# An Explicit Method for Fast Monocular Depth Recovery in Corridor
Environments

Yehao liu1, Ruoyan Xia1, Xiaosu Xu1,Zijian Wang1,Yiqing Yao1 , and Mingze Fan1
1Albert Author is with Key Laboratory of Micro-Inertial Instruments and
Advanced Navigation Technology, Ministry of Education, School of Instrument
Science and Engineering, Southeast University, Nanjing 210096, China

###### Abstract

Monocular cameras are extensively employed in indoor robotics, but their
performance is limited in visual odometry, depth estimation, and related
applications due to the absence of scale information.Depth estimation refers
to the process of estimating a dense depth map from the corresponding input
image, existing researchers mostly address this issue through deep learning-
based approaches, yet their inference speed is slow, leading to poor real-time
capabilities. To tackle this challenge, we propose an explicit method for
rapid monocular depth recovery specifically designed for corridor
environments, leveraging the principles of nonlinear optimization. We adopt
the virtual camera assumption to make full use of the prior geometric features
of the scene. The depth estimation problem is transformed into an optimization
problem by minimizing the geometric residual. Furthermore, a novel depth plane
construction technique is introduced to categorize spatial points based on
their possible depths, facilitating swift depth estimation in enclosed
structural scenarios, such as corridors. We also propose a new corridor
dataset, named Corr_EH_z, which contains images as captured by the UGV camera
of a variety of corridors. An exhaustive set of experiments in different
corridors reveal the efficacy of the proposed algorithm.

##  I INTRODUCTION

Monocular cameras play a pivotal role in indoor robotics[1, 2,
3];nevertheless, their performance is constrained in certain application
fields, such as visual odometry and 3D object detection, due to the absence of
scale information [1, 4, 5, 6]. To address this limitation and obtain scale
information from images, researchers often employ supplementary auxiliary
methods. While RGBD cameras offer the advantage of obtaining relatively
highprecision scene depth images, their resolution remains comparably low,
rendering them susceptible to the influence of deep black objects, translucent
materials, specular reflections, and parallax effects, thereby leading to
reduced accuracy. Conversely, stereo cameras can calculate pixel depth through
triangulation; however, this advantage comes at the expense of increased
computational overhead, and their distance estimation is subject to
limitations imposed by the baseline length.

Depth estimation refers to the process of estimating a dense depth map from
the corresponding input image[2]. Monocular depth estimation holds significant
research value [5, 3]. When combined with object detection, it can achieve the
effect of 3D reconstruction of general Lidar detection[7]. Furthermore,
through integration with semantic segmentation, the approach can be extended
from 2D to 3D, allowing the acquisition of both semantic and depth information
for pixels.

Monocular depth estimation is an ill-posed problem that requires the
introduction of sufficient prior information for its resolution. Monocular
depth estimation methods can be categorized into structure from motion (SFM)
based methods[8, 9, 10, 11], hand-crafted feature based methods, and deep
learning-based methods[13, 14, 15, 16]. Each approach explores different
strategies to address the challenge of recovering depth from a single camera
input. With the rapid advancement of deep neural networks, monocular depth
estimation based on deep learning has attracted considerable interest and
demonstrated remarkable accuracy[17, 5]. The impressive performance of deep
learning methods relies on thorough training on extensive datasets, and the
accuracy heavily hinges on the quality of precisely annotated data. Acquiring
highquality data for depth/parallax reconstruction involves substantial time
and labor costs [5]. Furthermore, deep learning-based methods exhibit limited
generalization capacity in depth estimation due to the influence of image size
and scene characteristics present in the training data[4]. Additionally, the
majority of deep learning approaches demonstrate slow inference speeds and
insufficient realtime performance.

SfM-based methods perform 3D scene reconstruction using multiple image
sequences from different perspectives. They extract feature points from the
images for feature matching, estimate camera motion and 3D positions of
pixels, and construct sparse depth maps by assembling point cloud information
of 3D space points[8, 9, 10, 11]. However, SfM-based methods require matching
alignment between multiple frames with continuous motion, and their accuracy
is highly dependent on the results of inter-frame registration, thereby
limiting their application in certain scenarios.

Long corridors/hallways are characteristic of challenging scenarios with
limited texture features, and a high degree of similarity between frames
hinders reliable inter-frame alignment. Thus, SfM-based methods are
susceptible to failure and reduced accuracy in such degraded scenes.
Nonetheless, long corridors/hallways manifest strong structured
characteristics, encompassing abundant geometric information, such as parallel
walls on both sides and maintaining parallel lines at the junction of the
floor and walls. By fully leveraging these structured features, Inverse
Projective IPM (Inverse Projective IPM)[18, 19, 20, 21] can be employed to
derive scale information for these characteristics, enabling depth plane
construction and subsequent scene depth recovery.

In this context, we propose a novel display method for rapid monocular depth
estimation. Our approach differs from existing methods as it eliminates the
need for inter-frame matching assistance and avoids extensive training on
large datasets, thereby saving on model training and transfer costs. By
leveraging the virtual camera assumption and minimizing geometric residuals,
we transform the depth estimation problem into an optimization task.
Furthermore, we introduce a depth plane construction method, which categorizes
spatial points based on their possible depths, enabling fast depth estimation
in enclosed structural scenes such as long corridors/hallways. Our proposed
method achieves state-of-the-art depth estimation accuracy in long
corridor/hallway scenarios while significantly accelerating the depth recovery
process. Moreover, it can achieve realtime monocular depth recovery on mid-to-
low-performance processors.

##  II Related Work

###  II-A Explicit Method for Depth Estimation

The explicit method for depth estimation refers to an approach in which the
entire process of depth estimation, from feature extraction and feature
transformation to the output of prediction results, can be explained using
mathematical formulas. This method is commonly employed in depth estimation
techniques based on SFM. On the other hand, implicit methods achieve the same
task through techniques such as convolutional neural networks (CNNs), where
the processes of feature extraction, feature space transformation, and depth
prediction are encapsulated within an end-to-end deep network model.

The SFM algorithm receives input image sequences taken from different viewing
angles and first extracts features such as Harris, SIFT or SURF from all
images. Feature matching is then performed to estimate the 3D coordinates of
the features and generate a point cloud that can be converted into a depth
map. In 2014, Prakash et al. [prakash2014sparse]proposed a sparse depth
estimation method based on SFM. Based on monocular image sequences from 5 to 8
different perspectives, the method used a multi-scale fast detector for
feature detection and 3D position solution based on geometric view relations
to obtain a sparse depth map.

In 2016, Ha et al. [8]proposed a Structure From Small Motion (SFM) recovery
method, which uses planar scanning technology to estimate depth maps. By using
Harris corner detection and optical flow tracking method to solve the 3D
position of the feature points, a relatively dense depth map can be obtained,
but this algorithm cannot run in real time in terms of speed.

In recent years, researchers have attempted to combine the strengths of
explicit and implicit methods. In 2022, Zhong et al. [22] introduced a method
that simultaneously conducts implicit reconstruction and extracts 3D feature
points, while others usually use explicit method to get 3D points. It replaces
manual feature extraction with an implicit description for 3D keypoint
detection.

In 2023, Wu et al. [23] demonstrated the equivalence of depth and height in
the 2D-to-3D mapping transformation and proposed an explicit height
description method applied to deep network models for transforming Bird’s Eye
View (BEV) space.

###  II-B Real-Time Monocular Depth Estimation

The overall development trend of monocular depth estimation is to push the
increase of accuracy using extremely deep CNNs or by designing a complex
network architecture, which are computationally expensive for current mobile
computational devices which have limited memory and computational capability.
Therefore, it is difficult for these networks to be deployed in small sized
robots which depend on mobile computational devices. Under this context,
researchers have begun to develop real-time monocular depth estiamtion
methods[2].

In 2018,Poggi et al. [24] stack a simple encoder and multiple small decoders
working in a pyramidal structure, which is capable to quickly infer an
accurate depth map on a CPU, even of an embedded system, using a pyramid of
features extracted from a single input image.The network was trained in an
unsupervised manner casting depth estimation as an image reconstruction
problem. The designed network only has 1.9M parameters and requires. 0.12s to
produce a depth map on a i7-6700K CPU, which isclose to a real-time speed.

