps: 1) we choose a
adaptive kernel [33] instead of the spatially invariant convolution. In our
work, the standard spatial convolution W is adapted at each pixel using
focuesness features f via the adaptive kernel. Therefore, when the focusness
features change, the parameters of the convolution kernel also change; 2) we
apply the generated adaptive convolution kernel to RGB features f, making the
whole network dynamically fuse multi-modal information to get an accurate
depth map.

For the output from the spatial-correlation perception module h, we have:

$\begin{array}[]{l}d_{i}=\sum\limits_{j\in\Omega(i)}{\exp(-\frac{1}{2}(h_{i}-h_{j})}^{T}(h_{i}-h_{j}))W\left[{p_{i}-p_{j}}\right]f_{j}+b\end{array}$
(6)

where $[pi-pj]$ is index of the spatial dimensions of an array with 2D spatial
offsets, i and j represent the pixel coordinates, $W$ is a standard spatial
convolution W, $f_{j}$ represents the output from the RGB stream, $b$ is bias
term, $d$ represents the final depth prediction map. Before we perform
filtering operation, the filter has a pre-defined form and depend on the
content of focusness features. It is hoped that the prediction map depends on
both the RGB features and the reliable focusness information. Finally, we
refine the depth map by adding two consecutive convolutional layers. As shown
in the Fig.4(b), in test stage, compared to traditional methods of fusing
multi-modal information which parameters do not change with input features,
our model dynamically selects filter parameters for convolution based on the
content of the focusness features and adaptively selects multimodal features
for fusion, avoiding information loss.

###  3.4 Loss Functions

In this section, we calculate the reconstruction error between the predicted
images and the ground truth. We define the loss by:

$\begin{array}[]{l}L=l_{depth}+\lambda l_{grad}+\mu l_{normal}\end{array}$ (7)

where $\lambda$, $\mu$ are weighting coefficients. Here we set as $\lambda=1$,
$\mu=1$. We use a logarithm of depth errors as $l_{depth}$ to minimize the
difference between the depth estimate map $d_{i}$ and its ground truth
$g_{i}$:

$\begin{array}[]{l}l_{depth}=\frac{1}{n}\sum\limits_{i=1}^{n}{\ln\left({\left\|{d_{i}-g_{i}}\right\|_{1}+\alpha}\right)}\end{array}$
(8)

where $\alpha(>0)$ is a parameter we set $\alpha=0.5$. In order to deal with
edge distortion problems caused by CNNs training, we consider the following
loss function of the gradients of depth:

$\begin{array}[]{l}l_{grad}=\frac{1}{n}\sum\limits_{i=1}^{n}{\left({F\left({\nabla_{x}\left({\left\|{d_{i}-g_{i}}\right\|_{1}}\right)}\right)+F\left({\nabla_{y}\left({\left\|{d_{i}-g_{i}}\right\|_{1}}\right)}\right)}\right)}\end{array}$
(9)

where $\bigtriangledown x(*)$ is the spatial derivative of $\|d_{i}-g_{i}\|$
computed at the ith pixel with respect to $x$, and so on. To deal with such
small depth structures and further improve fine details of depth maps, we
consider yet another loss for training, which measures accuracy of the normal
to the surface of an estimated depth map with respect to its ground truth:

$\begin{array}[]{l}l_{normal}=\frac{1}{n}\sum\limits_{i=1}^{n}{\left({1-\frac{{\left\langle{n_{i}^{d},n_{i}^{g}}\right\rangle}}{{\sqrt{\left\langle{n_{i}^{d},n_{i}^{d}}\right\rangle},\sqrt{\left\langle{n_{i}^{g},n_{i}^{g}}\right\rangle}}}}\right)}\end{array}$
(10)

where $<\textperiodcentered,\textperiodcentered>$ denotes the inner product of
vectors. Denoting the surface normal of an estimated depth map and its ground
truth by
$n_{i}^{d}=\left[{-\nabla_{x}\left({d_{{}_{i}}}\right),{\rm{}}-\nabla_{y}\left({d_{{}_{i}}}\right),1}\right]^{{}^{T}}$,
$n_{i}^{g}=\left[{-\nabla_{x}\left({g_{{}_{i}}}\right),{\rm{}}-\nabla_{y}\left({g_{{}_{i}}}\right),1}\right]^{{}^{T}}$

##  4 Experiments

###  4.1 Experiments Setup

####  4.1.1 Datasets.

To demonstrate the effectiveness of the proposed approach, we conduct
experiments on DUT-LFDD dataset, LFSD dataset and a Mobile Phone dataset. Note
that although there are some synthetic light field (LF) datasets proovided by
[12] which consist of multiview images and corresponding depth maps. The
models trained on such synthetic data have trouble generalizing to real-world
data. Furthermore, the focal stack can not be correctly synthesized from the
multi-view images provided in HCI, because the changes between views are not
transitional but rotational.

DUT-LFDD: This dataset contains 967 real-world light-field samples coming from
a large variety of indoor and outdoor scenes. We randomly select 630 samples
for training and the remaining 337 samples for testing. All images are
captured by commercial Lytro Illum camera in real life scenes and contains
many challenging scenes. Each light field sample consists of a RGB image, a
focal stack with 12 focal slices focusing at different depths and a depth
image corresponding to the RGB image.

LFSD:  The dataset proposed by [19] contains 100 light fields captured by
Lytro camera, including 60 indoor and 40 outdoor scenes. Each light field
consists of an RGB image, focal slices and depth map.

Mobile Phone dataset:  The Mobile Phone dataset provided by previous
researchers [32]. They continuously captured images of size 640$\times$360
pixels using a Samsung Galaxy S3 phone during auto-focusing. The dataset
provides focal stack and RGB image of different scenes (number of frames in
parenthesis): plants(23), bottles(31), fruits(30), metals(33), window(27),
telephone(33), etc. For each scene, we choose 12 focal slices and a RGB image
to evaluate our model.

To prevent the overfitting problem, we augment the training set by the
following operations:

  * •

Flipping: we only consider horizontal flipping (i.e. mirroring) of images at a
probability of 0.5.

  * •

Rotating: the RGB image, focal stack and the depth image are rotated by a
random degree r $\in$ [-5,5].

  * •

Color Jitter: brightness, contrast, and saturation values of the sample are
randomly scaled by c $\in$ [0.6,1.4].

####  4.1.2 Evaluation Metrics.

We adopt seven metrics for comprehensive evaluation, including Mean Eelation
error(Abs Rel), Squared Relative error (Sq Rel), Root Mean Squared error
(RMSE), Mean log 10 error (RMSE log), Accuracy with threshold
($thr=1.25,1.25^{2},1.25^{3}]$). They are universally-agreed and standard for
evaluating a depth estimation model and well explained in many literatures
[41, 34, 42].

