# Motion and Region Aware Adversarial Learning for Fall Detection with Thermal
Imaging

Vineet Mehta⋆ Abhinav Dhall†⋆ Sujata Pal⋆ Shehroz S. Khan+ ⋆Indian Institute
of Technology Ropar, India †Monash University, Australia +KITE – University
Health Network, Canada

###### Abstract

Automatic fall detection is a vital technology for ensuring the health and
safety of people. Home-based camera systems for fall detection often put
people’s privacy at risk. Thermal cameras can partially or fully obfuscate
facial features, thus preserving the privacy of a person. Another challenge is
the less occurrence of falls in comparison to the normal activities of daily
living. As fall occurs rarely, it is non-trivial to learn algorithms due to
class imbalance. To handle these problems, we formulate fall detection as an
anomaly detection within an adversarial framework using thermal imaging. We
present a novel adversarial network that comprises of two-channel 3D
convolutional autoencoders which reconstructs the thermal data and the optical
flow input sequences respectively. We introduce a technique to track the
region of interest, a region-based difference constraint, and a joint
discriminator to compute the reconstruction error. A larger reconstruction
error indicates the occurrence of a fall. The experiments on a publicly
available thermal fall dataset show the superior results obtained compared to
the standard baseline.

###### Index Terms:

Fall detection, adversarial learning, thermal

##  I Introduction

Automatic detection of falls is important due to the possibility of severe
injury, high cost to the health system, and the psychological effect of a
fall. However, due to their rarity of occurrence, traditional supervised
machine learning classifiers are ill-posed for this problem [1]. There are
also challenges in collecting realistic fall data as it can put people’s lives
in danger [1]. Therefore, in many realistic settings, there may be few, or no
falls data available during training. Due to these skewed data situations, we
adopted fall detection as an anomaly detection problem [2]. In this setting, a
classifier is trained on only normal activities; during testing, both normal
and fall samples are presented to the classifier.

Another challenge in video-based fall detection is preserving the person’s
privacy, which traditional RGB cameras cannot provide [3]. Thus, detecting
falls in videos without explicitly knowing a person’s identity is important
for the usability of such systems in the real world. Thermal Imaging can
partially or fully obfuscate a person’s identity and has been used in other
fall detection applications [3, 4, 2].

Most recent works have focused on reconstruction based networks for fall
detection using autoencoders [2] and adversarial learning [5]. The adversarial
learning framework presents a unique opportunity to train a network to not
only mimic the normal activities through a generator but also helps to
discriminate it from the abnormal events through discriminator ([6, 5]). For
video-based anomaly detection, normally, the generator is some variant of
autoencoder, and the discriminator is a feed-forward neural network, both of
which are trained in an adversarial manner. The most successful previous works
have mostly focused on learning spatio-temporal features by using 3D
Convolutional Autoencoders (3DCAE) and 3D Convolution Neural Network (3DCNN)
[5].

The performance of video-based fall detection may be marred by differences in
background. This may become more prominent in thermal cameras where the
intensities may change due to differences in heat (e.g., when a person enters
the scene). Therefore, it is important to focus on the region around the
person. The relative motion of the person and objects around it can also
provide useful information to detect falls. Region and motion-based methods
([7],[8]) have shown superior performance in action recognition tasks.
Therefore, we hypothesize that the learning spatio-temporal features utilizing
region and motion awareness in video sequences would improve the detection of
falls when trained in an adversarial manner. To this end, we propose a motion
and region aware adversarial framework which consists of two separate channels
optimized jointly. The first channel is input with thermal video sequence
(with the extracted region of interest) and the second channel is input with
corresponding optical flow. The outputs from both channels are combined to
give a discriminative score for adversarial learning. We assume that joint
training of thermal and optical flow channels can facilitate in the learning
of both motion and region-based discriminatory features.

##  II Related Work

There is scarce literature on detecting falls in videos in an adversarial
manner with thermal cameras. We will now review studies that closely match our
work.

Fall Detection:  With progress in economical camera sensors, there are several
works [9, 10, 11], which use RGB cameras for data capturing. One major
limitation of RGB sensors is the lack of privacy as the identity of the
subject is not preserved. To overcome this limitation, Vadivelu et al. [3]
were one of the first fall detection works on thermal data. Further, Nogas et
al. [4] proposed using the thermal cameras and recurrent Convolutional
AutoEncoder (CAE) for fall detection. Motivated by these works, we also use
the thermal camera modality data in our experiments. The readers are also
pointed to recent surveys [1, 12] in fall detection for more insights into
different techniques proposed in the literature. Most of the recent works on
fall detection using thermal and depth camera formulate it as anomaly
detection.

Anomaly Detection: Given the rare nature of fall events, we follow the works
in abnormal event detection that is conceptually similar to our work. Many of
the recent anomaly detection methods ([13], [14]) are based on one class
classification paradigm in which the distribution of normal events are learned
using autoencoders and the deviations from the learned distribution is
detected as anomaly during the test time. Hasan et al. [13] learnt the normal
motion patterns in videos using hand-crafted features and CAE. Ravanbakhsh et
al. [14] proposed a video to flow and vice-versa generation adversarial
approach for abnormal event detection. Khan et al. [15] proposed the use of
3DCAE for abnormal event detection applied to fall detection. Sabokrou et al.
[16] proposed an end-to-end adversarial network that consists of a generator
that reconstructs the input with added noise and the discriminator to
discriminate the reconstructed output from the actual input. Further, Khan et
al. [5] extend the work of Sabokrou et al. [16] from single image to a
sequence of images for fall detection using a spatio-temporal adversarial
learning framework.

Fall being a spatio-temporal change in a subject’s pose, a limitation of the
work is that the motion information is not explicitly added into the network.
We build upon the adversarial learning work of Khan et al. [5] and propose a
two channel network, with one channel explicitly learning the motion in the
form of the optical flow while the other takes raw video frames as input. Our
proposed approach can handle the situations where a person may not be present
in a frame, which may reduce false positive rate.

##  III Methods

Our proposed adversarial framework consists of two channels. The input to the
first channel is a window of thermal frames and the second channel is a window
of optical flow frames. Each channel consists of (i) a 3DCAE to reconstruct
the input window and (ii) a 3DCNN to discriminate them from the original
window of frames where both of the channels are joined by a single neuron as a
joint discriminator (Fig. 1). We trained this framework using only Activities
of Daily Living (ADL) from thermal frames. We perform person tracking and
extract the Region of Interest (ROI) from the thermal and the optical flow
frames for motion and region-based reconstruction.

###  III-A Advesarial Framework

####  III-A1 3DCAE-3DCNN

Our architecture of 3DCAE is similar to Khan et al. [5]. We extend their
network by adding a channel that takes optical flow as input (see Table I). We
use 3D filters of $3\times 3$ with a temporal depth of $5$ in all layers of
3DCAE as same as Khan et al. [5]. The operations are the same in the flow
3DCAE except for the second deconvolution layer, which uses filters of
$2\times 2$ with a temporal depth of $4$, to reconstruct the temporal depth of
odd length.

The architecture of the 3DCNN is the same as the encoder in 3DCAE, followed by
a neuron at the end with a sigmoid function to output a probability of whether
a sequence of frames is original or reconstructed. Batch normalization is used
in all the layers of the 3D discriminator except for the input layer.
LeakyRelu activation is set in all hidden layers, with a negative slope
coefficient set to 0.2 ([5, 17]).

TABLE I: Configuration of the 3DCAE.

| Thermal 3DCAE | Flow 3DCAE
---|---|---
Input | (8, 64, 64, 1) | (7, 64, 64, 1)
Encoder | 3D Conv - (8, 64, 64, 16) | 3D Conv - (7, 64, 64, 16)
3D Conv- (8, 32, 32, 8) | 3D Conv- (7, 32, 32, 8)
3D Conv - (4, 16, 16, 8) | 3D Conv - (4, 16, 16, 8)
3D Conv - (2, 8, 8, 8) | 3D Conv - (2, 8, 8, 8)
Decoder | 3D Deconv - (4, 16, 16, 8) | 3D Deconv - (4, 16, 16, 8)
3D Deconv - (8, 32, 32, 8) | 3D Deconv - (7, 32, 32, 8)
3D Deconv - (8, 64, 64, 16) | 3D Deconv - (7, 64, 64, 16)
3D Deconv - (8, 64, 64, 1) | 3D Deconv - (7, 64, 64, 1)

Figure 1: The proposed adversarial network: top channel takes window of
thermal frames and bottom channel takes window of the optical flow frames as
input.

####  III-A2 Adversarial Learning

In this section, we first explain the general adversarial training as
described in the work of Khan et al. [5]. This model consists of a 3DCAE
(represented as R), takes the input sequence I of window size T and
reconstructs the sequence where the output sequence is named as O, which is
then fed to fool 3DCNN (represented as D). R and D are trained using standard
GAN loss described as:

$L_{R+D}=E_{I\sim p}[\lg{D(I)}]+E_{O\sim p}[\lg{(1-D(O))}]$ (1)

We combine the adversarial loss with the Mean Squared Error (MSE) loss, which
is used only for R and defined as

$L_{R}=E[(I-O)^{2}]$ (2)

The total loss function to minimize R is defined as:

$L=L_{R+D}+\lambda L_{R}$ (3)

where $\lambda$ is a positive hyperparameter for weighted loss. The notations
used for thermal and flow networks are (IT,OT,RT,DT) and (IF,OF,RF,DF),
respectively.