In 2019, Wofk et al. [25] develop a lightweight encoderdecoder network for
monocular depth estimation.A low latency, high throughput, high accuracy depth
estimation algorithm running on embedded systems was designed. In addition,a
network pruning algorithm is applied to further reduce the amount of
parameters, which enables real-time depth estimation on embedded platforms
with an Nvidia-TX2 GPU.

In 2020, Wang et al. [14] design a highly compact network named DepthNet Nano.
DepthNet Nano applies densely connected projection batchnorm expansion
projection (PBEP) modules to reduce network architecture and computation
complexity while maintaining the representative ability.

In 2020, Liu et al. [26] introduce a lightweight model (named MiniNet) trained
on monocular video sequences for unsupervised depth estimation. The core part
of MiniNet is DepthNet, which iteratively utilizes the recurrent module-based
encoder to extract multi-scale feature maps. The obtained feature maps are
passed to the decoder to generate multi-scale disparity maps. MiniNet achieves
real-time speed about 54fps with 640 192 sized images on a single Nvidia
1080Ti GPU.

However, the accuracy of above is inferior to state-of-the-art methods.
Therefore, developing real-time monocular depth estimation network is assumed
to achieve the trade-off between accuracy and efficiency.

###  II-C Corridor Environments Perception and Localization

Long corridor is a typical degraded scene with a lack of texture features,
which brings new challenges to visual perception and localization tasks[27].
It is necessary to understand the characteristics of this scene, so as to make
full use of prior features and achieve high-precision perception and
localization. However, long corridor scenes are inevitably faced by mobile
robots, in recent years, some researchers have begun to focus on solving this
problem.

In 2021, Padhy et al.[28] introduce a localization method of Unmanned Aerial
Vehicles(UAV) in Corridor Environments, a Deep Neural Network(DNN) was trained
to understand corridor environmental information, andpredict the position of
the UAV as either on the left or center or right side of the
corridor.Depending upon the divergence of the UAV with respect to an imaginary
central line, known as the central bisector line (CBL) of the corridor, a
suitable command is generated to bring the UAV to the center, making UAV fly
safely in Corridors.

In 2023, Ge et al.[29] proposed a visual-feature-assisted localization methods
in long corridor environments.A novel coarse-to-fine paradigm was presented
that uses visual features to assist mobile robot localization in long
corridors.Sufficient keyframes are obtained through the camera, and a visual
camera map was created while the grid map built by the laser-based SLAM method
with a low accuacy in corridors, and the mobile robot captures images in a
proper perspective according to the moving strategy and matches them with the
image map to achieve a coarse localization.

##  III Materials and Methods

###  III-A System Overview

Figure 1: System framework.There are two threads.Thread 1 main function is to
extract ground edges combined along the line. Thread 2 mainly completes the
depth estimation of the scene.

The proposed method consists of two main threads: the edge extraction thread
,which main function is to extract ground edges line Sets, and the depth
recovery thread, witch mainly completes the depth estimation of the scene.

As shown in Figure 1. Images acquired from the visual sensor are first input
to the feature extraction thread. In this thread, line feature information is
extracted from the scene using Hough transform-based line feature detection.
The line features of the sense are then filtered based on the distribution of
line segment angles, leading to the construction of a set of ground edge
lines. The information about the edge line set is then sent to the depth
recovery thread.

In the depth recovery thread, the edge line set is first projected into the
virtual camera imaging space. The current camera-to-virtual-camera pose
transformation is estimated by minimizing symmetry geometric residuals.
Subsequently, the edge line set is transformed to the virtual camera imaging
plane using the computed transfor-mation matrix. Distance geometric residuals
are then constructed, and a non-linear optimization process is iteratively
performed to estimate the camera’s pitch angle and the depths of the edge
points.Based on the estimated depths of the edge points, depth planes are
constructed and transformed back to the original image plane. Pixel points in
the original image plane are classified based on the depth information, and
finally, depth estimation values for the pixel points are obtained through
approximate inter-polation.

###  III-B Edge Extration Thread

Thread 1 primarily engages in the extraction of structured scene features
through ground edge detection, thereby furnishing the depth recovery thread
with essential prior information about the scene. In this study, it is assumed
that the width of the long corridor/hallway remains constant, leading to the
representation of ground and side wall edges as two straight lines within the
image.

(a)

(b)

(c)

(d)

Figure 2: (a)The region of interestiong. (b) The results of Canny edge
detection. (c) The results of Hough line transform(after NMS). (d) The line
definetion in the sence.

####  III-B1 Construction of ROI

In the context of long corridor/hallway scenarios, aside from ground edges,
there often exist additional linear features such as door frames and objects,
which can introduce interference with the detection of ground edge lines.
Consequently, it becomes imperative to establish a Region of Interest (ROI)
within the scene. This selection process is based on empirical observations.
Due to geometric perspective, ground edge lines typically manifest in the
lower-to-middle section of the image, converging from the bottom sides towards
the center. Relying on this a priori knowledge, an ROI with a height of H
within the image is designated, encompassing rows from the middle to the
bottom of the image, as shown in Figure$~{}\ref{fig2}(\textbf{a})$.

####  III-B2 Hough Transform Based Line Feature Extration

For the extraction process, this study employs a line feature detection
algorithm based on the Hough transform. The Hough transform is a methodology
designed to extract linear features from images. It leverages the duality
between points and lines, mapping the discrete pixel points along a straight
line in image space to curves in Hough space through parameter equations.
Subsequently, the intersection points of multiple curves in Hough space are
mapped back to straight line equations in image space, thereby forming the
detected straight lines.

Before performing the Hough transform, the image is initially binarized and
subjected to Canny edge detection thus expediting the process of Hough
transformationas shown in Figure$~{}\ref{fig2}(\textbf{b})$. Subsequently,
linear features are extracted through the Hough transform,as shown in
Figure$~{}\ref{fig2}(\textbf{c})$, and the extracted features undergo linear
fitting to yield the set of linear features, denoted as $S_{l}$.

$S_{l}=\\{l_{k}\left({\theta}_{k},{pt}_{k}^{0}\right),k=1,2,...,n\\}$ (1)

Where,${\theta}_{k},{pt}_{k}^{0}$are the inclination angle of $l_{k}$,and the
the pixel coordinates of the starting point of $l_{k}$, respectively, as
illustrated in Figure$~{}\ref{fig2}(\textbf{d})$.

####  III-B3 Scene-Prior Based Line Feature Extraction

Based on scene priors, a reasonable range for setting the length and
inclination angle of the lines is established, which is used to filter the
lines within set $S_{l}$. If the image is strict symmetry of axis, the length
of the edge lines $L\left(l_{k}\right)$ and the angle ${\theta}_{k}$ should
satisfy the following conditions:

$\begin{split}\frac{H}{4}\leq
L\left(l_{k}\right)&\leq\frac{1}{2}\sqrt{H^{2}+W^{2}}\\\
\arctan{\frac{H}{W}}\leq{\theta}_{k}<\frac{\pi}{2}\;&\vee\;\frac{\pi}{2}<{\theta}_{k}\leq\arctan{-\frac{H}{W}}\end{split}$
(2)

According to (2), the left and right line are picked out. And the edge line
set is
$S_{le}=\\{l_{l}\left({\theta}_{l},{pt}_{l}^{0}\right),l_{r}\left({\theta}_{r},{pt}_{r}^{0}\right)\\}$,
where $l_{l}$ and $l_{r}$ are left and right line, respectively.

###  III-C Visual Camera

####  III-C1 Visual Camera Model

The primary function of the virtual camera is to ensure consistency in the
imaging process across different scenes, currently predominantly employed
within deep learning-based computer vision methodologies. In 2023, BEV-LaneDet
[30] introduced the concept of the virtual camera in the context of 3D lane
detection tasks on the Bird’s Eye View (BEV) plane. Due to variations in
camera intrinsic parameters, installation positions, and camera poses, images
captured by the same scene may have different dimensions and scaling ratios.
The BEV-LaneDet method employs a deep neural network that requires image
parameters to be as consistent as possible with those in the training dataset.
This necessitates aligning the camera position, pose, and height above the
ground during imaging to avoid substantial disparities between the predictive
accuracy of the deployed model and that observed during offline training.