Table 1:  Quantitative results of the ablation analysis for our SCPM. Methods | Error metric$\downarrow$ | Accuracy metric$\uparrow$
---|---|---
RMSE | RMSE Log | Abs Rel | Sq Rel | $\delta<1.25$ | $\delta<1.25^{2}$ | $\delta<1.25^{3}$
2DCNN | 0.3667 | 0.0501 | 0.1879 | 0.1015 | 0.7102 | 0.9435 | 0.9930
GRU | 0.3648 | 0.0492 | 0.1821 | 0.0976 | 0.7159 | 0.9444 | 0.9931
SCPM | 0.3457 | 0.0453 | 0.1707 | 0.0864 | 0.7342 | 0.9516 | 0.9952

  * •

Mean relation error (Abs Rel): $\frac{1}{T}\sum\nolimits_{i\in
T}{\frac{{\left|{d_{i}-d_{i}^{gt}}\right|}}{{d_{i}^{gt}}}}$

  * •

Root mean squared error (RMSE):

${\frac{1}{T}\sum\nolimits_{i\in T}{\left\|{d_{i}-d_{i}^{gt}}\right\|}^{2}}$

  * •

Mean log 10 error (RMSE log):

${\sqrt{\frac{1}{T}\sum\nolimits_{i\in
T}{\left\|{\log(d_{i})-\log(d_{i}^{gt})}\right\|}^{2}}}$

  * •

Squared relative error (Sq Rel): $\frac{1}{T}\sum\nolimits_{i\in
T}{\frac{{\left\|{d_{i}-d_{i}^{gt}}\right\|^{2}}}{{d_{i}^{gt}}}}$

  * •

Accuracy with threshold
($\delta$):$\max\left({\frac{{d_{i}}}{{d_{i}^{gt}}},\frac{{d_{i}^{gt}}}{{d_{i}}}}\right)=\delta<thr$

####  4.1.3 Implementation Details.

We implement our network based on Pytorch framework with one Nvidia GTX 1080Ti
GPU. We train the RGB stream and focal stream using Adam optimizer with an
initial learning rate of 0.0001, and reduce it to 10% for every 5 epochs. We
set $\beta 1$ = 0.9, $\beta 2$ = 0.999, and use weight decay of 0.0001. The
encoder module in the RGB stream is initialized by a model pretained with the
ImageNet dataset. The other layers in the network are randomly initialized.
The batchsize is 1 and maximum epoch is set 80.

Figure 6:  The architectures of static fusion and our fuse fusion. (a) Three
different static fusion methods: sum fusion, weighted fusion, concatenate
fusion. (b) our MDFM.

###  4.2 Ablation Studies

####  4.2.1 Effect of SCPM.

The SCPM is proposed to excavate the spatial correlation between different
focal slices. To verify the effectiveness of the SCPM, we first replace the
SCPM with 7 convolution layers (noted as $2DCNN$). Table 1 shows that the SCPM
improves the RMSE performances by nearly 2% than 2DCNN. We believe that this
improvement is due to the cyclic structure retaining the spatial relationship
between different focal features. Then we compare the performance of our
pyramid ConvGRU and ConvGRU. We replace pyramid ConvGRU with the ConvGRU. And
we can observe from Table 1 that the improvements in performance are achieved
by using our pyramid ConvGRU. These improvements are resonable since our model
can adapt to different sizes of focus area compared to ConvGRU. Furthermore,
we randomly select four different focus slices for visualization after
inputting them into pyramid ConvGRU in Fig.5. (a)-(c) and (e)-(g) represent
the first three of the 12 focus slices of each scene. (d) and (h) represent
the output of the last pyramid ConvGRU in each scene. We can see from Fig.5
that the feature maps of different focus slices contain different details and
are passed by our pyramid ConvGRU.

Table 2:  Quantitative results of the ablation analysis for our MDFM. $Concatenate$ represents Concatenate Fusion. $SUM$ represents Sum Fusion. $Weight$ represents Weighted Fusion. Type | Methods | Error metric | Accuracy metric
---|---|---|---
RMSE |  RMSE Log |  Abs Rel |  Sq Rel |  $\delta<1.25$ |  $\delta<1.25^{2}$ |  $\delta<1.25^{3}$
Our dataset | SUM |  0.6600 |  0.1564 |  0.3777 |  0.3211 |  0.4186 |  0.6838 |  0.9184
Weight |  0.4877 |  0.0921 |  0.2603 |  0.1652 |  0.5190 |  0.8584 |  0.9749
Concatenate |  0.4268 |  0.0705 |  0.1917 |  0.1065 |  0.6250 |  0.9122 |  0.9790
our |  0.3457 |  0.0453 |  0.1707 |  0.0864 |  0.7342 |  0.9516 |  0.9952
LFSD | SUM |  0.7037 |  0.1788 |  0.3977 |  0.3484 |  0.3971 |  0.6325 |  0.8889
Weight |  0.4667 |  0.0829 |  0.2422 |  0.1501 |  0.5777 |  0.8823 |  0.9756
Concatenate |  0.4284 |  0.0697 |  0.1858 |  0.1042 |  0.6570 |  0.9081 |  0.9752
our |  0.3612 |  0.0550 |  0.1796 |  0.0901 |  0.6973 |  0.9339 |  0.9874

####  4.2.2 Effect of MDFM.

The MDFM is proposed for dynamically fusing multi-modal information between
RGB features and focusness features. To demonstrate the effectiveness of the
MDFM, we compare the MDFM with a variety of conventional fusion methods,
including Concatenate Fusion(noted as $Concatenate$), Sum Fusion(noted as
$SUM$), Weighted Fusion(noted as $Weight$). For better comparison, we replace
the fusion block in the framework. The results of the comparison are shown in
Figure 4. It can be seen that the quality of depth map achieves accumulative
improvements by a large margin with the MDFM. Especially in regions of depth
discontinuities, the MDFM is able to recover the depth more accurately than
the concatenate fusion and preserves more structural information compared to
other static fusion methods. Numerically, our proposed MDFM reduces the RMSE
performances by nearly 8% than concatenate fusion and as shown in Table 2.