###  III-B ROI Extraction

The performance of video-based fall detection methods may get impacted by
background artifacts. This situation can get worse with a thermal camera
because changes in the heat can alter the background and pixel intensity of
frames in a video sequence [2]. Therefore, we reconstruct only the region
where the person is present, which is not affected much by the change in
background objects and intensity. We perform person tracking using an object
detector and image processing techniques to localize the person in an image.

####  III-B1 Person Detection

To the best of our knowledge, there are no pre-trained publicly available deep
learning-based models for person detection, specifically for thermal images.
To this end, we used the Region-based Fully Convolutional Network (R-FCN) [18]
trained on COCO dataset [19]. As the TSF dataset contains only one subject in
a frame, the bounding box with the highest confidence score is selected. There
are no false proposals by the detector; however, the localized bounding box is
found to fluctuate in size and position, which degrades the prediction by the
tracking method.

(a)

(b)

(c)

(d)

Figure 2: Contour box localization process (Section III-B2).

####  III-B2 Contour Box Localization

In thermal images, a person may appear brighter than the background due to
differences in heat emitted by as a person and object. Therefore, Otsu
thresholding[20] is applied to the thermal image to separate dark background,
as shown in Fig. 2. The thresholded image may still contain bright background
objects. We find the contours [21] on the thresholded image after applying
morphological operations and select the biggest contour on the basis of the
inside area. The smallest box containing that contour blob is chosen as a
candidate for the person bounding box.

####  III-B3 Tracking

We apply Kalman filtering on the top-left and right-bottom coordinates of the
bounding box with the constant velocity assumption. The tracker is initialized
with the person detector and predicts the bounding box for the next frame. We
compare the predicted box with the person detection bounding box (if detected
in the next frame) to check if the tracker drifts. We use a counter (age) to
track the number of continuous tracker predictions without detection. In the
case of no detection, the tracker’s age is increased, and when the age of the
tracker exceeds a limit of 20, the tracker is stopped. The Intersection of
Union (IoU) is used in many tracking methods to match the bounding boxes.
However, IoU is small when the size of one box is large compared to the other
box, which could be possible due to the bad localization by the detector.
Therefore, we also used other criteria such as the ratio of area and checking
for a subset. At a particular instant, there could be at most three possible
candidates for the person localization – Detect, Contour, and Track box.
Detect box confirms the presence of the person, but it does not fit the person
in most cases. Therefore, the contour box and track box are used to improve
the overall tracking. Algorithm 1 describes the whole tracking method.

Input : Frame

Output : FinalBox

FinalBox=None;

DetectBox=Detector.GetLocalization(Frame);

if  _DetectBox is not None_ then

ContourBox=GetBiggestContourBox(Frame);

if  _Tracker is not None_ then

TrackBox=Tracker.GetCurrentBox();

if  _TrackBox matches with DetectBox_ then

DetectBox=BoxSelection(DetectBox,TrackBox);

else

Tracker=None;

end if

end if

if  _ContourBox matches with DetectBox_ then

DetectBox=BoxSelection(DetectBox,ContourBox);

end if

FinalBox=DetectBox;

if  _Tracker is not None_ then

Tracker.KalmanFilter(DetectBox);

Tracker.Losses=0;

else

Tracker=InitializeTracker(DetectBox);

end if

else

if  _Tracker is not None_ then

TrackBox=Tracker.Predict();

ContourBox=GetBiggestContourBox(Frame);

if  _ContourBox matches with TrackBox_ then

Tracker.Losses+=0.5;

FinalBox=ContourBox;

Tracker.KalmanFilter(ContourBox);

else

Tracker.CurrentBox=TrackBox;

FinalBox=TrackBox;

Tracker.Losses+=1;

end if

end if

end if

if  _Tracker is not None and Tracker.Losses greater than MaxAge_ then

Tracker=None;

FinalBox=None;

end if

Return FinalBox;

Algorithm 1 Tracking Algorithm

####  III-B4 Region based reconstruction

We remove the frames in which a person is not localized post tracking, and
rest of the frames are masked by their corresponding bounding box (named as
ROI mask). For region-based reconstruction, the 3DCAE is fed with the window
of masked frames and a region-based reconstruction loss LROI (instead of LR)
is used:

$L_{ROI}=E_{ROI}[(ROI(I)-ROI(O))^{2}]$ (4)

where ROI(X) represents the masking of frames in window X with the
corresponding ROI masks, and the expectation is taken over pixels inside the
ROI.

###  III-C Motion Constraint and Reconstruction

Beside the appearance based constraint on R, we incorporate motion in the fall
detection system in two ways:

####  III-C1 Difference Constraint

Mathieu et al. [22] compute the difference and gradient-based losses for
future frame prediction, which increase the sharpness of the predicted frame.
We adapted a similar technique to add an additional loss term in the thermal
3DCAE, which is based on the MSE of the difference frames of I and O. A
difference frame is a residual map computed by subtracting two consecutive
frames. We mask the difference frames by their respective ROI, which is the
union of the ROIs of the two frames used to compute the difference frame.
Further, the difference loss is defined as:

$L_{Diff}=E_{ROI}[(ROI(DF(I))-ROI(DF(O)))^{2}]$ (5)

where DF(X) represents the difference frames for the window X. Therefore, the
final loss for $R$ in (3) with LROI and LDiff is defined as:

$L=L_{R+D}+\lambda_{S}L_{ROI}+\lambda_{D}L_{Diff}$ (6)

where $\lambda_{S}$ and $\lambda_{D}$ are positive constants for weighted
loss.

####  III-C2 Optical Flow Reconstruction

Liu et al. [23] use constraint on the optical flow of the ground truth and
reconstructed images for future frame prediction. They used CNN based model
for flow estimation, which can facilitate backpropagation for optical flow
loss, but these models are trained on RGB images and may generate noise due to
temperature changes in a thermal video. Therefore, we train a spatio-temporal
adversarial network (RF, DF) for flow reconstruction which takes into input a
window of optical flow frames. We compute the dense optical flow frames for
two consecutive frames [24]. We stack the flow in the x, y direction, and the
magnitude to form a 3-dimensional image (similar to [25]).

The flow images are masked with their ROI to remove noise due to temperature
variations. As defined earlier for the difference frame, the ROI for flow
image is the union of ROI of the two thermal frames used to compute the
optical flow.

###  III-D Thermal and Optical Flow Fusion

We propose a joint adversarial network for thermal and optical flow
reconstruction, which consists of two 3DCAE and a single joint discriminator.
In the joint discriminator network DTF, the individual 3DCNN networks - DT and
DF are used where the two individual sigmoid neurons are replaced by a single
sigmoid neuron (see Figure 1). For region based reconstruction, the joint loss
function is defined as:

$L=L_{R+D}+\lambda_{T}L_{T\\_ROI}+\lambda_{F}L_{F\\_ROI}$ (7)

where LT_ROI and LF_ROI are region-based reconstruction loss for thermal and
optical flow, respectively. The hyperparameters $\lambda_{T}$ and
$\lambda_{F}$ are their corresponding positive constants for weighted loss.
The joint loss function with region based reconstruction and difference
constraint can be written as:-

$L=L_{R+D}+\lambda_{T\\_S}L_{T\\_ROI}+\lambda_{T\\_D}L_{T\\_Diff}+\lambda_{F}L_{F\\_ROI}$
(8)

where LT_Diff (Eq. (5)) and LT_ROI (Eq. (4)) are difference constraint and
region based reconstruction loss.

##  IV Detecting Unseen Falls

The strategy to detect unseen falls is shown in Figure 3 (derived from [2]).
All the frames in the video, Fri, are broken down into windows of frames of
length, T = $8$, using the sliding window method with stride=$1$. For the
$i^{th}$ window $I_{i}$, the 3DCAE gives the output as a reconstruction of
this window, $O_{i}$. The reconstruction error (Ri,j) between the $j^{th}$
frame of $I_{i}$ and $O_{i}$ can be calculated as (similar to Eq. (2))

$R_{i,j}=E[(I_{i,j}-O_{i,j})^{2}]$ (9)

For the region based reconstruction models, it can be defined (similar to Eq.
(4)) as follows:

$R_{i,j}=E_{ROI}[(ROI(I_{i,j})-ROI(O_{i,j}))^{2}]$ (10)

There are two ways to detect unseen falls, at the frame level, or at the
window level, which are described below.

Figure 3: Temporal sliding window method- showing error (Ri,j) per frame (Frj)
with T = 8 (Figure Source: [5]).