The virtual camera is manually defined, with its intrinsic parameters,
installation position, and orientation preconfigured. Prior to training and
inference with deep neural networks, images are initially projected onto the
imaging space of the virtual camera through perspective transformation. This
process ensures that the input images to the model exhibit consistency.

In this study, the concept of the virtual camera is introduced to achieve
algorithmic consistency across various scenes. The intrinsic coordinate system
for both the real and virtual cameras is established as right-down-forward.
The coordinate definitions for the two types of cameras and their respective
images are depicted in Figure 3. Within this hypothesis, the width of the long
corridor remains constant, and the virtual camera is positioned at the center
of the scene, equidistant from the left and right walls. Its optical axis is
directed straight ahead along the length of the corridor, while maintaining
consistent pitch angles, camera intrinsic parameters, and mounting height as
the current real camera.

Figure 3: The visual camera defination and cordinate difination.

To achieve the pose transformation from the real camera to the virtual camera,
this study begins by projecting ground edge feature points onto the imaging
plane of the virtual camera. Subsequently, a geometric error model is
constructed, and the maximum likelihood estimation of the pose transformation
is computed through an iterative optimization process.

####  III-C2 Virtual Camera Pose Estimation

In the assumption of this study, the virtual camera is positioned at the
center of the scene, ensuring that the edge lines in the image maintain its
symmetry in the virtual camera. To fully utilize this structural feature and
ensure algorithm consistency across different input images, the original image
is initially transformed into the imaging space of the virtual camera. After
obtaining the reference depth plane of the virtual camera, the depth plane
information is then reprojected onto the current image. This process
ultimately yields the depth distribution of the current image. Therefore,
prior to conducting depth estimation, it is necessary to acquire the pose
transformation from the current camera to the virtual camera.

Let $P^{W}$ represent a spatial point within the current scene, the pixel
coordinate $P^{C}\left(u,v\right)$ of space point $P^{W}\left(x,y,z\right)$ is
calculated as

$\left[\begin{array}[]{c}P^{C}\\\
1\end{array}\right]=\left[\begin{array}[]{c}u\\\ v\\\
1\end{array}\right]=\frac{1}{z}\left[\begin{array}[]{ccc}f_{x}&0&c_{x}\\\
0&f_{y}&c_{y}\\\ 0&0&1\end{array}\right]\left[\begin{array}[]{c}x\\\ y\\\
z\end{array}\right]=\frac{1}{z}KP^{W}$ (3)

Where, K is the camera internal parameter matrix, $f_{x}$ and $f_{y}$ are the
pixel focal lengths in the x and y directions, respectively, corresponding to
the physical focal length f of the camera. And the cx and cy represent the
pixel offsets of the optical center in the image of the camera. In general,
the horizontal and vertical dimensions of the camera sensor have equal pixel
sizes, in which case $f_{x}$=$f_{y}$. The rotation matrix R and displacement t
are used to represent the pose transformation from the current camera to the
virtual camera, and the current frame is projected onto the plane of the
virtual camera. The projection result is calculated according to (3)

$s^{V}P^{C}=K\left(RP^{W}+t\right)$ (4)

Where $s^{V}$ is the scale in the transformed virtual camera. In the
assumption of the virtual camera, the disparity in pose between the virtual
camera and the real camera arises from the yaw angle $\phi$ and the
displacement $\tau$ in the x-direction. In which case, R and t are calculated
as

$R=\left[\begin{array}[]{ccc}\cos{\phi}&0&\sin{\phi}\\\ 0&1&0\\\
-\sin{\phi}&0&\cos{\phi}\end{array}\right],t=\left[\begin{array}[]{c}\tau\\\
0\\\ 0\end{array}\right]$ (5)

As depth has not been recovered yet, the images lack scale information. In
order to unify the scales in both cameras, a assumption is accepted of that
for every pixel in the original image, its corresponding original spatial
point lies in front of the camera at a distance of 1 meter. It means, for any
spatial point $P^{W}\left(x,y,z\right)$, $z\equiv 1$. Therefore, the scale
factor $s^{V}$ for the virtual camera is calculated as:

$s^{V}=\cos{\phi}-\frac{u-c_{x}}{f}sin{\phi}$ (6)

Based on equations (2) to (5), the transformation from the current frame image
pixel point $pt^{C}(u,v)$ to the virtual camera image pixel point
$pt^{V}(u^{V},v^{V})$ can be obtained as follows:

$\begin{split}\left[\begin{array}[]{c}u^{V}\\\
v^{V}\end{array}\right]&=\xi\left[\begin{array}[]{cc}cos\phi&0\\\
0&1\end{array}\right]\left[\begin{array}[]{c}u\\\ v\end{array}\right]\\\ \\\
&+\left[\begin{array}[]{c}f_{t}\left(\tau+\sin{\phi}\right)+\left(\xi-
cos{\phi}\right)\\\ \left(\xi+1\right)c_{y}\end{array}\right]\\\ \end{split}$
(7)

Where,$\xi={1}/\left({f_{t}\cos{\phi}-u\sin{\phi}-c_{x}\cos{\phi}}\right)$,$f_{t}=f_{x}=f_{y}$.Each
pixel of
$l_{l}\left({\theta}_{l},pt_{l}^{0}\right)$,$l_{r}\left({\theta}_{r},pt_{r}^{0}\right)$along
the edge can be transformed into the virtual camera according (7).

Figure 4: The symmetry geometric error.

Within the assumption, the optical center of the virtual camera oincides with
the symmetry axis of the corridor. As a result, the symmetry of the road edge
lines is preserved in the virtual camera imaging space. The axis of symmetry
is located at position $u=\frac{W}{2}$ on the image plane of virtual camera,
where W represents the image width. Based on this structured feature, the
geometric residual is calculated. As shown in Figure 4., Set
$pt_{k}^{lt}=\left(u_{k}^{l}.v_{k}^{l}\right)$ as a point in $l_{l}$, and its
symmetry point about $u=\frac{W}{2}$ is $pt_{k}^{lt}$. Substituting
$v=v_{k}^{l}$ into the equation of the right line, $pt_{k}^{lt1}$ can be
obtained, and the distance D between two points can be calculated. According
to (7), D is a function of both the yaw angle $\phi$ and the displacement
$\tau$ in the x-direction. By uniformly selecting N feature points at
intervals of $l_{l}$, the sum of D calculated for these N feature points
yields symmetry error the left line $E_{L}$. Similarly, the symmetry error of
the right line $E_{R}$ can be obtained. Ultimately, this process yields the
symmetry geometric error $E_{G}$.

$E_{G}=E_{L}+E_{R}=\sum_{k}^{N}{D\left(pt_{k}^{lt},pt_{k}^{lt1}\right)}+\sum_{k}^{N}{D\left(pt_{k}^{rt},pt_{k}^{rt1}\right)}$
(8)

Once the mathematical model for the symmetry-based geometric error $E_{G}$ is
established, leveraging the prior features of the scene, a range of
variability for the yaw angle $\phi$ and the displacement $\tau$ is defined.
Nonlinear optimization is employed to compute the maximum likelihood
estimation values $\widehat{\phi}$ and $\widehat{\tau}$ for the yaw angle
$\phi$ and displacement $\tau$. Iterative calculations are performed within
the range of variability for both parameters to minimize $E_{G}$. The
resulting yaw angle and displacement that yield the smallest $E_{G}$ are
considered as the maximum likelihood estimation results.

$\widehat{\phi},\widehat{\tau}=argmin\left\\{\sum_{k}^{N}{D\left(pt_{k}^{lt},pt_{k}^{lt1}\right)}+\sum_{k}^{N}{D\left(pt_{k}^{rt},pt_{k}^{rt1}\right)}\right\\}$
(9)

After obtaining the estimated value of the heading Angle and displacement of
the heading Angle, the edge points are projected to the virtual camera space
through the formula(4)(5), and
$S^{V}=\left(pt_{k}^{Vl},pt_{k}^{Vr}\right),k=0,1,2...,n$.