Table 3: Quantitative comparison with state-of-the-art methods. From top to bottom: our dataset, LFSD dataset. $*$ respects non-deep-learning methods. The best result results are shown in boldface.  Type | Methods | Error metric | Accuracy metric
---|---|---|---
RMSE |  RMSE Log |  Abs Rel |  Sq Rel |  $\delta<1.25$ |  $\delta<1.25^{2}$ |  $\delta<1.25^{3}$
Our dataset | Our |  0.3457 |  0.0453 |  0.1707 |  0.0864 |  0.7342 |  0.9516 |  0.9952
DDFF |  0.5255 |  0.1042 |  0.2666 |  0.1834 |  0.4944 |  0.8202 |  0.9667
EPINET |  0.4974 |  0.0959 |  0.2324 |  0.1434 |  0.5010 |  0.8375 |  0.9837
VDFF∗ |  0.7192 |  0.1887 |  0.3887 |  0.3808 |  0.4040 |  0.6593 |  0.8505
PADMM∗ |  0.4730 |  0.0912 |  0.2253 |  0.1509 |  0.5891 |  0.8560 |  0.9577
LF∗ |  0.6897 |  0.1436 |  0.3835 |  0.3790 |  0.4913 |  0.7549 |  0.8783
| LF$\\_$OCC∗ |  0.6233 |  0.1432 |  0.3109 |  0.2510 |  0.4524 |  0.7464 |  0.9127
LFSD | Our |  0.3612 |  0.0550 |  0.1796 |  0.0901 |  0.6973 |  0.9339 |  0.9874
DDFF |  0.4255 |  0.0717 |  0.2128 |  0.1204 |  0.6185 |  0.8916 |  0.9860
VDFF∗ |  0.5747 |  0.1258 |  0.3320 |  0.2660 |  0.4730 |  0.7823 |  0.9359
PADMM∗ |  0.4238 |  0.0722 |  0.2153 |  0.1336 |  0.6536 |  0.8880 |  0.9770
LF∗ |  - |  - |  - |  - |  - |  - |  -
LF$\\_$OCC∗ |  - |  - |  - |  - |  - |  - |  -
| EPINET |  - |  - |  - |  - |  - |  - |  -

###  4.3 Comparison with State-of-the-arts

We compare our method with 6 other state-of-the-arts light field depth
estimation method, containing both deep-learning-based methods and traditional
methods (marked with $*$). Deep-learning-based method is DDFF [11], EPINET
[30], traditional methods are VDFF∗ [24], PADMM∗ [13], LF∗[14], LF$\\_$OCC∗
[35]. For fair comparisons, the results from competing methods are generated
by authorized codes. To verify the generalization and applicabilityof our
network, we conduct experiments on three dataset.

Figure 7: Visual comparisons of our method over some other approaches. Our
method produces more accurate depth maps than others, and our results are more
consistent with the ground truths (denoted as ’GT’).

####  4.3.1 Quantitative Evaluation.

As shown in Table 3, our method is able to clearly outperform the other state-
of-the-art methods on our dataset in terms of seven evaluation metrics. Not
only that, we also apply the model parameters trained on our dataset directly
to the LFSD dataset for testing, and our method achieves significant
advantages, such as Top-1 accuracies (Sq Rel) and Top-1 accuracies (RMSE).
Note that the LFSD dataset only provides the focal slices, RGB images and
depth maps, therefore, we only compare our method with 3 focus-based methods
on LFSD dataset.

####  4.3.2 Qualitative Evaluation.

Figure 5 provides some challenging samples of results comparing our method
with other state-of-the-art methods. It can be seen that our method can
achieve accurate prediction, when foreground and background are similar as
shown in the $1^{th}$ and $4^{th}$ rows, when some areas are textureless as
shown in the $3^{th}$ row, when smooth surface as shown in the $2^{th}$ row.

We further illustrate the visual results of our method on LFSD dataset in
Figure 6. From the $5^{th}$ line and the $6^{th}$ , we can find that compared
with traditional methods, our network retains more detailed information and
better maintains the structure of the depth map. From the $7^{th}$ line, we
can find the results of DDFF contain a lot of noise. This is beacuse DDFF used
a standard 2D CNN to learns filters that extend across the entire focal stack.
In contrast, our model preserves the spatial correlation between focal slices
and effectively fuses the focusness feature and RGB information to reduce
information loss.

Figure 8:  Comparison with deep learning-based methods on LFSD dataset: RGB
images, the corresponding ground truth, our estimated depth maps and other
state-of-the-arts light field depth estimation results.

###  4.4 Mobile Phone Results

The common consumer level cameras also can capture focal stack. To expand the
practical application of our network, we test our network on the focal stack
captured captured by a Samsung Galaxy S3 phone, provided by the authors of
[32]. For ease of testing, we choose 12 focal slices and a RGB image from each
scene. Figure 9 provides a qualitative comparison of different deep learning-
based methods. It can be seen that the proposed method allows to recover high
quality depth maps with less noise and sharper object boundaries, although the
model was not trained on this specific dataset. The impressive results show
that the proposed method can be used for further applications in our daily
life. Further, our model is not limited by the light field camera because the
smart-phone camera can capture focal stacks by manipulating the focus distance
programmatically and it represents the dominant modality of image capture.

Figure 9:  Comparison with deep learning-based methods on Mobile Phone
dataset:(a) RGB images (b) our results (c) DDFF[11]

##  5 Conclusion

In this paper, we propose a multimodal learning which incorporates RGB data
and the focal stack for focus-based depth estimation. Our SCPM excavates the
spatial correlation between different focal slices and sequently pass multi-
scale focusness information along the depth direction by exploiting our
proposed pyramid ConvGRU. The MDFM dynamically fuse the RGB features and
focusness features in an adaptive manner. In this module, the filter varies
with focusness features is used to convolve with RGB information. Our
experiments show that the proposed method achieves superior performance,
especially in challenging scenes. Our extensive evaluation shows that the
proposed network can be applied on common consumer level cameras data
successfully.

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# Phase coexistence and associated non-equilibrium dynamics under
simultaneously applied magnetic field and pressure

Sudip Pal, Kranti Kumar and A. Banerjee  UGC DAE Consortium for Scientific
Research
Khandwa road, University campus
Indore-452001, M.P., India

###### Abstract

A quantitative estimation of the effect of simultaneously applied external
pressure (P) and magnetic field (H) on the phase coexistence has been
presented for Pr0.5Ca0.5Mn0.975Al0.025O3 and La0.5Ca0.5MnO3, where the
ferromagnetic (FM)-metal and antiferromagnetic (AFM)-insulator phases compete
in real space. We found that the nonequilibrium dynamics across the FM-AFM
transition is primarily dictated by the effect of P and H on the supercooling,
superheating temperatures, and the nucleation and growth rate of the
equilibrium phase. These effects across the transition is also responsible for
the relative volume fraction of the competing phases at low temperature.
Importantly in the entire magnetic field-pressure-temperature range of phase
coexistence, the interface between the two competing phases having different
spin and structural order plays a very important role in controlling the non-
equilibrium dynamics.