###  IV-A Frame Level Anomaly Score

In the frame level anomaly detection, the reconstruction error (Ri,j)
(obtained from the 3DCAE) is computed for every $j^{th}$ frame across
different windows. The average (Cj$\mu$) and standard deviation (Cj$\sigma$)
of the Ri,j across different windows are used as an anomaly score for the
$j^{th}$ frame as follows [2]:

$\displaystyle
C_{\mu}^{j}=\begin{cases}\frac{1}{j}\sum_{i=1}^{j}R_{i,j}&j<T\\\
\frac{1}{T}\sum_{i=j-T+1}^{j}R_{i,j}&j\geq T\end{cases}$ (11) $\displaystyle
C_{\sigma}^{j}=\begin{cases}\sqrt{\frac{1}{j}\sum_{i=1}^{j}(R_{i,j}-C_{\mu}^{j})^{2}}&j<T\\\
\sqrt{\frac{1}{T}\sum_{i=j-T+1}^{j}(R_{i,j}-C_{\mu}^{j})^{2}}&j\geq
T\end{cases}$

###  IV-B Window Level Anomaly Score

For window level anomaly detection, the anomaly score is calculated for the
entire window. For a particular window Wi of length T, the mean of
reconstruction error of all the T frames (Wi$\mu$) and their standard
deviation (Wi$\mu$) are used as an anomaly score [5]:

$\displaystyle W_{\mu}^{i}=\frac{1}{T}\sum_{j=i}^{T+i-1}R_{i,j}$ (12)
$\displaystyle
W_{\sigma}^{i}=\sqrt{\frac{1}{T}\sum_{j=i}^{T+i-1}(R_{i,j}-W_{\mu}^{i})^{2}}$

While using difference constraint in thermal reconstruction models, we also
compute another window level anomaly score using the reconstruction error of
difference frames (Eq. (5)) by taking the mean and standard deviation of
reconstruction error over T-$1$ frames (similar to Eq. (12) with window length
T-$1$). Similarly, we compute the window level anomaly scores for the optical
flow reconstruction models using the reconstruction error of the optical flow
frames.

For region-based reconstruction models with difference constraint, we name the
anomaly scores computed using thermal frames as ROI-score and anomaly scores
computed using difference frames as Diff-score. For fusion models, we use the
terms Thermal ROI-score, Flow ROI-score and Thermal Diff-score for comparisons
of the different anomaly scores calculated for the same model. We define
tolerance ($\alpha$) as the number of fall frames required in a window to set
the ground-truth label of the entire window as a fall. We varied $\alpha$ from
$1$ to T to understand its impact on the results.

##  V Experiments

Dataset: We use the publicly available TSF dataset [3] containing 44 thermal
videos of resolution 640x480. There are 9 videos with normal ADL and 35 videos
containing falls and other normal activities. The ADL frames include different
scenarios such as an empty room, a person entering a room, sitting on a chair,
or lying in bed, whereas the fall frames include a person falling from the
chair, bed, or falling while walking. Some of the ADL and fall frames are
shown in Fig. 4. ADL videos contain a total of 22,116 frames.

(a)

(b)

(c)

(d)

Figure 4: TSF dataset [3] samples: top-left shows empty room frame; bottom-
left shows the fall while during walking and the right column shows ADL
frames-walking and lying on bed respectively.

(a) 3DCAE input

(b) 3DCAE output

(c) ROI-3DCAE input

(d) ROI-3DCAE output

(e) ROI-3DCAE masked output

(f) 3DCAE flow input

(g) 3DCAE flow output

(h) ROI-3DCAE flow input

(i) ROI-3DCAE flow output

(j) ROI-3DCAE masked flow output

Figure 5: Qualitative analysis- The middle frame of the input and output
window reconstructed by different models: the top row images [(a)-(e)] and
bottom row images [(f)-(j)] are the inputs and corresponding outputs of the
thermal data channel and flow magnitude channels, respectively.

Data Processing: We perform person tracking on thermal videos; when a person
is not detected continuously for $10$ frames, we break the video and create
sub videos, resulting in $22$ sub videos, which further contain 12,454 frames
from ADL videos for training. For ROI computation, the detection threshold is
empirically set to $0.3$. We perform the sliding window method on sub videos
with stride=1 to create windows of length T=8, which creates a total of 12,300
windows from ADL sub videos. We normalize frame in the range [-1, 1] [17]
followed by resizing to 64x64.

ROI processing:  For the input to region-based networks, we perform
normalization on the pixels inside the ROI and mask it by ROI mask. We set the
pixels outside the ROI to $-1$. Finally, an ROI masked frame contains pixel
value in the range $[-1,1]$ inside the ROI and $-1$ outside the ROI.

Optical flow image pre-processing: Each channel of a flow image is re-scaled
in the range of $[-1,1]$ by min-max normalization. For region-based
reconstruction in flow images, we perform ROI masking as described earlier for
the normalized thermal image, where the ROI of the optical flow frame is the
union of the corresponding consecutive frames.

###  V-A Network Implementations

We train two adversarial models, one each for for thermal window
(Thermal-3DCAE) and optical flow window (Flow-3DCAE) reconstruction. For
region-based reconstruction, we train two adversarial models on thermal data,
one with ROI masking and ROI loss (Eq. (4)) described earlier and the other
one with the addition of difference constraint in the region based
reconstruction (named as Thermal-ROI-3DCAE and Thermal-Diff-ROI-3DCAE) We
train one adversarial model for optical flow with region-based reconstruction,
named as Flow-ROI-3DCAE. For the fusion networks, we train two models: one
without the difference constraint (Fusion-ROI-3DCAE) and one with it (Fusion-
Diff-ROI-3DCAE).

We use the SGD optimizer with the learning rate as $0.0002$ for the 3DCNN
discriminator and adadelta optimizer for the 3DCAE in all the adversarial
models. All the models are trained for 300 epochs. The hyperparameters
($\lambda$’s) used for the weighted loss (Eq-(3), (6) and (7)) are varied
between three values [0.1, 1 and 10]. We found that the large value of these
hyperparameters led to mode collapse. The best hyperparameter setting for
Thermal-Diff-ROI-3DCAE and Fusion-Diff-ROI-3DCAE have all the constants equal
to 1, whereas the hyperparameter setting for the rest of the models has all
the constants equal to 0.1. The full code of our implementation is available
at https://github.com/ivineetm007/Fall-detection.

Figure 6: Frame level anomaly score for a fall video. Figure 7: Plots of AUC
values of ROC and PR curve computed using window level anomaly scores with the
variation in tolerance- W$\mu$(Left) and W$\sigma$(Right).

###  V-B Evaluation Metrics

For assessing the performance of detecting falls as anomaly, the Area Under
Curve (AUC) of Receiver Operating Characteristics (ROC) and Precision-Recall
(PR) curve is used. The latter is used to specifically focus on the detection
of minority ‘fall’ class. We compute the anomaly scores at the frame and
window level, which consists of two types - mean and standard deviation
anomaly scores. We calculate and compare the AUC of ROC and PR curve using all
these scores.

###  V-C Ablation Studies

####  V-C1 3DCAE

The thermal input and the reconstructed output by Thermal-3DCAE can be seen in
Fig. 5. The basic techniques to utilize the ROI in the deep learning models
are resizing and ROI pooling. We also train two different Thermal-3DCAE models
by changing the thermal input (1) resizing input ROI to 64x64, and (2) ROI
pooling to 64x64 dimension. We observe that these techniques increase the
false-positives, and the results for them are not reported. We argue that the
resizing the ROI leads to geometric distortions and introduces false motion on
the borders even if the subject is not moving.

Optical Flow: As described earlier, optical flow images contain noise due to
temperature variation due to which the reconstruction quality is also noisy
(Fig. 5(g)).

####  V-C2 ROI-3DCAE

The output of Thermal-ROI-3DCAE and Flow-ROI-3DCAE is shown in Fig. 5. We
observe that the region-based method improves the reconstruction quality in
the ROI region as the model learns to reconstruct only the ROI (see Fig. 5
(c), (d), and (e)). Similar behaviour is observed for Flow-ROI-3DCAE (see Fig.
5 (h), (i) and (j)).

####  V-C3 Difference constraint

To understand the impact of difference constraint, we compare the results
computed at window level (see Fig. 7) for the AUC of ROC and PR. On the
comparison between the Diff score and the ROI score of the Thermal-Diff-
ROI-3DCAE, we found that the results using the Diff score are better than the
ROI score in all the plots, which suggests that the use of Diff score is more
suitable for window level analysis. We also compare the results of Thermal-
ROI-3DCAE and Diff score of Thermal-Diff-ROI-3DCAE, we found that the addition
of constraint increase the AUC for both ROC and PR curve. Similar behaviour
was observed on the comparison of Fusion-ROI-3DCAE and Fusion-Diff-ROI-3DCAE.
This suggests that the difference constraint makes the model more
discriminative in temporal direction and increases the overall performance.

TABLE II:  AUC of ROC and PR based frame level anomaly comparison.