###  III-D Fast Monocular Dpth Recovery

####  III-D1 3D Coordinates Estimation of Ground Edge Points

In the virtual camera space, there are now pixel coordinates along two ground
edges. Using the ground plane hypothesis, the points along each edge are
assumed to be coplanar, and the 3D spatial coordinates of each point meet
$\forall P^{W}_{k}\left(x_{k},y_{k},z_{k}\right)\in S^{W},y_{k}=0$. According
to the installation position and aperture Angle information of the camera, the
3D spatial coordinates of each edge point corresponding to the virtual camera
can be solved by geometric method. The depth plane can then be constructed.

Figure 5: The geometry of the $\Delta z$ computation.

The space coordinate system is selected as the lower left front, the camera
height is known as h, the vertical aperture Angle is $\theta_{v}$ . Assume
that the pitch Angle of the virtual camera in the current scene is
$\theta_{p}$ , and for a pixel point $pt_{0}$ at the bottom of the image, its
pixel coordinate is $\left(u_{t},v_{t}\right)$ meet $0\leq u_{t}\leq
W$,$v_{t}=H$ , W,H are the image height and image width respectively, and its
depth is calculated as

$z_{0}=\frac{h}{\tan{\left(\theta_{v}+\theta_{p}\right)}}$ (10)

The theoretical method of Inverse Perspective Transform (IPM) is used to
calculate the depth of each edge point, as shown in Figure 5. For any point in
the set of edge points
$S^{V}=\left(pt_{k}^{Vl},pt_{k}^{Vr}\right),k=0,1,2...,n$, its pixel
coordinate is $\left(u_{k}^{V},v_{k}^{V}\right)$, and its depth is calculated
according to the inverse perspective transform.

$z=z_{0}+\Delta z$ (11)

The equation is constructed to solve according to the perspective geometry as
(12)

$\frac{\eta_{k}\cos{\left(\theta_{k}-\theta_{p}\right)}}{\Delta
z\sin{\theta_{p}}}=\frac{z_{1}}{z_{0}}$ (12)

Where , $\eta_{k}=f_{y}\Delta v_{k},\Delta v_{k}=H-v_{k}^{V}$is the depth in
space at the bottom of the imaging plane 1m away from the optical center of
the camera,

$z_{1}=cos\theta_{v}-sin\theta_{v}sin\theta_{p}$ (13)

Solving (9),$\Delta z$ is calculated as

$\Delta
z=z_{0}\frac{z_{0}\cos{\theta_{p}}+h\sin{\theta_{p}}}{z_{1}h-\eta_{k}z_{0}\cos{\theta_{p}}}\eta_{k}$
(14)

####  III-D2 3D Coordinates Optimization $\&$ Pitch Angle Estimation

As shown in (14), the calculation of edge point depth $\Delta z$ is related to
camera elevation Angle $\theta_{p}$. In this paper, $\Delta z$ and
$\theta_{p}$ are estimated simultaneously by nonlinear optimization. In the
corridor, the road surface in the scene is flat and the pitch Angle variation
range is small. The iterative optimization algorithm is designed to
iteratively calculate the geometric residual of 3D space points within the
interval $\left[\lambda_{1},\lambda_{2}\right]$, taking $\alpha$ as the step
length, and calculate the optimal $\theta_{p}$,to estimate the result.

Set the initial value of $\theta_{p}$, according to (14) for the edge point
pixel set $S^{V}$, in which all pixels recover the depth distribution,
according to camera imaging model (3), the 3D spatial coordinates of edge
point $pt_{k}^{Vl}$ are calculated as

$\left\\{\begin{aligned} &z^{W}_{k}=z_{0}+\Delta z_{k}\\\
&x_{k}^{W}=z_{k}\left(u_{k}^{V}-c_{x}\right)/f_{t}\\\
&y_{k}^{W}=z_{k}\left(v_{k}^{V}-c_{y}\right)/f_{t}\end{aligned}\right.$ (15)

And the geometry residual is calculated as

$\displaystyle E_{G}^{W}$
$\displaystyle=\sum_{k}^{L}{\left\|P_{k}^{Wl},P_{k}^{Wr}\right\|_{2}}$ (16)
$\displaystyle\Longleftrightarrow\sum_{k}^{L}{\left|x_{k}^{Wl},x_{k}^{Wr}\right|}_{y_{k}^{Wl}=y_{k}^{Wr}}$
$\displaystyle=\sum_{k}^{L}{\frac{z_{k}}{f_{t}}\left(1+\frac{z_{0}\cos{\theta_{p}}+h\sin{\theta_{p}}}{z_{1}h-\eta_{k}z_{0}\cos{\theta_{p}}}\eta_{k}\right)}\left(u_{k}^{Vl}-u_{k}^{Vr}\right)$

The approximate optimal estimate of $\theta_{p}$ can be obtained by iterative
optimization

$\hat{\theta}_{p}=\arg\min\left\\{\sum_{k}^{L}{\frac{z_{k}}{f_{t}}\left(1+\frac{z_{0}\cos{\theta_{p}}+h\sin{\theta_{p}}}{z_{1}h-\eta_{k}z_{0}\cos{\theta_{p}}}\eta_{k}\right)}\left(u_{k}^{Vl}-u_{k}^{Vr}\right)\right\\}$
(17)

####  III-D3 Depth Plane Construction $\&$ Spatial Point Depth Recovery

In the image space of the virtual camera, the spatial points in the same depth
plane retain the parallel characteristics, and the contour points of the same
depth are connected to build a depth plane. The depth plane $\Gamma_{k}^{V}$
is guided by the contour lines. The depth plane $\Gamma_{k}^{V}$ is determined
by the left and right edge points
$\left\\{pt_{k}^{Vl}\left(u_{k}^{Vl},v_{k}^{Vl}\right),pt_{k}^{Vr}\left(u_{k}^{Vr},v_{k}^{Vr}\right)\right\\}$,
and is uniquely determined, $Dp\left(\Gamma_{k}^{V}\right)$ is the depth of
$\Gamma_{k}^{V}$, $Dp\left(\Gamma_{k}^{V}\right)=z_{k}$.

The depth plane $\Gamma_{k}^{V}$, obtained in the virtual camera imaging
space, is inversely transformed according to (4-5), and the set of depth
planes in the real camera image can be obtained:
$S_{\Gamma}^{C}=\left\\{\Gamma_{k}^{C}\right\\},k=1,2,3,...,L$. In a real
camera image, if the pixel $pt_{k}^{C}\left(u_{k}^{C},v_{k}^{C}\right)$
belongs to the depth plane $\Gamma_{k}^{C}$ then equation (18) is a necessary
condition.

$\forall pt_{k}^{C}\in\Gamma_{k}^{C}\Rightarrow\left\\{\begin{aligned}
&f\left(u_{k}^{C}\right)\leq u_{i}^{C}\leq u_{k}^{C}\\\ &v_{k}^{Cl}\leq
v_{i}^{C}\leq v_{k}^{Cr}\\\ &u_{k-1}^{C}\leq u_{i}^{C}\;\vee\;v_{i}^{C}\leq
v_{k-1}^{Cl}\;\vee\;v_{i}^{C}\geq v_{k-1}^{Cr}\end{aligned}\right.$ (18)

Where $f\left(u_{k}^{Cl}\right)=\alpha u_{k}^{Cl}-\beta$ is a linear transform
of $u_{k}^{Cl}$ , and $\alpha$ , $\beta$ are empirical coefficients.

In the corridor scene, the spatial point distribution is relatively ideal,
ignoring the influence of dynamic objects, occlusion, etc., and taking (18) as
sufficient and necessary conditions of $pt_{k}^{cl}\in\Gamma_{k}^{C}$ , each
pixel in the image is classified.