††preprint: APS/123-QED

##  I Introduction:

The coexistence of competing phases in spatially separated regions is a common
phenomenon across first order phase transitions. Such coexistence has been
studied in the context of various class of promising materials, for instance,
multiferroics showing strong magnetodielectric coupling, Mott insulators,
artificial spin ice etc. and of course manganites, shape memory alloys showing
colossal magneto-resistance, magnetocaloric effects Dagotto2002 ; Ahn2004 ;
SBR2004 ; Dagotto2005 ; Lin2018 ; Zhu2016 . The increasing research interest
in this direction is driven firstly, due to a possibility of technological
application arising from tunability of contrasting phases and secondly, these
phenomena are not yet fully understood. Control over the volume fraction of
the competing phases and identification of the critical parameters are
required to rein in their physical properties Zang2015 ; Pandey2003 ;
Lourembam2014 ; Song2021  and utilize them in spin valves, memory devices etc
Yang2019 . In this context, besides magnetic and electric fields, external
pressure can be a very good control parameter due to strong involvement of the
underlying lattice in these systems, which has been recently exemplified using
strain in La0.7Ca0.3MnO3 Hong2020 . In recent times, advancement in
experimental techniques have turned pressure (P) into an additional, as well
as important state-of-the-art experimental tool. However, investigation
involving the variation of both magnetic field (H) and P along with necessary
quantitative understanding is still missing. In this backdrop, manganites can
serve as an excellent prototype Hong2020 ; Yang2019 ; Baldini2012 ; McLeod2017
; Joseph2019 ; Hong2018 ; Udalov2020 ; Zhou2020 .

Colossal magnetoresistance in Manganites, large magnetocaloric and
magnetostriction in various alloys and intermetallics are directly related to
the coexistence of metallic-ferromagnetic (FM) and insulating-
antiferromagnetic (AFM) phases appearing due to the first order magnetic
transition (FOMT) SBR2013 ; Markovich2017 ; Banik2018 . In our study, we have
combined both H and P to tune and determine the origin of the tunability of
the phase coexistence in two well known manganites Pr0.5Ca0.5Mn0.975Al0.025O3
(PCMAO) and La0.5Ca0.5MnO3 (LCMO). Our aim is twofold: How simultaneous
application of H and P influence the phase coexistence across the first order
transition i.e., across the H-T region where magnetization exhibits thermal
hysteresis due to supercooling and superheating phenomena. It is interesting
to note further that PCMAO and LCMO show phase coexistence even below the
thermal hysteresis which persists down to lowest temperature due to kinetic
arrest of the first order phase transition. Therefore, we are also interested
on the effect of P and H at the low temperature regime below the hysteresis.
Throughout the study, we emphasize on an important fact that in both of these
regions (across and below the thermal hysteresis), the phase coexistence is a
nonequilibrium phenomenon and the time scale associated with the non-
equilibrium dynamics can be tuned by H and P. In addition, a detail
quantitative understanding of the volume fraction of the coexisting phases is
necessary to exploit the phase coexistence as a device. Here, we have used
bulk magnetization value to estimate the volume fraction of the FM-metal and
AFM-insulator phases. It may be noted that, until now mostly the surface
sensitive techniques, like Hall probe, scanning tunneling microscopy, magnetic
force microscopy, photo electron spectroscopy etc. have been used as a probe
of phase coexistence SBR2004 ; Dagotto2005 ; Lin2018 ; Zhu2016 , but the
surface often contains additional disorder and strain. As a result, the nature
of the phase coexistence in the bulk may be different.

A first order transition can be distinguished from a continuous transition by
the appearance of supercooling (H∗, T∗) and superheating (H∗∗, T∗∗) lines
which give rise to a hysteresis across the transition in the H-T plane.
However, quenched disorder present in the sample broadens these lines into
bands Imry1979  and results into the phase coexistence in a wide range of the
control parameters. In several materials, the kinetics of a FOMT gets arrested
below a heuristic band, called kinetic arrest band (HK, TK). The
transformation from the high-T phase to low-T phase is arrested below that
part of the [HK, TK] band which lies above the [H∗, T∗] band and the system
below the thermal hysteresis remains as a composite of the transformed
fraction of the low-T phase and untransformed fraction of the high-T phase
Chattopadhyay2005 ; AB2006 ; AB2008 ; Chaddah2008 ; Dash2010 ; AB2011 ;
EPL2013 ; Sudip2021 ; Pal2021 . The phenomena is called as the “kinetic
arrest” (KA) and the magnetic state at low temperature is called as the
magnetic glass. In this state, the high-T phase persists down to lowest
temperature because of KA, and the phenomenon of KA provides a systematic
framework to understand the bizarre properties of phase separation in many
systems. This phenomenon has been observed across various transitions,
including magnetic, structural, metal-insulator transitions, in multiferroics
etc SWC2006 ; Choi2010 ; MnFePSi ; Mn2PtGa ; Cakir2020 ; Pampi2020 .

PCMAO is in the paramagnetic state at room temperature and undergoes
transition to CE type antiferromagnetic (AFM) phase with decrease in
temperature AB2006 . The high-T AFM state is eventually followed by a FOMT to
low-T FM phase on further cooling. At low strength of the magnetic field, the
AFM to FM phase transition is partially arrested during cooling and the system
contains a large fraction of the AFM phase down to lowest temperature AB2006 .
On the other hand, in LCMO, it undergoes a first order transition from high-T
FM to low-T CE type AFM phase which is arrested at high fields AB2008 . The
arrested FM fraction increases with increase in H. Present work includes the
magnetization measurements under simultaneous application of P and H and a
quantitative investigation on the effects of H and P on the volume fraction
and the non-equilibrium dynamics of the competing phases associated with the
first order transition. We explain the observation in terms of the interplay
of H and P induced changes in the supercooling and superheating across the
first order transition and the kinetic arrest. We also underline that the
interface between the two competing phases plays a crucial role in controlling
the dynamics of the phase coexisting state.

##  II Experimental details:

Polycrystalline bulk samples of PCMAO and LCMO have been prepared by standard
solid state method and chemical route, known as pyrophoric method,
respectively Chaddah2008 ; AB2006 . Further details on the sample preparation
and characterization can be found in Ref. 24 and 26. Magnetic measurements
have been carried out in commercial 7 Tesla superconducting quantum
interference device (SQUID) magnetometer (M/S Quantum Design, USA). For the
measurements under pressure, a Cu-Be cell (Mcell 10, easy lab) has been used,
where a fixed pressure was applied and locked at room temperature. The
pressure cell can be mounted inside the magnetometer to carry out the
magnetization measurements, which have been performed at different H at a
fixed pressure. Daphne oil has been used as the pressure transmitting medium
to maintain the hydrostatic condition of pressure on the sample. Pressure
values have been determined from the variation of superconducting transition
temperature of Sn wire that has been loaded inside the pressure cell Dash2010
. Field cooled cooling (FCC) magnetization (M) versus T curves are recorded
while cooling the sample is presence of a constant magnetic field. The field
cooled warming (FCW) curves are recorded in the subsequent warming cycle
without changing the field. For zero field cooled (ZFC) measurements, the
sample has been cooled down to T = 5 K in absence of external field. Then, M
versus T data has been recorded during warming after applying external field
at the lowest temperature. Time ($t$) dependent magnetization (M) data have
been recorded in field cooling (FC) mode where the sample is first cooled from
T = 320 K to the measurement temperature in presence of the external field.
After temperature becomes stable, then isothermal variation of M with time
recorded keeping the field unchanged.