Method | ROC | PR
---|---|---
| C$\mu$ | C$\sigma$ | C$\mu$ | C$\sigma$
Thermal-3DCAE | 0.88 | 0.90 | 0.47 | 0.48
Thermal-ROI-3DCAE | 0.89 | 0.92 | 0.55 | 0.57
Thermal-Diff-ROI-3DCAE (ROI score) | 0.90 | 0.92 | 0.57 | 0.56
Fusion-ROI-3DCAE (Thermal ROI score) | 0.90 | 0.93 | 0.56 | 0.58
Fusion-Diff-ROI-3DCAE (Thermal ROI score) | 0.90 | 0.93 | 0.57 | 0.57

####  V-C4 Fusion models

To understand the effect of fusion, we first compare the results of Fusion-
ROI-3DCAE with the ROI based models at frame level as shown in Table II; we
found that there are minor improvements in frame-level results. We compare the
window level results ( Fig. 7) of Thermal-ROI-3DCAE and Flow-ROI-3DCAE with
Thermal-ROI score and Flow-ROI score of Fusion-ROI-3DCAE respectively, we
observe that there is a small increment in the results by ROI score of thermal
and a substantial increment in the results by ROI score of flow, which
indicates that joint learning improved the learning of flow reconstructor.

Qualitative Analysis\- We use the frame based anomaly scores to visualize the
performance of the proposed method (see Fig. 6). Although our model is able to
detect fall events, we observed a false peak when the person enters the room
and our tracking method misses some frames when person enters.

##  VI Results

The AUC values of ROC and PR curves computed using frame level anomaly scores
are shown in Table II. For window level results, we plot the AUC values with
the variation of tolerance ($\alpha$) from $1$ to $8$ (see Fig. 7). The frame
level and window level results can be summarized as:

  1. 1.

There is a improvement in the AUC of the ROC and PR curve by region-based
construction, which confirms the importance of region awareness.

  2. 2.

Addition of difference constraint increases the AUC values using window level
scores, which indicates its importance for the learning of spatio-temporal
autoencoders.

  3. 3.

Fusion models leads to an increase in the performance.

Comparison with the existing methods: In the previous works using DSTCAE-C3D
[2], Conv-LSTM AE [4] and 3DCAE-3DCNN [5], AUC values of ROC computed using
frame-level anomaly score are reported (see Table III ‘All frames’ column).
The previous methods do not perform person tracking in the video, due to which
the number of frames used for training and testing is different. Therefore, we
train and test these methods using the frames on which a person is localized
by our tracking method. The comparison of these methods with the proposed
model on tracked frames only is shown in Table III and summarized as:

  1. 1.

AUC values of previous methods decrease when only tracked frames are used.
Furthermore, the empty frames in train and test set videos are similar. Low
reconstruction error on these frames may give high AUC values in previous
methods (Table III-‘All frames’) during testing.

  2. 2.

The proposed methods achieves similar and better AUC of ROC than previous
methods and higher AUC of PR against all previous methods. The method focuses
on region in the frame where a person is present; therefore, it can facilitate
learning of background agnostic models.

TABLE III: Comparison with the previous methods based on AUC of ROC and PR
curve calculated on frame level anomaly scores.

Method | All frames | Tracked frames
---|---|---
| ROC | ROC | PR
| C$\mu$ | C$\sigma$ | C$\mu$ | C$\sigma$ | C$\mu$ | C$\sigma$
Conv-LSTM AE[4] | 0.76 | 0.83 | 0.63 | 0.73 | 0.26 | 0.37
DSTCAE-C3D[2] | 0.93 | 0.97 | 0.85 | 0.90 | 0.46 | 0.53
3DCAE-3DCNN[5] | 0.95 | 0.95 | 0.90 | 0.88 | 0.47 | 0.48
Fusion-Diff-ROI-3DCAE | — | — | 0.90 | 0.93 | 0.57 | 0.57

##  VII Conclusion and Future Work

Fall detection is a non-trivial problem due to large imbalance in data; thus,
we formulate it as an anomaly detection problem. In this problem, we trained
the model on only normal ADL and predicts whether the test sample is normal
ADL or a fall. Building upon the advantages of adversarial learning paradigm,
we present a two channel adversarial learning framework to learn spatio-
temporal features by extracting ROI and its generated optical flow followed by
a joint discriminator. We note that the introduction of person-ROI and
difference loss function increases the performance. The major improvement in
comparison to previous methods is the increase in AUC of PR curve. The optical
flow information is also useful in the network and the fused method performs
better than the raw thermal analysis only. In the future, we plan to extend
the proposed techniques to detect falls using multiple camera modalities,
including depth and IP cameras.

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# Reference frame-independent model of a collective excitation atom
interferometer

B.J. Mommers  Centre for Engineered Quantum Systems, The University of
Queensland, St Lucia, Australia<EMAIL_ADDRESS>School of Mathematics and
Physics, The University of Queensland, St Lucia, Australia  M.W.J. Bromley
School of Sciences, University of Southern Queensland, Toowoomba, Australia
School of Mathematics and Physics, The University of Queensland, St Lucia,
Australia

###### Abstract

We theoretically analyze the operating principles of a proposed matter-wave
Sagnac interferometer utilizing Bose-Einstein condensate (BEC) phonon modes as
an interference medium. Previous work found that the orbital angular momentum
phonon modes of a ring-trapped BEC are split in frequency by rotations,
leading to a measurable rotation signal. We develop an alternate description
in which an imbalance in the counter-propagating modes’ amplitudes
(populations) is induced by the rotation of the system during condensation.
This description gives analytic forms for the interferometic phase shift in 1D
and is readily generalized to include mean-field interactions. To validate our
findings, we simulate a ring-trapped BEC Sagnac interferometer in one
dimension and demonstrate that measurement of an unknown rotation rate can be
performed using a modified analysis. Our simulation data show strong agreement
with our analytic results, and we further employ simulations to explore and
clarify the role of superfluidity in this matter-wave Sagnac interferometer.

##  I Introduction

The Sagnac effect links the phase shift between waves counter-propagating
within an enclosed loop with the external rotation of such a system [1]. This
has been exploited to allow high-precision interferometric measurement of
rotations, useful in inertial sensing and navigation (for examples see Ref.
[2]). Current state-of-the-art Sagnac interferometers utilize counter-
propagating light, with a large enclosed area to boost sensitivity [3].
Matter-wave systems appear to have a sensitivity advantage when considering
the energy difference between optical and atomic systems [4]. Despite this,
matter-wave Sagnac interferometers are yet to surpass their optical
counterparts in terms of precision.

Recent proposals have made use of advances in Bose-Einstein condensate (BEC)
research and experimental techniques, providing an alternative medium to
atomic beam-based and guided matter-wave interferometry schemes [5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. In particular,
proposals to measure the Sagnac phase shift of an interference pattern
produced by counter-propagating orbital angular momentum (OAM) modes in a
trapped BEC offer a way to utilize the high levels of control and coherence
available in modern experimental systems [24, 25, 23]. This can be achieved
through imparting optical OAM onto the condensate [24] or by exciting standing
wave collective excitations [25] – the latter being the focus of this work.

The use of collective excitation modes is expected to ameliorate technical
difficulties often associated with BEC interferometric protocols that require
condensate splitting or spin-dependent transitions [26, 27, 7, 28, 29, 11, 13,
15]. A protocol for collective excitation Sagnac interferometry in a ring-
trapped BEC has been proposed and tested for rotation sensing [25]. This
protocol imprints a standing wave excitation on ring-trapped atoms through a
weak azimuthally-modulated optical potential whilst cooling through the BEC
transition. By imposing this potential during condensate formation then
releasing the BEC to freely evolve in the ring, rotation of the standing wave
pattern is observable in the rotating laboratory frame, as illustrated in Fig.
1.

Ref. [25] analyze the standing wave rotation in terms of a frequency splitting
between the standing wave’s constituent counter-propagating travelling wave
components in a three-mode model. However, the frequency splitting effect
requires observation in an inertial frame, something that cannot be achieved
when attempting to experimentally measure unknown rotations.

Figure 1: Marti et al. proposal schematic illustration, with 2D top-down view
(top) and 1D representation (bottom). (a) Begin with atoms confined to a ring
with constant density and impose an optical potential with azimuthally-
dependent modulations $V(\theta)$ while condensing to form a BEC with density
modulations. At $t=0$, remove the optical potential and allow the density
modulations to undergo standing wave oscillation, where at each half-cycle,
the density modulation pattern is inverted. (b) In the presence of rotation,
the modulation peaks shift position with each half-cycle.

This work is an alternate analysis of the theory in Ref. [25], in which we
develop a frame-independent theory of operation for this protocol. By
analyzing the dynamics in the context of mode amplitudes we reveal multiple
features of the interferometric design that were not previously apparent. The
splitting of counter-propagating modes in a rotating waveguide via the Sagnac
effect has been exploited in optical systems to produce non-reciprocal
propagation of light, analysed in terms of both mode amplitude- [30] and
frequency-splitting [31, 32]. For the purposes of matter-wave Sagnac
interferometry for precision rotation sensing, we show that a mode amplitude-
splitting analysis provides further insight into achieving high precision.