After pixel classification is completed, it is determined whether it is a
ground point according to the coordinates of each pixel. The condition that
$pt_{i}^{V}\left(u_{i}^{V},v_{i}^{V}\right)$ is a ground point is

$pt_{i}^{C}\in\Gamma_{k}^{C}\;\wedge\;u_{k-1}^{C}\leq u_{i}^{C}\leq
u_{k-1}^{C}$ (19)

For the pixel point
$pt_{k}^{C}\left(u_{k}^{C},v_{k}^{C}\right)\in\Gamma_{k}^{C}$, its depth is
calculated by interpolation method (20).

$Dp\left(pt_{i}^{C}\right)=Dp\left(\Gamma_{k}^{C}\right)+\sigma\left(Dp\left(\Gamma_{k-1}^{C}\right)-Dp\left(\Gamma_{k}^{C}\right)\right)$
(20)

In the formula, $\sigma$ is a nonlinear coefficient. When the depth plane is
sufficiently dense, linear interpolation method is adopted. If the point is a
ground point, then

$\sigma=\frac{u_{i}^{C}-u_{k-1}^{C}}{u_{k}^{C}-u_{k-1}^{C}}$ (21)

Otherwise,

$\sigma=\left\\{\begin{aligned}
&\frac{v_{i}^{C}-v_{k-1}^{Cl}}{v_{k}^{Cl}-v_{k-1}^{Cl}}\,,\,v_{i}^{C}\leq
v_{k-1}^{Cl}\\\
&\frac{v_{i}^{C}-v_{k-1}^{Cl}}{v_{k}^{Cr}-v_{k-1}^{Cr}}\,,\,v_{i}^{C}\geq
v_{k-1}^{Cr}\end{aligned}\right.$ (22)

##  IV Results

###  IV-A Experiment Overview

Figure 6: The Unmanned Ground Vehicle(UGV) and ZED2 camera

The algorithm proposed in this paper is validated by collecting data in real
scenarios. The ZED2 camera sensors mounted on an Unmanned Ground Vehicle(UGV)
were used to collect images in different scenes to build data sets, and the
RGB images and 16-bit depth images output by the cameras were recorded by
ROSbag. The size of the output RGB image and depth image is 420x360 for the
ZED2 camera. Camera mounting height are 0.66m, 0.62m respectively. The
experimental equipment is shown Figure LABEL:figt ZED2 camera is used to
collect 135 images of 9 kinds corridor with length range $[0-100]m$ and width
range $[2-4]m$ and different lighting conditions, from which a corridor
dataset, named Corr_EH_z were constructed. There are two parts of Corr_EH_z,
$Cord\\_Exx\\_z$ , $Cord\\_Hxx\\_z$. The $Cord\\_Exx\\_z$ subset is a simple
condition while $Cord\\_Hxx\\_z$ is a complex condition. In Figure 7, scene
images and corresponding depth images in two types of subsets are shown
respectively.

(a)

(b)

(c)

(d)

Figure 7: (a)Source img of Cord_E05_z sence. (b) Depth truth of Cord_E05_z
sence. (c) Source img of Cord_H02_z sence. (d) Depth truth of Cord_H02_z
sence.

###  IV-B Experiment 1: Verification of Algorithm Accuracy

In the two types of sub-datasets, the heading Angle estimation interval is set
as $[-0.314,0.314]rad(18)$ interval, and the accuracy of the pitch Angle
estimation and corridor width estimation in this paper is tested with 0.05rad
as the step length. The estimation results are shown in Table 1. The relative
error of the output corridor width estimation value of the method in this
paper is less than 0.0427. The mean relative error is 0.0221.

TABLE I: Accuracy of corridor width estimation.

Scene | Estimated(m) | Groundtruth(m) | relative error %
---|---|---|---
Cord_E01_z | 2.16 | 2.11 | 2.3697
Cord_E02_z | 2.08 | 2.13 | 2.3474
Cord_E03_z | 2.20 | 2.11 | 4.2654
Cord_E04_z | 2.17 | 2.09 | 3.8278
Cord_E05_z | 1.98 | 2.02 | 1.9802
Cord_E06_z | 1.88 | 1.86 | 1.0753
Cord_H01_z | 2.16 | 2.13 | 1.4085
Cord_H02_z | 3.04 | 3.09 | 1.6181
Cord_H03_z | 2.95 | 2.98 | 1.0018

1 Tables may have a footer.

The depth estimation accuracy of the proposed method is tested in the
$Cord\\_Exx\\_z$ and $Cord\\_Hxx\\_z$ datasets, as shown in Table
$~{}ref{tab2,tab3}$. Four types of general accuracy indexes of the proposed
method are calculated under the two evaluation conditions of depth truth value
and depth truth value respectively, and the four types of accuracy indexes are
defined as [13]

$\displaystyle AbsRel:$
$\displaystyle\frac{1}{N}\sum_{p}^{N}\left|\frac{y_{p}-{\hat{y}_{p}}}{y_{p}}\right|$
$\displaystyle Log10:$
$\displaystyle\frac{1}{N}\left|\log_{10}{y_{p}}-\log_{10}{\hat{y}_{p}}\right|$
$\displaystyle RMSE:$
$\displaystyle\sqrt{\frac{1}{N}\sum_{p}^{N}\left({y_{p}}-{\hat{y}_{p}}\right)^{2}}$
$\displaystyle RMSElog:$
$\displaystyle\sqrt{\frac{1}{N}\sum_{p}^{N}\left(\log_{10}{y_{p}}-\log_{10}{\hat{y}_{p}}\right)^{2}}$

TABLE II: Accuracy of depth estimation while depth is less than 5m.

Scene | AbsRel | Log10 | RMSE | RMSElog
---|---|---|---|---
Cord_E01_z | 0.06338 | 0.02908 | 0.30892 | 0.05119
Cord_E02_z | 0.06094 | 0.02630 | 0.26311 | 0.04062
Cord_E03_z | 0.05813 | 0.02522 | 0.27610 | 0.04183
Cord_E04_z | 0.07106 | 0.03279 | 0.32257 | 0.05431
Cord_E05_z | 0.06011 | 0.02518 | 0.35607 | 0.04817
Cord_E06_z | 0.07028 | 0.03156 | 0.32098 | 0.05422
Cord_H01_z | 0.08588 | 0.04104 | 0.45468 | 0.07382
Cord_H02_z | 0.11233 | 0.05968 | 0.65776 | 0.11828
Cord_H03_z | 0.10867 | 0.04709 | 0.46969 | 0.06848

1 Tables may have a footer.

TABLE III: Accuracy of depth estimation while depth is less than 40m.

Scene | AbsRel | Log10 | RMSE | RMSElog
---|---|---|---|---
Cord_E01_z | 0.09740 | 0.05185 | 1.26336 | 0.11411
Cord_E02_z | 0.08256 | 0.04079 | 1.04763 | 0.09274
Cord_E03_z | 0.09026 | 0.01348 | 1.29909 | 0.11610
Cord_E04_z | 0.10603 | 0.05551 | 1.89761 | 0.12333
Cord_E05_z | 0.08386 | 0.04342 | 1.28335 | 0.10847
Cord_E06_z | 0.09879 | 0.05225 | 1.25565 | 0.11414
Cord_H01_z | 0.12416 | 0.07247 | 1.61772 | 0.15947
Cord_H02_z | 0.16391 | 0.10121 | 1.57186 | 0.19957
Cord_H03_z | 0.12647 | 0.06797 | 1.43041 | 0.14808

1 Tables may have a footer.

As can be seen from Table II, under the condition of depth truth value ¡5m,
the AbsRel index $<7.106\%$ and RMSE index $<0.35607$ output depth estimates
in the $Cord\\_Exx\\_z$ dataset of this method reach the advanced level of
accuracy. At the same time, the accuracy of the proposed method can be
predicted in the depth range of 40m, as shown in Table III. In the
$Cord\\_E03\\_z$ scenario, the AbsRel of the proposed method in the depth
range of 40m can reach $8.256\%$. Figure 8 shows the depth recovery effect of
the proposed method in several scenario.It should be noted that the ceiling
part is eliminated to save computing resources.