Figure 1: (a) Temperature variation of magnetization of PCMAO at H = 2, 5 and
7 Tesla recorded in FCC and FCW modes. The thermal hysteresis indicates the
transition between high-T AFM and low-T FM phases and it shifts towards higher
temperature with magnetic field. $\Delta$T is the width of the thermal
hysteresis. (b) Variation of FM volume fraction ($f_{FM}$) with H at T = 5 K.
The line is a guide to eye. Figure 2: Temperature variation of magnetization
in different pressure of (a) PCMAO at H = 2 Tesla. In the FCC curve at P = 5
kbar, we have drawn a straight line to determine the slope of the curve, which
should be propertional to the transformation rate from high-T AFM to low-T FM
phase across the thermal hysteresis. (b) LCMO at H = 1 Tesla. The legends
indicate the P values applied on the sample in the unit of kbar. In case of
LCMO also, the transformation rate from high-T FM to low-T AFM phase can be
found from the slope of the FCC curves at respected pressure values.

##  III Results:

Fig. 1(a) presents the temperature variation of M (FCC and FCW) across the
transition between high-T AFM and low-T FM phases in PCMAO at three different
magnetic fields. Magnetization shows thermal hysteresis across the transition
and the hysteresis shifts towards higher temperature with increase in H. Also
note that, the width of the thermal hysteresis ($\Delta$T, see Fig. 1(a))
varies non-monotonically with H. At H = 5 Tesla, the width $\Delta$T $\approx$
20 K, which is significantly larger than $\Delta$T $\approx$ 12 and 10 K at H
= 2 and 7 Tesla respectively. On the other hand, the value of M at low
temperatures, say at T = 5 K increases by a large amount with increase in H
which can only be explained with the phase separation scenario. It should be
mentioned here that the saturation field of the FM phase in PCMAO is a few
hundred Oe AB2006 . At low temperature, the system contains both FM and AFM
volume fractions, and the amount of the FM phase increases with H at the
expense of the competing AFM phase AB2006 . In Fig. 1(b), we have estimated
the variation of the FM phase fraction at T = 5 K with the applied magnetic
field. It has been estimated from the magnetization value at T = 5 K
($M_{5K}$) following the method described in Ref. 30. If the FM volume
fraction is $f_{FM}$ at T = 5 K [so the AFM volume fraction is (1-$f_{FM}$)],
then $M_{5K}$ is given by the relation-

$M_{5K}=f_{FM}(M_{0}+\alpha\times H_{m})+(1-f_{FM})\times\beta\times H_{m}$
(1)

Here, $\alpha$ and $\beta$ are the dc susceptibility (dM/dH) of the FM and AFM
phases respectively which are obtained from the slope of the M-H curve at T =
5 K. $M_{0}$ is the spontaneous magnetization of the FM phase. As H increases
from H = 2 to 7 Tesla, $f_{FM}$ also increases by more than eight folds from
around 7 to 60 %.
In Figs. 2(a) and 2(b), we have shown the FCC and FCW curves of PCMAO and LCMO
respectively. Different curves correspond to the different applied pressure at
a constant H = 2 and 1 Tesla for PCMAO and LCMO respectively. It may be
mentioned here that a part of this data of LCMO has been reported earlier in
Ref. 28 but has been reproduced here to make this paper self content. The
broad thermal hysteresis between the FCC and FCW curves evince the coexistence
of the high and low temperature phases across the hysteresis. Besides, due to
kinetic arrest, the phase coexistence prevails down to the lowest temperature
in both these systems. In PCMAO, the M at T = 5 K ($M_{5K}$) shows non-
monotonic variation with pressure. It initially decreases as we increase the
pressure from ambient to P = 2.5 kbar. As pressure increases further, $M_{5K}$
starts to increase again. On the other hand in LCMO, $M_{5K}$ monotonically
increases with pressure. Such variation of $M_{5K}$ indicates that the volume
fraction of the two coexisting phases (FM and AFM) change with pressure in
both systems, but the dependence is qualitatively very much different. In the
next section, we will estimate the volume fraction of the FM phase at T = 5 K
in H-P landscape and show that the efficiency of pressure in tuning the volume
fraction of the coexisting phases also depends upon the applied magnetic
field.

Figure 3:  Change in volume fraction of the metallic-FM phase ($f_{FM}$) with
H and P in (a) PCMAO and (b) LCMO. Note that in case of PCMAO, $f_{FM}$
initially decreases and finally starts to increase again with pressure. On the
other hand in LCMO, $f_{FM}$ monotonically increases.

###  III.1 Effect of pressure and magnetic field on the FM and AFM phase
fractions at low temperature:

In Figs. 3(a) and 3(b), we present the variation of the FM volume fraction
($f_{FM}$ in %) at T = 5 K in PCMAO and LCMO, respectively at different P and
H, which we have calculated using eqn. 1. It is interesting that in case of
PCMAO [Fig. 3(a)], pressure initially suppresses the FM phase fraction. At
higher pressure, $f_{FM}$ increases again. Moreover, the change in phase
fraction with the application of pressure is small. For example (see H = 5
Tesla, Fig. 3(a), black filled square), at H = 5 Tesla $f_{FM}$ is around
50.8% at ambient pressure which is consistent with Fig. 1(b). As pressure is
increased to P = 2.5 kbar, $f_{FM}$ reduces to $f_{FM}$ = 48% which is
followed by monotonic rise to 50.1% at P = 9.1 kbar. Such kind of non-
monotonic dependence of $f_{FM}$ on pressure is interesting. It points towards
the existence of two competing mechanisms occurring simultaneously, one trying
to enhance the $f_{FM}$ and the other mechanism tending to reduce the $f_{FM}$
at each pressure. As we will show later that this is actually the situation
and the net volume fraction of the two phases is decided by which of these two
mechanism is dominant. On the other hand, in case of LCMO as shown in Fig.
3(b), the change in the FM volume fraction is comparatively large and it
monotonically increases with increase in the pressure. For example, at H = 1
Tesla, $f_{FM}$ = 21.1% at ambient pressure which monotonically rises to 34.5%
at P = 9.12 kbar. Also note that at low fields say H = 1 Tesla, $f_{FM}$
varies rather smoothly as a function of pressure. However at higher H say 4
Tesla, $f_{FM}$ initially increases sharply and tends to saturate with
increase in the applied pressure. The variation of $f_{FM}$ in both PCMAO and
LCMO reveal a complex dependence of the volume fraction of the competing
metallic-FM and insulating-AFM phases on H and P.