In Section II we describe the three-mode model and use it to derive the
relationship between the rotation rate and standing-wave mode amplitudes,
showing that rotation induces a splitting in the amplitudes of counter-
propagating modes. Section III details our numerical simulations of the
interferometer, which show that rotation measurements can be performed by
examining the evolution of the Fourier phase of the imprinted density modes.
In Section IV we show numerically that the imbalanced state is the ground
state of the ring system in the presence of both rotation and an imprinting
potential. In Section V we extend our analysis to low-temperature BECs
described by the Gross-Pitaevskii equation (GPE), and examine practical
considerations of our results.

##  II  Three-mode model

In this section we introduce the three-mode model and use it to derive the
Sagnac phase shift for the ring geometry in both inertial and rotating frames.
In the rotating frame we directly relate the mode imbalance to the rotation
rate.

Our initial analysis considers a simple model of the proposed phonon-mode
interferometer: a non-interacting system confined to a one-dimensional ring.
This ideal phonon-mode interferometer with wavefunction $\psi(\theta)$
populates only three angular momentum eigenstates,

$\displaystyle\psi(\theta)$
$\displaystyle=\phi_{0}+\phi_{+l}+\phi_{-l}\leavevmode\nobreak\ ,$ (1)
$\displaystyle\phi_{0}$ $\displaystyle=a_{0}\leavevmode\nobreak\ ,$ (2)
$\displaystyle\phi_{l}$ $\displaystyle=\left(a+\Delta\right)\
e^{il\theta}\leavevmode\nobreak\ ,$ (3) $\displaystyle\phi_{-l}$
$\displaystyle=\left(a-\Delta\right)\ e^{-il\theta}\leavevmode\nobreak\ .$ (4)

Here $\phi_{k}$ is an angular momentum eigenstate with quantized circulation
$k\in\\{0,l,-l\\}$, $a$ is the mean amplitude of the of the $k=\pm l$ modes,
$\Delta$ is the mode splitting, and $\theta$ is the positional parameter
(angle) around the 1D ring. The wavefunction is normalized such that
$\int\left|{\psi}\right|^{2}dr=1$. Therefore $\phi_{0}$ is the constant
background on which the angular momentum modes (i.e. phonon modes) with OAM
quantum number $\pm l$ are imprinted. Non-zero values of $\Delta$ result in an
imbalance between counter-propagating mode amplitudes.

###  II.1  In the inertial frame

To begin, we first note that Ref. [25] assumes an equal superposition of
counter-propagating modes, i.e. $\Delta=0$. Our analysis proceeds without this
assumption. For each mode (eqs. 2, 3 and 4), the time evolution operator
$U_{t}=e^{-i\hat{H}t/\hbar}$ is given by,

$\displaystyle U_{t}\phi_{0}$ $\displaystyle=a_{0}\
e^{-iE_{0}t/\hbar}\leavevmode\nobreak\ ,$ (5) $\displaystyle U_{t}\phi_{l}$
$\displaystyle=(a+\Delta)\ e^{i(l\theta-Et/\hbar)}\leavevmode\nobreak\ ,$ (6)
$\displaystyle U_{t}\phi_{-l}$ $\displaystyle=(a-\Delta)\
e^{-i(l\theta+Et/\hbar)}\leavevmode\nobreak\ .$ (7)

Where $E_{0}$ is the energy of the background mode, and
$E=E_{l}=l^{2}\hbar^{2}/2mr^{2}$ is the degenerate energy of the counter-
propagating modes, with $m$ representing particle mass and $r$ the ring’s
radius, which we both set to 1 without loss of generality.

The time-dependent density profile can be decomposed into the sum of a
standing wave and a travelling wave oscillating on a time-independent
background,

$\displaystyle\left|{\psi(t)}\right|^{2}=\leavevmode\nobreak\ $ $\displaystyle
a_{0}^{2}+(a+\Delta)^{2}+(a-\Delta)^{2}$
$\displaystyle+2(a+\Delta)(a-\Delta)\cos(2l\theta)$
$\displaystyle+4a_{0}\Delta\cos[l\theta-(E-E_{0})t/\hbar]$
$\displaystyle+2a_{0}(a-\Delta)\left(\cos[l\theta-(E-E_{0})t/\hbar]\right.$
$\displaystyle+\left.\cos[l\theta+(E-E_{0})t/\hbar]\right)$ (8)

Note when $\Delta=0$ the travelling wave component of Eqn. 8,
$4a_{0}\Delta\cos[l\theta-(E-E_{0})t/\hbar]$, is zero, consistent with a non-
rotating standing wave state. The rotation rate is determined through the
evolution of the complex phase shift in the single-frequency Fourier transform
of the density profile,

$\displaystyle\varphi(t)$
$\displaystyle=\arctan\left[{\int\left|{\psi(t)}\right|^{2}e^{-il\theta}d\theta}\right]\leavevmode\nobreak\
,$ (9)

for a single mode $l$.

From Eqns. 8 and 9, the phase shift in the inertial frame takes the form,

$\displaystyle\varphi(t)$
$\displaystyle=\arctan\left[{-\frac{\Delta}{a}\tan((E-E_{0})t/\hbar)}\right]\leavevmode\nobreak\
.$ (10)

In the limit $\frac{\Delta}{a}\rightarrow 1$, Eqn. 10 gives a constant phase
gradient - a linear phase accumulation consistent with a pure travelling wave.
Conversely, taking the limit $\Delta\rightarrow 0$ results in a constant zero
phase gradient - as expected for a pure standing wave.

Considering Eqn. 10 at stroboscopic measurement times,

$\tau_{s}=\frac{n\pi\hbar}{E-E_{0}},$ (11)

where $n$ is a natural number encoding the number of oscillations since $t=0$,
the phase shift is always zero. The rotation of the density profile relative
to the measurement frame (in this case the inertial frame) is given by the
time derivative of the phase shift at stroboscopic times – when the standing
wave component is at its maximum amplitude,

$\displaystyle\frac{d\varphi}{dt}$
$\displaystyle=-\frac{\Delta(E-E_{0})}{\hbar a}\leavevmode\nobreak\ .$ (12)

In the inertial frame this depends only on the mode imbalance and choice of
excited mode number. In a sensing application, the measurement frame is
expected to rotate with the system at an unknown rate. By analysing the phase
shift $\varphi(t)$ in the rotating frame, we can determine the relationship
between rotation rate $\Omega$ and mode imbalance $\Delta$.

Figure 2: Space-time density profiles and extracted phase for different
rotating reference frame configurations from simulations without interactions.
(a)–(d): space-time density profiles for the ideal 1D ring system. Each case
is labelled with the rotation rate of the system during imaginary time
evolution ($\Omega_{I}$) which corresponds to condensate formation, and the
real-time rotation rate ($\Omega_{R}$) that corresponds to the free evolution
of the system once the imprinting potential is removed. Dotted vertical lines
indicate full oscillations of the standing wave component from $t=0$, whilst
dotted horizontal lines indicate angular positions around the ring, to guide
the eye. (e)–(h): plots of the extracted phase for each of the simulation
conditions matching the density profile plot immediately above. The purple
curve is from simulations using the ground state found via imaginary time,
while the green points are from simulations that numerically optimize Eqn. 22.
The numerics in Ref. [25] correspond to subfigures (b) and (f), whilst a
physical experiment necessarily corresponds to (d) and (h).

###  II.2  In the rotating frame

In the rotating frame, we can perform a similar analysis as above with the
inclusion of a time-dependent coordinate transform,

$\displaystyle\theta$ $\displaystyle\rightarrow\theta^{\prime}+\Omega
t\leavevmode\nobreak\ .$ (13)

Repeating the calculation of the time-dependent density, the Fourier transform
result for a rotating system is given by,

$\displaystyle\int\left|{\psi(\theta^{\prime},t)}\right|^{2}e^{-il\theta^{\prime}}d\theta^{\prime}=\leavevmode\nobreak\
$ $\displaystyle
a_{0}\left(a+\Delta\right)e^{-i(E-E_{0})t/\hbar\leavevmode\nobreak\ +il\Omega
t}$
$\displaystyle+a_{0}\left(a-\Delta\right)e^{i(E-E_{0})t/\hbar\leavevmode\nobreak\
+il\Omega t}\leavevmode\nobreak\ .$ (14)

The phase angle subsequently depends on the rotation rate,

$\displaystyle\varphi(t)=\text{arctan}\leavevmode\nobreak\ $
$\displaystyle\left[\frac{-\Delta\sin[(E-E_{0})t/\hbar]\cos(l\Omega
t)+a\cos[(E-E_{0})t/\hbar]\sin(l\Omega
t)}{\Delta\sin[(E-E_{0})t/\hbar]\sin(l\Omega
t)+a\cos[(E-E_{0})t/\hbar]\cos(l\Omega t)}\right]\leavevmode\nobreak\ .$ (15)

For $\Omega=0$, this reduces to Eqn. 10. At stroboscopic times $\tau_{s}$
(Eqn. 11) we find explicit dependence of the phase shift on the rotation rate,

$\displaystyle\varphi(\tau_{s})$
$\displaystyle=\frac{n\,\pi\,\hbar\,l\,\Omega}{E-E_{0}}\leavevmode\nobreak\ .$
(16)

This phase shift increases with each oscillation due to the $n$-dependence of
Eqn. 16. At exactly $t=\tau_{s}$, the rotation of the density profile is zero
relative to the rotating frame. At other times, it follows the general form of
the phase shift time derivative in the rotating frame,

$\displaystyle\frac{d\varphi}{dt}$
$\displaystyle=l\Omega-\frac{\Delta(E-E_{0})}{\hbar a}\leavevmode\nobreak\ .$
(17)

As $\frac{d\phi}{dt}=0$ at stroboscopic measurement times, we can directly
relate the mode imbalance $\Delta$ to the rotation rate $\Omega$ using Eqn.
17,

$\displaystyle\frac{\Delta}{a}(E-E_{0})=l\hbar\Omega\leavevmode\nobreak\ .$
(18)

This result demonstrates that a rotating system develops an imbalance in the
amplitudes of imprinted counter-propagating OAM modes which is intrinsically
linked to the rate of rotation – with non-zero rotation there is always a non-
zero imbalance.