###  IV-C Experiment 2: Comparison with the State-of-the-art Method

In the $Cord\\_Exx\\_z$ dataset, a precision comparison test was conducted
between the proposed method and the ADABINS[15] method based on deep learning.
ADABINS method is based on codec-decoding network +ViT for depth
classification estimation. This method was published in CVPR in 2021, and
currently ranks 14th in Kitti[31] list and 25th in NYU[32] list, which is at
the advanced level among existing methods.

First, 25epochs of ADABINSS method was trained on $Cord\\_Exx\\_z$ sub-
dataset, and then the accuracy of depth prediction between the proposed method
and the ADABINS method was compared, as shown in Table IV.

TABLE IV: Precision comparison experiment with Adabins.

Method | AbsRel | Log10 | RMSE | RMSElog
---|---|---|---|---
ADABINS1 | 0.079 | 0.033 | 0.299 | 0.101
Ours1 | 0.083 | 0.036 | 0.363 | 0.053
ADABINS2 | - | - | - | -
Ours2 | 0.098 | 0.054 | 1.425 | 0.121

1 Depth range 0-5m.
2 Depth range 0-40m.

AbsRel, Log10 and RMSE accuracy indexes of the proposed method are similar to
those of ADABINS under the condition of depth truth value ¡5m, and RMSElog is
0.053, which is better than ADABINS (0.101). AbsRel and RMSE were $9.8\%$ and
1.425 respectively under the condition of depth truth value ¡40m, while the
effective depth prediction range of ADABINS indoor depth estimation is only
10m.

Compare the speed of reasoning/recovering a single image between ADABINS and
the proposed method in different computing platforms. Computing platform 1 is
AMDThreadripperPRO 5975W 32-core high-performance processor with single-core
main frequency 3.6GHz, and GPU is Nvidia RTX4090 graphics processing unit,
which is used to assist deep neural network reasoning. Computing platform 2 is
a medium-low performance Intel Core i5-7300HQCPU with a main frequency of
2.5GHz; Repeated experiments were used to calculate the average execution time
of the two methods, and the results were shown in Table V.

TABLE V: Running time comparison experiment with Adabins.

Method | Average running time
---|---
ADABINS1 | 0.0507
Ours1 | 0.0097
ADABINS2 | -
Ours2 | 0.0482

1 On platform 1.
2 On platform 2.

The average execution time of the proposed method for depth recovery of a
single image on computing platform 1 is 0.0097s, which is less than 1/5 of the
average execution time of the ADABINS method under GPU acceleration, and the
average execution time of the proposed method on processor 2 is 0.048s. The
results show that the proposed method can achieve a real-time processing speed
of 20FPS in low and medium performance processors.

##  V Discussion

In this paper, an explicit method for fast monocular depth recovery in long
corridor scenes is proposed.

By extracting key information in long corridor scenes, the method optimizes
pitch angle estimates and scene edge point depths by minimizing geometric
residuals, and classification of space points by constructing depth planes,
the monocular depth estimation problem is transformed into a solvable
optimization problem. Fast monocular depth estimation for long corridor
scenarios is achieved. In this paper, we collected the corridor image
construction datasets under various scenarios, tested the performance of this
method experimentally, and conducted precision comparison test with the method
based on deep learning. The test results show that:

  1. 1.

The accuracy of the explicit method for fast monocular depth recovery in long
corridor scenarios reaches the advanced level of the existing monocular depth
estimation algorithms;

  2. 2.

The fast monocular depth recovery method greatly accelerates the depth
recovery process of a single image. In low and medium performance processors,
this method can perform real-time depth estimation of corridor scenes at a
speed of 20FPS.

Although the method work well in the experimental scenarios, there are still
some limitations:

  1. 1.

Due to the use of more scene prior information, the use scenarios of the
proposed method are limited. It is more suitable for corridors with
characteristics of closed and straight;

  2. 2.

Due to the ground plane assumption, the performance of the method decreases
when a small slope exists in the corridor, although this scenario is rare in
reality;

  3. 3.

Because of perspective geometry, the accuracy is low at farther distances,
where there is a cumulative drift in height and transversality, as shown in 8.
And the drift needs to be compensated empirically.

We are trying to replace straight lines with curve detection and simply model
the height of the ground to tackle the above limitations, so that the proposed
method can also perform well in scenes with a curvature.

The proposed algorithm can be used as an auxiliary module in the autonomous
navigation and positioning system of Unmanned Aerial Vehicle(UAV) and Unmanned
Ground Vehicle(UGV), such as the Simultaneous Localization and Mapping(SLAM)
system, and other applications of monocular camera systems. There are several
possible applications as we can thought:

  1. 1.

Inner corridor UAV safe flying;

  2. 2.

3D positioning of the monitor camera in the corridor;

  3. 3.

SLAM of delivery robots in the corridor.

In the future, we will try to apply this method to the above applications
effectively. In addition, a small deep neural network for segmentation and
classification can be trained to combine with this method to solve the problem
of limited scenarios. Finally, the theory of our method is suitable for
unsupervised training of depth estimation models.

Figure 8:  (a)-(k) are estimation results in several sences. For each line,
from left to right, there are edge extraction results, 3D edge estimation
results and depth estimation results, respectively .