Figure 4: (a) and (b) shows the variation of the T∗ and T∗∗s with P at
different applied field in PCMAO. (c) and (d) shows the same in case of LCMO.
The solid lines are guide to eye.

###  III.2 Effect of pressure on the supercooling, superheating and kinetic
arrest temperatures:

Thermal hysteresis in PCMAO progressively shifts toward higher temperature as
P increases [see Fig. 2(a)] i.e., the supercooling (T∗) and superheating (T∗∗)
temperatures increase with P. We want to mention here that we have performed
magnetization measurements at different P and H, but have not shown here for
conciseness. Some data on LCMO can be found in Ref. 28. The variation of the
T∗ and T∗∗ with H and P are shown in Figs. 4(a) and 4(b). There are different
ways to define T∗ and T∗∗ because the thermal hysteresis is broad. In case of
PCMAO, we have considered the temperature where the (dM/dT) in FCC (FCW) shows
the minima as T∗ (T∗∗). In the FCC curve, M start to increase rapidly at TNC,
which indicates the rapid nucleation and growth of the FM phase below TNC.
Similarly, in the FCW curve, nucleation of the AFM phase has started above
TNW. Interestingly note that, TNC $>$ TNW, which means that onset of the
nucleation of the AFM phase during heating has started at a lower temperature
than the onset temperature of nucleation of FM phase during cooling. This
indicates towards a wide distribution of the T∗ and T∗∗ in the material, such
that they form bands instead of lines in H-T plane, and overlap with each
other. This occurs due to presence of quenched disorder, which results into a
landscape of the transition temperature, and thereby broadening of the first
order transition SBR2004 ; Imry1979 . In LCMO, the thermal hysteresis shifts
towards lower temperature as higher P is applied. In this case, we have taken
a different approach to find out the T∗ and T∗∗ values. In the FCC curve, the
temperature where M shows peak i.e., M is maximum is taken as the T∗.
Similarly, T∗∗ has been taken where M has a peak in the FCW curve. We have
shown the variation of T∗ and T∗∗ in Figs. 4(c) and 4(d). The effect on T∗ by
P and H is quite dramatic in case of LCMO. T∗ falls sharply at very tiny
pressure of P = 0.68 kbar (Fig. 4(c), remains comparatively unaffected at
higher pressure. It is important to note that, there are a few common features
if we compare the effect of pressure on PCMAO and LCMO. First of all,
application of P suppresses the AFM phase in both the systems, which is
evident from the variation of T∗ and T∗∗ with pressure. Secondly, the
variation of these transition temperatures are not linear in P.

Besides, T∗ and T∗∗, the temperature TK corresponding to kinetic arrest band
is also important in case of both PCMAO and LCMO. The phase coexistence
persist even below the thermal hysteresis down to lowest temperature due to
kinetic arrest band, [HK, TK], and therefore, it is necessary to check the
position of the [HK, TK] band at higher pressure. In case of PCMAO, the effect
of P on the [HK, TK] band at a fixed H can be found from the measurement of
the ZFC curves as shown in the Fig. 5 Lakhani2010 . In the case of zero field
cooling, as T is increased, magnetization sharply increases with temperature
due to transformation of arrested AFM phase into the FM phase-which is called
dearrest. M increases up to the end point of [HK, TK] band, where the value of
M is maximum, which we have marked as vertical line in Fig. 5. Note that, this
temperature is not significantly affected by pressure. Above this temperature,
no significant conversion of the phase fraction happens until [H∗∗, T∗∗] band
is reached, where the FM phase converts into AFM phase and M sharply
decreases. We have also indicated the T∗∗ values by the downward arrows for
the ambient and highest applied pressures. This data reveals that the external
pressure does not affect the [HK, TK] band significantly as compared to the
[H∗∗, T∗∗] band. It should be mentioned here that [HK, TK] is practically a
band, i.e. it is spread over certain H-T window and the ZFC curve as shown
Fig. 5 only gives one end of the band. Nonetheless, it can be shown that
pressure does not significantly affect the width of the band as well (data no
shown here). In case of LCMO also, P does not affect the [HK, TK] band
significantly. The data has not been shown here for conciseness.

Figure 5: ZFC curves of PCMAO at different P. The measurement H = 5 Tesla. The
vertical line marks the end of KA band. It indicates that [HK, TK] remains
unaffected by P.

###  III.3 Effect of pressure on the nucleation and growth across the thermal
hysteresis:

A first order transition during cooling proceeds through the nucleation and
subsequent growth of the low-T phase at the cost of the high-T parent phase.
Therefore, in case of PCMAO, as we gradually reduce the sample temperature,
the high-T AFM phase transforms into the FM phase through the creation and
growth of the FM droplets which are larger than a critical size Rawat2013 .
This is manifested as the increase in M with decrease in temperature across
the thermal hysteresis shown in Figs. 1(a) and 2(a). Therefore, legitimately,
the transformation rate of the FM phase should be proportional to the slope of
the M-T curve in the hysteresis region. The slope of the FCC curves shown in
Fig. 2(a) in the hysteresis region decreases with the increase in P. The
decrease in the slope indicates that the transformation rate of the FM phase
declines with increasing P at a fixed temperature and field. To further
confirm this, we have measured the isothermal relaxation of M with time at a
fixed T in the hysteresis region, but at different P. We have recorded the
time dependence of M in the field cooled state, i.e. the sample has been
cooled to the probing temperature in presence of the field and after
stabilizing the temperature, magnetization has been recorded for next few
hours at the same field. The data has been shown in Fig. 6(a) for T = 65 K at
H = 4 Tesla. First of all, note that M increases with time, which confirms
that high-T AFM phase transforms into the FM phase during cooling across the
hysteresis region, which is expected. Now the important observation is, as P
increases the relaxation rate, i.e. how fast M changes with time, becomes
slower, which indicates that the growth rate of the FM phase fraction
decreases at higher pressure. This is consistent with the fact that the slope
of the M-T curve decreases with P. Similarly, in LCMO, the transition during
cooling proceeds through the transformation of the high-T FM phase into the
low-T AFM phase. It can be shown that the slope of the M-T curves in Fig. 2(b)
decreases with increasing P, which indicates towards the suppression of the
growth of the AFM phase with increase in the pressure. In Fig. 6(b), we have
also shown the time dependence of M after the sample has been cooled in the
field cooled protocol to T = 100 K in presence of H = 6 Tesla, which is within
the thermal hysteresis region. In this case also, the transformation rate
reduces at higher pressure, and hence conforms that P suppresses the
transformation rate from FM to AFM phase across the thermal hysteresis. Now,
note that these observations are consistent with the fact that pressure favors
the FM phase in both the systems, which we have concluded in the last section
(Fig. 4).