Experimentally, the rotation rate is measured by imaging the atomic density at
one or more stroboscopic measurement times and determining the Fourier phase
shift. The linear gradient between stroboscopic phase shifts according to Eqn.
16 is proportional to $l\Omega$ – therefore an unknown rotation rate can be
measured in a frame where the laboratory is also rotating.

##  III  Numerical Simulations

To test and extend our analytic results, we simulate the dynamics of a one-
dimensional ring-trapped condensate in two parameter regimes: the phonon
regime and the mean-field regime. In the phonon regime, excitations are of
sufficiently small amplitude that inter-particle interactions are negligible –
allowing simulation of dynamics using the Schrödinger equation for
wavefunction $\psi(\theta,t)$,

$\displaystyle-i\hbar\frac{\partial{\psi(\theta,t)}}{\partial{t}}$
$\displaystyle=\left({\frac{-\hbar^{2}}{2mr^{2}}\frac{\partial{{}^{2}}}{\partial{\theta^{2}}}+V(\theta,t)}\right)\psi(\theta,t)\leavevmode\nobreak\
.$ (19)

In the mean-field regime, interactions are non-negligible and described by the
Gross-Pitaevskii equation (GPE),

$\displaystyle-i\hbar\frac{\partial{\psi(\theta,t)}}{\partial{t}}$
$\displaystyle=\left({\frac{-\hbar^{2}}{2mr^{2}}\frac{\partial{{}^{2}}}{\partial{\theta^{2}}}+V(\theta,t)+g\left|{\psi(\theta,t)}\right|^{2}}\right)\psi(\theta,t)\leavevmode\nobreak\
,$ (20)

where $g$ is the interaction strength parameter, $r$ is the radius of the
ring, $V(\theta,t)$ the time-dependent imprinting potential, and in the GPE
$\psi(\theta,t)$ is the 1D order parameter for the bosonic field. Simulation
is performed using a three-point Crank-Nicolson method on a grid of 105
points. We simulate the full protocol: condensation (via imaginary time
evolution) in an $|l|=5$ imprinting potential, and removal of the imprinting
potential at $t=0$ for free evolution of the system. The Fourier components at
select frequencies corresponding to OAM modes $l=[-15,15]$ are calculated at
each timestep, including the phase shift extracted from the density profile.
Rotation of the system was independently set for the imaginary time and real
time portions of the simulation via the rotation parameters
$\Omega_{I},\Omega_{R}$, respectively. Density plots and extracted phase
shifts for the non-interacting system are presented in Fig. 2.

Several features of our numerical results support our analytic solution from
Sec. II. In the inertial frame $(\Omega_{I}=\Omega_{R}=0)$ the phase shift at
stroboscopic times is zero, as shown in Fig. 2 (e). Similarly, Fig. 2 shows
the time derivative of the phase shift in the rotating frame
$(\Omega_{I}=\Omega_{R}=\Omega)$ is zero, and the phase shift accumulates by a
fixed amount proportional to $l\Omega$ each oscillation, as predicted by Eqn.
16.

Figure 2 highlights the differences in both phase shift and density
oscillations under different rotation conditions. The rotating-frame analysis
in Sec.II corresponding to Fig. 2 (d) and (h) matches the conditions of an
experimental measurement of an unknown rotation – the condensate is prepared
under rotation and freely evolves in the ring trap under rotation.

##  IV  Numerical calculation of ground state

The protocol in Ref. [25] requires an imprinting potential during condensation
to form the initial state. According to the three-mode model, the ground state
in the imprinting potential will depend on model parameters $a_{0},a,\Delta$.
In this section, we calculate the expectation value of the imprinting
Hamiltonian using a three-mode state and compare its numerically-calculated
minimum value to the ground state found using imaginary time evolution.

We simulate the 1D particle-in-a-ring Hamiltonian with imprinting potential in
a rotating frame,

$\displaystyle\hat{H}=-\frac{\hbar^{2}}{2mR^{2}}\frac{\partial{{}^{2}}}{\partial{\theta^{2}}}+[V_{0}-\alpha\cos(l\theta)]+i\hbar\Omega\frac{\partial{}}{\partial{\theta}}\leavevmode\nobreak\
.$ (21)

Here $m$ is the particle mass, $R$ the ring radius, $\Omega$ the rotation
rate, and $V=(1-\alpha\cos(l\theta))$ the imprinting potential with modulation
amplitude $\alpha$. We choose as our _ansatz_ $\psi$ from Eqn. 1, with
normalization factor
$\mathcal{N}=[2\pi(a_{0}^{2}+2a^{2}+2\Delta^{2})]^{-1/2}$. We then calculate
the expectation value of the Hamiltonian, $\langle{\hat{H}}\rangle$,

$\displaystyle\langle{\hat{H}}\rangle$
$\displaystyle=\frac{\frac{\hbar^{2}l^{2}}{2mR^{2}}\left(2a^{2}+2\Delta^{2}\right)+4l\Omega
a\Delta-2\alpha a_{0}a}{a_{0}^{2}+2a^{2}+2\Delta^{2}}\leavevmode\nobreak\ .$
(22)

Full details of the calculation are available in Appendix A. The ground state
of the system is defined by the values of $a_{0},a,\Delta$ that minimize this
expectation value for given values of the ring radius, rotation rate, and
imprinting potential amplitude. Note that for non-zero rotation
$\left|{\Omega}\right|>0$, there is an explicit coupling of $a$ and $\Delta$,
therefore the ground state of a non-rotating system will have a different
value for $a$ than that of an identical system that is rotating. This can be
seen by careful examination of Fig. 3. As the normalization factor couples the
derivatives with respect to our three main parameters, we use a numerical
gradient-descent method to obtain values for $a_{0},a,\Delta$ and compare to
the converged ground state obtained by simulating the system in 1D with
imaginary time evolution. The main result of this comparison is shown in Fig.
2(e–h), where the extracted phase of both the three-mode model and imaginary
time simulations are shown with strong agreement.

(a) $g=0$

(b) $g=1$

(c) $g=10$

(d) $g=100$

Figure 3: Mode decomposition of the simulated state following imaginary time
evolution with and without rotation for increasing interaction strength $g$.
The mode splitting is clear between the non-rotating and rotating cases. The
dominant $l=0$ mode has been omitted for visual clarity. Note that the
amplitude in the non-rotating case is not the mean amplitude for the
corresponding rotating case due to coupling between parameters $a$ and
$\Delta$ in Eqns. 22 and 23. As $g$ is increased, we see a suppression of the
imprinted mode.

##  V  Mean-field excitations

To investigate the effects of imprinting deeper modulations, we consider the
1D Gross-Pitaevskii equation (GPE) from Eqn. 20 to describe mean-field
condensate dynamics.

As we increase the interaction parameter $g>0$, the amplitude of the primary
($l$) mode Fourier component decreases as the repulsive interactions of the
mean-field potential suppress modulations, as shown in Fig. 3.

The ground state for the interacting case is determined by using the GPE
Hamiltonian from Eqn. 20. This gives a modified expectation value to that in
Sec. IV, though it can be numerically minimized in the same way as Eqn. 22,

$\displaystyle\langle{\hat{H}}\rangle=$ $\displaystyle\leavevmode\nobreak\
\frac{1}{a_{0}^{2}+2a^{2}+2\Delta^{2}}\bigg{(}\frac{\hbar^{2}l^{2}}{2mR^{2}}\left(2a^{2}+2\Delta^{2}\right)+4l\Omega
a\Delta$ $\displaystyle-2\alpha
a_{0}a+g[8a_{0}^{2}a^{2}+(a^{2}-\Delta^{2})^{2}a^{4}]\bigg{)}\leavevmode\nobreak\
.$ (23)

As shown in Fig. 4, the interacting ground state result provides strong
agreement between optimized parameters and those found via imaginary time
evolution in numerical simulations.