## References

  * [1] F. Khan, S. Salahuddin, and H. Javidnia, “Deep learning-based monocular depth estimation methods—a state-of-the-art review,” _Sensors_ , vol. 20, no. 8, p. 2272, 2020.
  * [2] X. Dong, M. A. Garratt, S. G. Anavatti, and H. A. Abbass, “Towards real-time monocular depth estimation for robotics: A survey,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 23, no. 10, pp. 16 940–16 961, 2022.
  * [3] Y. Ming, X. Meng, C. Fan, and H. Yu, “Deep learning for monocular depth estimation: A review,” _Neurocomputing_ , vol. 438, pp. 14–33, 2021.
  * [4] H. Laga, L. V. Jospin, F. Boussaid, and M. Bennamoun, “A survey on deep learning techniques for stereo-based depth estimation,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 44, no. 4, pp. 1738–1764, 2020.
  * [5] P. Vyas, C. Saxena, A. Badapanda, and A. Goswami, “Outdoor monocular depth estimation: A research review,” _arXiv preprint arXiv:2205.01399_ , 2022\.
  * [6] R. Mur-Artal and J. D. Tardós, “Orb-slam2: An open-source slam system for monocular, stereo, and rgb-d cameras,” _IEEE transactions on robotics_ , vol. 33, no. 5, pp. 1255–1262, 2017.
  * [7] D. Park, R. Ambrus, V. Guizilini, J. Li, and A. Gaidon, “Is pseudo-lidar needed for monocular 3d object detection?” in _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , 2021, pp. 3142–3152.
  * [8] H. Ha, S. Im, J. Park, H.-G. Jeon, and I. S. Kweon, “High-quality depth from uncalibrated small motion clip,” in _Proceedings of the IEEE conference on computer vision and pattern Recognition_ , 2016, pp. 5413–5421.
  * [9] H. Javidnia and P. Corcoran, “Accurate depth map estimation from small motions,” in _Proceedings of the IEEE International Conference on Computer Vision Workshops_ , 2017, pp. 2453–2461.
  * [10] C. Wang, J. M. Buenaposada, R. Zhu, and S. Lucey, “Learning depth from monocular videos using direct methods,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2018, pp. 2022–2030.
  * [11] N. Yang, R. Wang, J. Stuckler, and D. Cremers, “Deep virtual stereo odometry: Leveraging deep depth prediction for monocular direct sparse odometry,” in _Proceedings of the European conference on computer vision (ECCV)_ , 2018, pp. 817–833.
  * [12] A. Saxena, S. Chung, and A. Ng, “Learning depth from single monocular images,” _Advances in neural information processing systems_ , vol. 18, 2005\.
  * [13] D. Eigen, C. Puhrsch, and R. Fergus, “Depth map prediction from a single image using a multi-scale deep network,” _Advances in neural information processing systems_ , vol. 27, 2014.
  * [14] L. Wang, M. Famouri, and A. Wong, “Depthnet nano: A highly compact self-normalizing neural network for monocular depth estimation,” _arXiv preprint arXiv:2004.08008_ , 2020.
  * [15] S. F. Bhat, I. Alhashim, and P. Wonka, “Adabins: Depth estimation using adaptive bins,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2021, pp. 4009–4018.
  * [16] H. Fu, M. Gong, C. Wang, K. Batmanghelich, and D. Tao, “Deep ordinal regression network for monocular depth estimation,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2018, pp. 2002–2011.
  * [17] C. Zhao, Q. Sun, C. Zhang, Y. Tang, and F. Qian, “Monocular depth estimation based on deep learning: An overview,” _Science China Technological Sciences_ , vol. 63, no. 9, pp. 1612–1627, 2020.
  * [18] J. Jeong and A. Kim, “Adaptive inverse perspective mapping for lane map generation with slam,” in _2016 13th International Conference on Ubiquitous Robots and Ambient Intelligence (URAI)_. IEEE, 2016, pp. 38–41.
  * [19] J. Lin and J. Peng, “Adaptive inverse perspective mapping transformation method for ballasted railway based on differential edge detection and improved perspective mapping model,” _Digital Signal Processing_ , vol. 135, p. 103944, 2023.
  * [20] Z. Wang, X. Wu, Y. Yan, C. Jia, B. Cai, Z. Huang, G. Wang, and T. Zhang, “An inverse projective mapping-based approach for robust rail track extraction,” in _2015 8th International Congress on Image and Signal Processing (CISP)_. IEEE, 2015, pp. 888–893.
  * [21] W. Yang, B. Fang, and Y. Y. Tang, “Fast and accurate vanishing point detection and its application in inverse perspective mapping of structured road,” _IEEE Transactions on Systems, Man, and Cybernetics: Systems_ , vol. 48, no. 5, pp. 755–766, 2016.
  * [22] C. Zhong, P. You, X. Chen, H. Zhao, F. Sun, G. Zhou, X. Mu, C. Gan, and W. Huang, “Snake: Shape-aware neural 3d keypoint field,” _Advances in Neural Information Processing Systems_ , vol. 35, pp. 7052–7064, 2022.
  * [23] Y. Wu, R. Li, Z. Qin, X. Zhao, and X. Li, “Heightformer: Explicit height modeling without extra data for camera-only 3d object detection in bird’s eye view,” _arXiv preprint arXiv:2307.13510_ , 2023.
  * [24] M. Poggi, F. Aleotti, F. Tosi, and S. Mattoccia, “Towards real-time unsupervised monocular depth estimation on cpu,” in _2018 IEEE/RSJ international conference on intelligent robots and systems (IROS)_. IEEE, 2018, pp. 5848–5854.
  * [25] D. Wofk, F. Ma, T.-J. Yang, S. Karaman, and V. Sze, “Fastdepth: Fast monocular depth estimation on embedded systems,” in _2019 International Conference on Robotics and Automation (ICRA)_. IEEE, 2019, pp. 6101–6108.
  * [26] J. Liu, Q. Li, R. Cao, W. Tang, and G. Qiu, “Mininet: An extremely lightweight convolutional neural network for real-time unsupervised monocular depth estimation,” _ISPRS Journal of Photogrammetry and Remote Sensing_ , vol. 166, pp. 255–267, 2020.
  * [27] R. P. Padhy, S. Verma, S. Ahmad, S. K. Choudhury, and P. K. Sa, “Deep neural network for autonomous uav navigation in indoor corridor environments,” _Procedia computer science_ , vol. 133, pp. 643–650, 2018.
  * [28] R. P. Padhy, S. Ahmad, S. Verma, S. Bakshi, and P. K. Sa, “Localization of unmanned aerial vehicles in corridor environments using deep learning,” in _2020 25th International Conference on Pattern Recognition (ICPR)_. IEEE, 2021, pp. 9423–9428.
  * [29] G. Ge, Y. Zhang, W. Wang, L. Hu, Y. Wang, and Q. Jiang, “Visual-feature-assisted mobile robot localization in a long corridor environment,” _Frontiers of Information Technology & Electronic Engineering_, vol. 24, no. 6, pp. 876–889, 2023.
  * [30] R. Wang, J. Qin, K. Li, and D. Cao, “Bev lane det: Fast lane detection on bev ground,” _arXiv preprint arXiv:2210.06006_ , 2022.
  * [31] A. Geiger, P. Lenz, C. Stiller, and R. Urtasun, “Vision meets robotics: The kitti dataset,” _The International Journal of Robotics Research_ , vol. 32, no. 11, pp. 1231–1237, 2013.
  * [32] N. Silberman, D. Hoiem, P. Kohli, and R. Fergus, “Indoor segmentation and support inference from rgbd images,” in _Computer Vision–ECCV 2012: 12th European Conference on Computer Vision, Florence, Italy, October 7-13, 2012, Proceedings, Part V 12_. Springer, 2012, pp. 746–760.


# GENERALIZED WATSON TRANSFORMS II: THE COMPLEMENTARY SERIES OF
$\boldsymbol{GL(2,\mathbb{R})}$

Qifu Zheng  Department of Mathematics and Statistics, The College of New
Jersey, Ewing, NJ 08618<EMAIL_ADDRESS>
###### Abstract.

We apply the theory of generalized Watson transforms developed in [5] to
construct the complementary series of $GL(2,\mathbb{R})$.

###### Key words and phrases:

Complementary series, unitary representations, Watson transform.

###### 2010 Mathematics Subject Classification:

Primary 22E30, 43A32, 44A15; Secondary 43A65, 42A38

##  1\. Introduction

This article is the second in a series of articles in which we develop the
theory of generalized Watson transforms and make applications of those results
to the representation theory of the general linear groups over $\mathbb{R}$.
It is well known [1, 2] that the irreducible representations of
$GL(2,\mathbb{R})$, the general linear group of $2\times 2$ real matrices, are
classified according to three distinct constructions: (1) The principal series
are usually constructed by unitary induction from its parabolic subgroup. (2)
The complementary series are constructed by a form of analytic continuation
from the principal series. (3) The (relative) discrete series are usually
constructed in spaces of holomorphic functions on the unit disk or upper half
complex plane.

On the other hand, by applying the results developed in [3], we can obtain all
three series using the method of generalized Watson transforms. That this
method is able to achieve these results is due to the fact that the group
$G=GL(2,\mathbb{R})$ is generated by its upper triangular (Borel) subgroup $Q$
and the Weyl reflection,

(1.1) $p=\begin{bmatrix}0&1\\\ -1&0\\\ \end{bmatrix}$

then, any irreducible unitary representation $\pi$ of $G$ is determined by its
restriction to $Q$ and $p$, which in fact corresponding to a generalized
Watson transform. In this paper, we will illuminate this approach by applying
the method of generalized Watson transforms to construct the complementary
series of $G$, and in a subsequent article [6], we will use the generalized
Watson transform method to construct unitary representations of higher rank
groups.

This paper is organized as follows: In Section 2, we review briefly some
concepts and theorems related to generalized Watson transforms from [5]. In
Section 4, we will describe the subgroups of $G$ and its non-unitary
representations realized on the Hilbert space
$L^{2}(\mathbb{R},(1+x^{2})^{s}dx)$. Then, in Section 3 we will use Pitt’s
theorem [3] to realize the representations on the space
$H_{s}=L^{2}(\mathbb{R},|x|^{-s}dx)$ where $0<s<1$. Finally, in Section 5 we
will show that the representations realized on $H_{s}$ in Section 4 are
unitary.

##  2\. Some remarks on the generalized Watson transforms

Let $G_{0}$ be a topological group, $R$ and $L$ be unitary representations of
$G_{0}$ on a Hilbert space $H$, and let $I$ denote the identity operator on
$H$. A unitary operator $W$ that intertwines $R$ and $L$ is called a
generalized Watson transform with respect to $R$ and $L$ if $W^{2}=I$. The
operator $W$ is called a generalized skew Watson transform with respect to $R$
and $L$ if $W^{2}=-I$. The results in [5] provide several theorems on the
construction of generalized Watson transforms. Here, we list one corollary
that is needed in the proof of the unitarity of the complementary series.