Now, we will probe the effect of pressure on the metastable behaviour across
the entire thermal hysteresis region in case of LCMO. In Fig. 6(c), we have
shown the time dependence of M (relaxation) at different temperatures within T
= 100 to 40 K at ambient pressure and H = 6 Tesla which is the thermal
hysteresis region. To measure these relaxation curves, we have cooled the
sample from room temperature (i.e. paramagnetic state) to the measurement
temperatures in presence of the H = 6 Tesla magnetic field. After the
temperature becomes stable, magnetization has been recorded for next few hours
without changing the field. First of all note that, M decreases with time at
all the temperatures because the high-T FM phase transforms into the AFM
phase. Now, the change of M with time i.e. the relaxation rate initially
increases as we reduce the temperature from 100 K. For example, the relaxation
at T = 80 K is faster than at 100 K. However, the relaxation again decreases
with further decrease in the temperature down to 40 K. The increase in the
relaxation rate with decreasing temperature can be understood in the framework
of kinetically arrested first order transition Chaddah2008 ; Pal2021 . As
temperature reduces, the free energy barrier between the high and low-T phases
decreases, which results into the initial increase in the transformation rate
Chaikin1994 ; Chaddah2008 . However, the suppression of the relaxation rate on
further reduction of temperature can be understood by the interplay between
kinetic arrest and the free energy landscape Chaddah2008 ; Pal2021 .

Figure 6: Variation of M with time (a) of PCMAO at T = 65 K (b) of LCMO at 40
K measured in the FC protocol. The measurement fields are, H = 4 and 6 Tesla
for PCMAO and LCMO respectively. Solid lines are fitted curve using eq. 2. D
is the rate constant. The legends are the P values in the unit of kbar (c) the
relaxation of M at various temperature across the thermal hysteresis at H = 6
Tesla and P = 0 in the FC protocol. The legends indicate the measurement
temperatures. The curves are fitted using eqn. 2 and temperature dependence of
D is shown in (d). It also presents the D(T) at higher P. The solid lines in
(d) are guide to eye.  Figure 7: Relaxation in the magnetic glass state in (a)
PCMAO measured at T = 25 K and H = 4 Tesla and (b) LCMO measured at T = 40 K
and H = 6 Tesla. Solid lines are the fitted curves using eqn. 2. The inset of
(a) shows the D values at different P at T = 25 and 30 K in PCMAO obtained
from fitting. The solid line in the inset is guide to eye. For LCMO, the
curves at P = 0 and 1.94 kbar are fitted using same equation.

The relaxation curves can be satisfactorily described by logarithmic
dependence of M on time, which is given by:

$M(t)=M(0)[1+Dlog(1+t/\tau)]$ (2)

Here, M(0) is the magnetization at time $t$ = 0, and $D$ is known as the rate
constant which defines the relaxation time Amir2009 ; Kustov2010 ; AB2011 ;
Sudip2017 ; Sudip2019 ; Sudip2022 . Such logarithmic function has been
successfully used earlier to investigate the metastable behavior of the
nonequilibrium state in spin glasses, structural glasses, magnetic glass
etc.,AB2011 ; Kustov2010 ; Sudip2017 ; Sudip2019 ; Sudip2022 . We have fitted
all the time dependence of M data in Figs. 6(a) and 6(b) using eqn. 2. We have
shown the fitted curves as the solid lines and mentioned the $D$ values in the
respective figures. As can be observed here, in both PCMAO and LCMO, $D$
decreases with increasing P which further confirm that higher pressure
inhibits the growth dynamics of the low temperature phases in both the systems
in the course of the FM-AFM first order transition. Similarly, we have fitted
the relaxation curves at various temperatures which are shown in Fig. 6(c) for
ambient P and plotted the obtained $D$ values in Fig. 6(d)(black filled
circles). Also, we have measured the relaxation of M at higher pressure, but
we have not shown here for conciseness. We have fitted those data by using
eqn. 2 and plotted only the $D$ values at a few selected higher P in Fig.
6(d). The variation of $D$ with temperature at different P is qualitatively
similar. But, quantitatively,it is evident that external P suppresses the
growth rate in the entire temperature range of thermal hysteresis.

###  III.4 Effect of pressure on the transformation kinetics in the magnetic
glass state:

In the magnetic glass state, the non-equilibrium high-T phase which persists
because of the KA gradually transforms into the equilibrium phase with time.
The rate of transformation depends on the H and T. In Figs. 7(a) and 7(b), we
have shown the change in M as a function of time at temperatures T = 25 and 40
K following field cooled (FC) protocol in PCMAO and LCMO respectively. To
record the M($t$) curves, the sample is initially cooled from T = 320 K to the
measurement temperatures at different P in presence of the cooling fields of H
= 4 and 6 Tesla for PCMAO and LCMO respectively. After temperature becomes
stable, M has been recorded as a function of time for the next few hours
without changing the temperature and field. The thermal hysteresis at H = 4
and 6 Tesla in PCMAO and LCMO close at T = 30 and 56 K respectively (data are
not shown here for conciseness). Therefore, the measurement temperatures lie
below the closer of the thermal hysteresis and situate in the magnetic glass
regime of the respective materials. The relaxation of M with time indicates
the non-equilibrium nature of the low-T state which can not be described in
the framework of the metastable supercooled high-T phase. In case of PCMAO, M
increases with time at ambient P which indicates that the AFM phase is
unstable at this temperature and it transforms to the equilibrium FM phase. As
P increases, the increase in M with time i.e., the relaxation rate
monotonically decreases. It implies that the transformation rate monotonically
reduces with the increase in P. On the other hand, in case of LCMO, the M
decreases with time. It means that here the FM phase is the unstable phase and
it gradually transforms into the equilibrium AFM phase. In this case, as P
increases, the relaxation rate strongly decreases and becomes negligible at
higher P. Therefore, P stabilizes the phase coexisting state in both the
systems because it suppresses the relaxation rate of the FM-AFM
transformation. However, at these field and temperatures, the suppression of
the relaxation rate by P is pronounced in LCMO, compared to PCMAO. The
relaxation curves can be fitted by using eqn. 2. We have mentioned the
obtained $D$ values for PCMAO in the main panel of Fig. 7(a). In the inset, we
have shown the the variation of $D$ with P at T = 30 and 25 K. At T = 30 K, P
suppresses the relaxation and $D$ decreases with increase in P. However, the
effect of P is more pronounced at T = 25 K, which indicates that P strongly
suppresses the transformation kinetics at lower temperatures. $D$ values for
LCMO at P = 0 and 1.94 kbar have been mentioned in the Fig 7(b). At higher P,
there is no appreciable relaxation of M.