Figure 4: Same as Fig. 2 but with mean field interaction strength $g=10$. Note
the lighter color in the density plots (a)-(d) due to interactions suppressing
the imprinted mode amplitudes. The temporal oscillation frequency is also
increased due to the extra energy in the system. The Fourier phase evolution
of the three-mode model (green crosses, subfigures (e)-(h)) agrees with that
of the ground state found through imaginary time (purple curve).

An additional feature of Fig. 4 when viewed in comparison to Fig. 2 is the
dependence of the density oscillation period on the interaction strength. In
Fig. 5 we compare the oscillation period of simulations with varying
interaction strengths and find that in the mean-field approximation the
oscillation frequency is dependent on the Bogoliubov mode energy,

$\displaystyle\epsilon(p)$
$\displaystyle=\sqrt{\left({\frac{p^{2}}{2m}}\right)^{2}+\frac{p^{2}gn}{m}}\leavevmode\nobreak\
,$ (24)

where $p$ is the mode’s momentum and $n$ the atomic density.

Figure 5: Dependence of the temporal oscillation period on interaction
strength $g$. Points are the oscillation periods extracted from numerical
simulations, the curve corresponds to the analytic Bogoliubov mode energy from
Eqn. 24. Natural units are set such that $\hbar=m=c=1$.

###  V.1  The Hess-Fairbank effect

The Hess-Fairbank effect describes the formation of a non-rotating superfluid
condensate in a sufficiently-slowly rotating container [33]. It is in this
slowly-rotating regime that the three-mode model is obtained exactly, as shown
in Fig. 6 – higher external rotation rates induce a global circulation in the
condensed state, shifting the mode occupation. This corresponds to a global
phase factor in the wavefunction, which does not alter the mode structure of
the density, nor the evolution of the Fourier phase (except for the expected
scaling with rotation rate). This is explored further in Appendix B.

A condensate formed in a rotating Sagnac interferometer is not irrotational as
described by the Hess-Fairbank effect. In the slowly-rotating regime, the
rotation of the imprinted density profile indicates flow without global
circulation. This is where classical fluid analogies break down, as they
cannot describe the superposition state of the condensate. A simple
experimental test of the Hess-Fairbank effect in the collective excitation
interferometer can be performed using a rotating imprinting potential to
simulate different external rotation rates as in Fig. 6 without requiring
rotation of the entire experimental apparatus.

Figure 6: Heatmap of normalized mode amplitudes for quantized angular momentum
modes formed under rotation. Imaginary time evolution of an initial Gaussian
state under external rotation at $\Omega$ produces the expected three-mode
state at low rotation rates, and induces a global rotation of the system at
higher rates.

###  V.2  Experimental considerations

The rotating reference frame and mean-field effects described above impact the
feasibility of the original interferometric scheme. In a typical inertial
sensing application, the time-dependence of the phase shift requires
stroboscopic measurements to be precisely timed for maximum accuracy. This is
complicated by the dependence of the stroboscopic measurement time on the
interaction strength – which in an experiment is related to the atom number,
and cannot be precisely controlled between iterations. Using the Bogoliubov
mode energy as a guide to ideal measurement time can mitigate this where
interactions are sufficiently weak.

##  VI Conclusion

In conclusion, the three-mode model with imbalanced mode amplitudes is a
frame-independent model for studying the rotation signal in a phonon-mode
Sagnac interferometer. We have shown that this model is accurate in describing
the one-dimensional behavior of this system, and that absolute rotation
measurement is possible. The three-mode state is not irrotational per the
Hess-Fairbank effect even at low rotation rates, and we have suggested a
simple experiment to demonstrate this.

Given the results of this work, the ideal rotation measurement scheme involves
measuring the extracted phase at as close to the first stroboscopic
measurement time as possible. We have shown that there are multiple subtleties
in reference frame and interaction effects that must be considered when
designing ring-trapped interferometric schemes.

###### Acknowledgements.

The authors would like to thank M.C. Kandes, R. Carretero-Gonzalez, S.A.
Haine, T.W. Neely, and M.J. Davis for useful discussions, and especially thank
H. Rubinsztein-Dunlop for her support. This research was funded by the
(Australian) Defence Science and Technology Next Generation Technologies Fund
(QT95). This research was partially funded through an Australian Research
Council Future Fellowship (ARC, FT100100905), supported by the Australian
Research Council Centre of Excellence for Engineered Quantum Systems (EQUS,
CE170100009), and Australian Government Research Training Program
Scholarships.

##  Appendix A Numerical ground state calculation

In order to numerically determine the three-mode model parameters that define
the ground state in a rotating ring system, we calculate the expectation value
of the Hamiltonian for numerical minimization. The 1D particle-in-a-ring
Hamiltonian with imprinting potential is given in a rotating frame by:

$\displaystyle\hat{H}=-\frac{\hbar^{2}}{2mR^{2}}\frac{\partial{{}^{2}}}{\partial{\theta^{2}}}+[V_{0}-\alpha\cos(n\theta)]+i\hbar\Omega\frac{\partial{}}{\partial{\theta}}$
(25)

Where $m$ is the particle mass, $R$ the ring radius, $\Omega$ the rotation
rate, and $V=(1-\alpha\cos(n\theta))$ the imprinting potential with modulation
amplitude $\alpha$. We choose as our _ansatz_ $\psi$ from the three-mode
model, with normalization factor
$(2\pi(a_{0}^{2}+2a^{2}+2\Delta^{2}))^{-1/2}$. We can then determine the
expectation value of the Hamiltonian, $\langle{\hat{H}}\rangle$:

$\displaystyle\langle{\hat{H}}\rangle=$
$\displaystyle\int_{0}^{2\pi}\psi^{*}\hat{H}\psi\ d\theta$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\left({\left|{a_{0}}\right|^{2}+2\left|{a}\right|^{2}+2\left|{\Delta}\right|^{2}}\right)}$
$\displaystyle\times\int_{0}^{2\pi}\left[{V_{0}-\alpha\cos(m\theta)}\right]\left|{a_{0}}\right|^{2}$
$\displaystyle+\left({\frac{\hbar^{2}l^{2}}{2mR^{2}}+\left[{V_{0}-\alpha\cos(n\theta)}\right]-\hbar
l\Omega}\right)\left|{a+\Delta}\right|^{2}$
$\displaystyle+\left({\frac{\hbar^{2}l^{2}}{2mR^{2}}+\left[{V_{0}-\alpha\cos(n\theta)}\right]+\hbar
l\Omega}\right)\left|{a-\Delta}\right|^{2}$
$\displaystyle+\bigg{[}\left({\frac{\hbar^{2}l^{2}}{2mR^{2}}+\left[{V_{0}-\alpha\cos(n\theta)}\right]-\hbar
l\Omega}\right)$ $\displaystyle\qquad\times
a_{0}^{*}\left({a+\Delta}\right)e^{il\theta}\bigg{]}$
$\displaystyle+\bigg{[}\left({\frac{\hbar^{2}l^{2}}{2mR^{2}}+\left[{V_{0}-\alpha\cos(n\theta)}\right]+\hbar
l\Omega}\right)$ $\displaystyle\qquad\times
a_{0}^{*}\left({a-\Delta}\right)e^{-il\theta}\bigg{]}$
$\displaystyle+\left({V_{0}-\alpha\cos(n\theta)}\right)\left({a+\Delta}\right)^{*}a_{0}\
e^{-il\theta}$
$\displaystyle+\left({V_{0}-\alpha\cos(n\theta)}\right)\left({a-\Delta}\right)^{*}a_{0}\
e^{il\theta}$
$\displaystyle+\bigg{[}\left({\frac{\hbar^{2}l^{2}}{2mR^{2}}+\left[{V_{0}-\alpha\cos(n\theta)}\right]-\hbar
l\Omega}\right)$
$\displaystyle\qquad\left({a-\Delta}\right)^{*}\left({a+\Delta}\right)e^{-2il\theta}\bigg{]}$
$\displaystyle+\bigg{[}\left({\frac{\hbar^{2}l^{2}}{2mR^{2}}+\left[{V_{0}-\alpha\cos(n\theta)}\right]+\hbar
l\Omega}\right)$
$\displaystyle\qquad\left({a+\Delta}\right)^{*}\left({a-\Delta}\right)e^{-2il\theta}\bigg{]}\
d\theta$ (26)

This integral is simplified by noting that all terms with a single oscillatory
factor integrate to zero, leaving only the non-oscillatory terms, and terms
with a product of $\cos(n\theta)$ and an exponential. The latter can be
calculated as follows:

To determine the contribution of the imprinting potential modulations to the
integral in Equation 26, we calculate the potential matrix element for generic
OAM eigenstates $|{k}\rangle=\psi_{k}=(2\pi)^{-1/2}e^{ik\theta}$:

$\displaystyle\langle{l}|V_{m}|{n}\rangle=\leavevmode\nobreak\ $
$\displaystyle\int_{0}^{2\pi}\psi^{*}_{l}(\theta)\
[V_{0}-\alpha\cos(m\theta)]\ \psi_{n}(\theta)\ d\theta$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\int_{0}^{2\pi}\psi^{*}_{l}(\theta)\ V_{0}\ \psi_{n}(\theta)\
d\theta\ $
$\displaystyle-\frac{\alpha}{2\pi}\int_{0}^{2\pi}\psi^{*}_{l}(\theta)\
\cos(m\theta)\ \psi_{n}(\theta)\ d\theta$ $\displaystyle=\leavevmode\nobreak\
$ $\displaystyle V_{0}\
\delta_{l,n}-\frac{\alpha}{2\pi}\int_{0}^{2\pi}\cos(m\theta)\
e^{-i(l-n)\theta}\ d\theta$ $\displaystyle=\leavevmode\nobreak\ $
$\displaystyle V_{0}\
\delta_{l,n}-\frac{\alpha}{2\pi}\int_{0}^{2\pi}\frac{1}{2}\big{(}\cos[(l-n+m)\theta]$
$\displaystyle\hskip 71.13188pt+\cos[(l-n-m)\theta]\big{)}$
$\displaystyle\hskip 71.13188pt-\frac{i}{2}\big{(}\sin[(l-n+m)\theta]$
$\displaystyle\hskip 71.13188pt-\sin[(l-n-m)\theta]\big{)}\ d\theta$ (27)

The integral is non-zero if and only if either $l-n+m=0$, or $l-n-m=0$. In
either of these cases, the matrix element reduces to:

$\displaystyle\langle{l}|V_{m}|{n}\rangle$ $\displaystyle=-\frac{\alpha}{2}$
(28)

For the three-mode model, there are four such terms, resulting in the
$-2\alpha a_{0}a$ term in the final expression for the expectation value of
the Hamiltonian:

$\displaystyle\langle{\hat{H}}\rangle=\leavevmode\nobreak\ $
$\displaystyle\frac{1}{a_{0}^{2}+2a^{2}+2\Delta^{2}}$
$\displaystyle\times\bigg{(}\frac{\hbar^{2}l^{2}}{2mR^{2}}\left({2a^{2}+2\Delta^{2}}\right)+4\hbar
l\Omega a\Delta-2\alpha a_{0}a\bigg{)}$ (29)

This process is repeated using the Gross-Pitaevskii equation to obtain the
nonlinear expectation value presented in the main text.

##  Appendix B The Hess-Fairbank effect

First observed in liquid Helium [33], the Hess-Fairbank effect describes the
formation of a non-rotating superfluid in a container rotating sufficiently
slowly during the transition to the superfluid phase. In a simply-connected
cylindrical container of radius $R$, rotation proportional to
$\frac{n\hbar}{mR^{2}}$ is sufficient to generate $n$ vortices of quantized
circulation. This places an upper limit on the rotation rate of such a
container such that the ground state has zero circulation.

In the ring geometry of the phonon-mode interferometer, it is straightforward
to calculate the rotation rate at which an energy crossing between two OAM
modes occurs for a non-interacting superfluid by equating their eigenenergies:

$\displaystyle E_{l}$ $\displaystyle=E_{k}$ (30)
$\displaystyle\frac{\hbar^{2}l^{2}}{2mR^{2}}-\hbar l\Omega$
$\displaystyle=\frac{\hbar^{2}k^{2}}{2mR^{2}}-\hbar k\Omega$ (31)
$\displaystyle\frac{\hbar}{2mR^{2}}(l^{2}-k^{2})$ $\displaystyle=\Omega(l-k)$
(32) $\displaystyle\Omega$
$\displaystyle=\frac{\hbar}{mR^{2}}\frac{l^{2}-k^{2}}{2(l-k)}$ (33)

For the $l^{th}$ mode and the mode directly adjacent (in the direction of
rotation), the crossing occurs at $\Omega=\frac{\hbar}{mR}\leavevmode\nobreak\
\frac{2l+1}{2}$. However, in the three-mode model each mode does not shift to
the adjacent mode, instead there is a patterned shift where the $\pm
l^{\text{th}}$ mode shifts to the $\mp(l+1)^{\text{th}}$ mode for rotation
about the positive $z$-axis (or shifts to the $\mp(l-1)^{\text{th}}$ mode for
rotation about the negative $z$-axis), as shown in Table 1.

$\leavevmode\nobreak\ k$ $l\leavevmode\nobreak\ $ |  $-5$ |  $-4$ |  $-3$ |  $-2$ |  $-1$ |  $0$ |  $1$ |  $2$ |  $3$ |  $4$ |  $5$ |  $6$
---|---|---|---|---|---|---|---|---|---|---|---|---
$-5$ |  |  $-\frac{9}{2}$ |  $-4$ |  $-\frac{7}{2}$ |  $-3$ |  $-\frac{5}{2}$ |  $-2$ |  $-\frac{3}{2}$ |  $-1$ |  $-\frac{1}{2}$ |  $0$ |  $\frac{1}{2}$
$-4$ |  $-\frac{9}{2}$ |  |  $-\frac{7}{2}$ |  $-3$ |  $-\frac{5}{2}$ |  $-2$ |  $-\frac{3}{2}$ |  $-1$ |  $-\frac{1}{2}$ |  $0$ |  $\frac{1}{2}$ |  $1$
$-3$ |  $-4$ |  $-\frac{7}{2}$ |  |  $-\frac{5}{2}$ |  $-2$ |  $-\frac{3}{2}$ |  $-1$ |  $-\frac{1}{2}$ |  $0$ |  $\frac{1}{2}$ |  $1$ |  $\frac{3}{2}$
$-2$ |  $-\frac{7}{2}$ |  $-3$ |  $-\frac{5}{2}$ |  |  $-\frac{3}{2}$ |  $-1$ |  $-\frac{1}{2}$ |  $0$ |  $\frac{1}{2}$ |  $1$ |  $\frac{3}{2}$ |  $2$
$-1$ |  $-3$ |  $-\frac{5}{2}$ |  $-2$ |  $-\frac{3}{2}$ |  |  $-\frac{1}{2}$ |  $0$ |  $\frac{1}{2}$ |  $1$ |  $\frac{3}{2}$ |  $2$ |  $\frac{5}{2}$
$0$ |  $-\frac{5}{2}$ |  $-2$ |  $-\frac{3}{2}$ |  $-1$ |  $-\frac{1}{2}$ |  |  $\frac{1}{2}$ |  $1$ |  $\frac{3}{2}$ |  $2$ |  $\frac{5}{2}$ |  $3$
$1$ |  $-2$ |  $-\frac{3}{2}$ |  $-1$ |  $-\frac{1}{2}$ |  $0$ |  $\frac{1}{2}$ |  |  $\frac{3}{2}$ |  $2$ |  $\frac{5}{2}$ |  $3$ |  $\frac{7}{2}$
$2$ |  $-\frac{3}{2}$ |  $-1$ |  $-\frac{1}{2}$ |  $0$ |  $\frac{1}{2}$ |  $1$ |  $\frac{3}{2}$ |  |  $\frac{5}{2}$ |  $3$ |  $\frac{7}{2}$ |  $4$
$3$ |  $-1$ |  $-\frac{1}{2}$ |  $0$ |  $\frac{1}{2}$ |  $1$ |  $\frac{3}{2}$ |  $2$ |  $\frac{5}{2}$ |  |  $\frac{7}{2}$ |  $4$ |  $\frac{9}{2}$
$4$ |  $-\frac{1}{2}$ |  $0$ |  $\frac{1}{2}$ |  $1$ |  $\frac{3}{2}$ |  $2$ |  $\frac{5}{2}$ |  $3$ |  $\frac{7}{2}$ |  |  $\frac{9}{2}$ |  $5$
$5$ |  $0$ |  $\frac{1}{2}$ |  $1$ |  $\frac{3}{2}$ |  $2$ |  $\frac{5}{2}$ |  $3$ |  $\frac{7}{2}$ |  $4$ |  $\frac{9}{2}$ |  |  $\frac{11}{2}$
$6$ |  $\frac{1}{2}$ |  $1$ |  $\frac{3}{2}$ |  $2$ |  $\frac{5}{2}$ |  $3$ |  $\frac{7}{2}$ |  $4$ |  $\frac{9}{2}$ |  $5$ |  $\frac{11}{2}$ |
Table 1: Rotation rates as multiples of $\frac{\hbar}{mR^{2}}$ that produce
energy crossings from OAM mode $k$ to mode $l$; determined using Equation 33.
Highlighted entries show the mode shifting for the three-mode model studied in
the text. Note that the observed shift is not due to adjacent modes coupling
for $l=\pm 5$. Figure 7: Occupation of select orbital angular momentum (OAM)
modes after condensation into an $l=5$ imprinting potential for increasing
external rotation rate during condensation. Values below $10^{-6}$ are
equivalent to zero due to a finite convergence threshold. Figure 8: Amplitudes
of select density Fourier modes after condensation into an $l=5$ imprinting
potential for increasing external rotation rate during condensation. Values
below $10^{-6}$ are equivalent to zero due to a finite convergence threshold.

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