##  IV Discussion:

In case of PCMAO, P monotonically increases the supercooling and superheating
temperatures which indicate that P favours the FM phase. However, P does not
significantly affect the [HK, TK] band. It implies that the temperature
window, where the first order transition occurs before it is arrested, expands
with increasing P. This should increase the equilibrium FM volume fraction at
low temperature, say T = 5 K. However, the transformation rate from the high-T
AFM to the low-T FM phase across the thermal hysteresis is simultaneously
suppressed by P [see Fig. 6(a)]. Therefore, two competing mechanism acting
together decides the phase fraction of the competing phases at any
temperature. The variation of volume fraction of the FM phase at T = 5 K
indicates that initially the second mechanism must be dominating over the
first, so that the FM phase fraction initially decreases at low P but finally
increases, when the first mechanism dominates. Note that, because of these two
competing interactions, the change in the volume fraction with pressure is
also small. On the other hand, in case of LCMO, $T^{*}$ reduces with
increasing P and the $T_{K}$ band is unchanged. Hence, the temperature window
where the first order transition from the high-T FM to low-T AFM phase occurs
reduces with the increase in P. In addition, the nucleation and growth of the
AFM phase is also suppressed by P. Therefore, both of these mechanisms
together should result into the increase in the FM phase fraction at T = 5 K,
which is observed in Fig. 2(b). Note that, in this case the change in the
volume fraction is very large, which further support this scenario.

We would like to mention here that in PCMAO, the suppression of the
transformation rate from the high-T AFM to low-T FM phase across the thermal
hysteresis [see Fig. 6(a)] by P is nontrivial. Because, as P increases the
transition temperatures, the size of the critical nucleus of the FM phase at a
particular temperature should be smaller at higher P Chaikin1994 ; Chaddah .
This should, in principle, increase the growth rate of the FM phase. We
believe that this contradiction can be understood from the extremely different
spin structure of the FM and AFM phases in PCMAO. In the course of the
transition, the nucleus of a critical size (RC) grows in the matrix of CE-AFM
phase. The interface, where the two extremely different kind of spin structure
co-exist, is expected to have considerable spin disorder that may cause
hindrance to growth. Now, with increase in P, the RC decreases, the number of
the FM nucleus as well as the interface will increase, which may suppress the
growth rate at higher P. Note that, in LCMO, where AFM is also CE type and
grows within the FM matrix, the transformation rate decreases with P. The same
thing has been found at low temperatures i.e., in the magnetic glass state
[see Figs. 7(a) and 7(b)]. In PCMAO, the transformation from the AFM to FM
phase at T = 30 K is suppressed by P. Whereas, in LCMO the transformation from
FM to AFM is hindered by P at T = 65 K. This reveals that higher P suppresses
the transformation kinetics of the phase coexisting state in both systems.
Basically, the growth of one phase in the matrix of other phase appears to be
suppressed at the interface of FM and CE-AFM. This conjecture needs to be
verified by some other microscopic techniques.

Figure 8: (a) Isothermal M-H loop of PCMAO at T = 2 K. The sample has been
cooled at H = 2 Tesla, and the hysteresis loop has been measured between $\pm$
4 Tesla (b) Isothermal M-H loop of LCMO at T = 2 K. The sample has been cooled
at H = 1 Tesla, and the hysteresis loop has been measured between $\pm$ 3
Tesla. In the insets, we have shown the close view of the loop near origin.

Nonetheless, the importance of the AFM-FM interface is evident from the
observation of significant horizontal shift of the field cooled M-H loop in
the phase coexistence region of both PCMAO and LCMO. Such shift in the M-H
loop is popularly known as the exchange bias, and occurs due to pinning of the
surface spins at the interface EB1999 . In Fig. 8(a), we have shown isothermal
variation of magnetization with field (M-H curves) of PCMAO recorded after the
samples has been cooled down to T = 2 K in presence of H = 2 Tesla, and the
field cycle has been performed between $\pm$4 Tesla. Similarly, Fig. 8(b)
presents the field cooled M-H curve of LCMO at T = 2 K where the sample has
been cooled at H = 1 Tesla, and the loop has been recorded between $\pm$3
Tesla. In the inset of the figures, we have shown the close view of the
hysteresis loop near origin, which shows that the loop is not symmetric with
respect to origin along the field axis, rather it is shifted towards the
negative field axis. The exchange bias defined as $H_{EB}$ =
$|(H_{C1}+H_{C2})|$/2, where $H_{C1}$ and $H_{C2}$ are the coercive fields
along the positive and negative field axes respectively. In case of PCMAO and
LCMO, the $H_{EB}$ has been found to be around $H_{EB}$ $\sim$ 350 and 245 Oe
respectively. Such shift in the hysteresis loop arises due to coexistence of
FM and AFM volume fractions in close proximity with each other.

##  V Summary:

In brief, we have studied the effect of simultaneously applied hydrostatic
pressure and magnetic field on the phase coexistence in two well known
perovskite manganites Pr0.5Ca0.5Mn0.975Al0.025O3 and La0.5Ca0.5MnO3. These two
samples undergo first order phase transition between AFM and FM phases. In
addition to the phase coexistence across the thermal hysteresis associated
with conventional first order transition, the FM and AFM phases coexist even
below the thermal hysteresis due to the phenomenon of kinetic arrest. During
cooling, PCMAO undergoes transition from high-T AFM to low-T FM phase.
External P increases the supercooling and superheating temperatures, however,
the kinetic arrest band is not significantly affected. The nucleation and
growth rate of the FM phase from the AFM across the transition during cooling
reduces at higher pressure. All these effects in combination results into a
non-monotonic variation of the volume fraction of FM-metal and AFM-insulator
phases at low temperature. At low temperature, i.e. below the thermal
hysteresis, the AFM phase gradually transforms into the FM phase, and the
transformation rate is suppressed at higher pressure. On the hand, LCMO has
opposite phase diagram and it undergoes a transition from high-T FM phase to
low-T AFM phase while cooling. Pressure lowers the supercooling and
superheating limits and increases the FM to AFM transformation rate across the
transition, i.e. favors the AFM phase. With increase in P, the FM volume
fraction at T = 5 K rises monotonically. In addition, the FM phase gradually
transforms to the AFM phase and the transformation rate from FM to AFM slows
down with increase in the pressure. The coexistence of different lattice and
spin structure at the interface may be a possible reason behind such similar
effect on two contrasting systems. Observation of exchange bias in both PCMAO
and LCMO under field condition indicates the coupling of the FM and AFM phase
at the interface.

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