######  Proposition 2.1.

(Zheng [5]) Suppose that $G_{0}$ is Abelian, let $R$ be a unitary
representation of $G_{0}$ on a Hilbert space $H$, and set $L(g)=R(g^{-1})$ for
all $g\in G_{0}$. For $\phi\in H$, suppose that
$\phi^{\circ}=\left\\{R(g)\phi:g\in G_{0}\right\\}$ spans a dense subspace of
$H$. Then there exists a generalized Watson transform $W$ on $H$ with respect
to $R$ and $L$ such that $W\phi=\pm\phi$ if and only if
$\langle\phi|R(g)\phi\rangle$ is real for all $g\in G_{0}$.

##  3\. Subgroups of $G$ and its non-unitary representations

Denote the elements of $G$ by

$g=\begin{bmatrix}a&c\\\ b&d\\\ \end{bmatrix},$

where $a,b,c,d\in\mathbb{R}$ and $ad-bc\neq 0$. Let $Q$ be the full upper-
triangular subgroup of matrices

(3.1) $q:=q(a,c,d)=\begin{bmatrix}a&c\\\ 0&d\\\ \end{bmatrix},$

and $Q^{t}$ be the analogous full lower-triangular subgroup. Then $Q$ is the
semi-direct product of the normal Abelian subgroup $N$ of unipotent matrices

(3.2) $n:=n(c)=\begin{bmatrix}1&c\\\ 0&1\\\ \end{bmatrix},$

$c\in\mathbb{R}$, and the diagonal subgroup $D$ of matrices

(3.3) $\gamma(a,d)=\begin{bmatrix}a&0\\\ 0&d\\\ \end{bmatrix},$

with $a\neq 0,d\neq 0$. The matrix $p$ in (1.1), called the Weyl element,
plays a special role in the representation theory of $G$. Since $p^{2}=-I$,
the generalized Watson transforms are operators associated by representations
to $p$. This explains the importance of generalized Watson transforms in the
representation theory of $G$, and more generally, of reductive Lie groups.

The following result is well known [2].

######  Proposition 3.1.

The subgroups $N$ and $D$ and the Weyl element $p$ generate the group $G$.

The proof of the above result rests on two identities. First, when $b=0$,

$g=\begin{bmatrix}a&c\\\ 0&d\\\
\end{bmatrix}=q(a,c,d)=\gamma(a,d)\,n(\frac{c}{a});$

and when $b\neq 0$,

$g=\begin{bmatrix}a&c\\\ b&d\\\
\end{bmatrix}=n(a/b)\,p\,\gamma(-b,-(\det{g})/b)\,n(d/b).$

Therefore, any representation of the group $G$ is completely determined by its
restrictions to $N,D$ and $p$.

For $d\in\mathbb{R}$, let $\sigma(d)$ denote the sign of $d$; also, let
$\epsilon$ equal 0 or 1. Starting from the one-dimensional character of
$Q^{t}$,

$\tau_{s}\left(\begin{bmatrix}a&0\\\ b&d\\\
\end{bmatrix}\right)=\sigma(d)^{\epsilon}|a|^{-(\Re(s)+1)/2}|d|^{(\Re(s)+1)/2}$

we can obtain the bounded representation $\pi_{s}$, of $G$ on the weighted
norm space $V_{s}=L^{2}(\mathbb{R},(1+x^{2})^{\Re(s)}dx)$, defined by

(3.4) $(\pi_{s}(g)f)(x)=(\sigma(\det
g))^{\epsilon}(\sigma(\det(a+xb)))^{\epsilon}\\\ \times|\det
g|^{(\Re(s)+1)/2}|\det(a+xb)|^{-(\Re(s)+1)}f\left(\frac{c+xd}{a+xb}\right),$

The representation $\pi_{s}$ on $V_{s}$ is unitary if and only if $s$ is pure
imaginary, i.e., $\Re(s)=0$. Since in this article we concentrate on the
complementary series, we will only consider the case in which $\epsilon=0$ and
$0<s<1$, and in that case, (3.4) becomes

(3.5) $(\pi_{s}(g)f)(x)=|\det
g|^{(s+1)/2}|\det(a+xb)|^{-(s+1)}f\left(\frac{c+xd}{a+xb}\right)$

##  4\. New realizations of non-unitary representations of $G$

In this section, we will make use of a well-known theorem of Pitt [3] to
realize the non-unitary representation $\pi_{s}$ on the weighted norm space
$H_{s}=L^{2}(\mathbb{R},|x|^{s}dx)$ when $0<s<1$. We first state a special
case of Pitt’s theorem and refer readers to Stein [4] for more general
versions of Pitt’s theorem.

######  Theorem 4.1.

(Pitt [3]) Let $s\in(0,1)$ and let $\hat{f}$ denote the Fourier transform of a
function $f:\mathbb{R}\to\mathbb{C}$. Then there exists a constant $C$ such
that

(4.1) $\int_{-\infty}^{\infty}|\hat{f}(x)|^{2}|x|^{-s}dx\leq
C\int_{-\infty}^{\infty}|\hat{f}(x)|^{2}|x|^{s}dx$

for every function $f\in H_{s}$ for which the integral on the right side of
(4.1) is convergent.

By applying Proposition 4.1, we can define in terms of $\pi_{s}$ a
representation $\rho_{s}$ on $H_{s}$. Indeed, denoting by $\mathcal{F}$ the
Fourier transform, we have the following result.

######  Lemma 4.2.

Define, for $g\in G$, the operator

(4.2) $\rho_{s}(g)=\mathcal{F}\pi_{s}(g){\mathcal{F}}^{-1}.$

Then $\rho_{s}$ is a well-defined representation of $G$ on $H_{s}$, $0<s<1$.

###### Proof.

By Pitt’s theorem, there exists a constant $C>0$ such that for any $f\in
V_{s}$

$\displaystyle\int_{-\infty}^{\infty}|(\mathcal{F}{f})(x)|^{2}\,|x|^{-s}dx$
$\displaystyle\leq C\int_{-\infty}^{\infty}|f(x)|^{2}\,|x|^{s}dx$
$\displaystyle\leq C\int_{-\infty}^{\infty}|f(x)|^{2}|1+x^{2}|^{s}dx,$

$0<s<1$. Hence, the Fourier transform $\mathcal{F}$ is a continuous map from
$V_{s}$ to $H_{s}$, and this shows that $\mathcal{F}V_{s}\subset H_{s}$.

Define the functions $\lambda,\mu:\mathbb{R}\to\mathbb{R}$ by
$\lambda(x)=e^{-x^{2}/2}$ and $\mu(x)=xe^{-x^{2}/2}$. Then it is a simple
calculation to show that both $\lambda$ and $\mu$ are in $V_{s}$ and $H_{s}$,
and also that $\mathcal{F}\lambda=\mu,\mathcal{F}\mu=-i\mu$. Define a
homomorphism $T$ of $\mathbb{R}^{\times}$, the multiplicative group of non-
zero real numbers, by

$(T(a)f)(x)=a^{1/2}f(ax)$

for any $a\in\mathbb{R}^{\times}$ and any function $f$ on $\mathbb{R}$. Then
for $f\in V_{s}$, we obtain the relation,

(4.3) $\mathcal{F}T(a)=T(a^{-1})\mathcal{F}.$

By the uniqueness property of Laplace transform, it follows that if $f\in
V_{s}$ is such that for any $a\in\mathbb{R}^{\times}$

$\int_{-\infty}^{\infty}f(x)(T(a)\lambda)(x)(1+x^{2})^{s}dx=0$

and

$\int_{-\infty}^{\infty}f(x)(T(a)\mu)(x)(1+x^{2})^{s}dx=0$

then $f=0$. Hence,
$\hbox{Span}\\!\left\\{T(a)\lambda,T(a)\mu|a\in\mathbb{R}^{\times}\right\\}$
is a dense subspace of $V_{s}$. Similarly, the space generated by its image,
viz.,
$\hbox{Span}\\!\left\\{T(a^{-1})\lambda,T(a^{-1})\mu|a\in\mathbb{R}^{\times}\right\\}$,
is dense in $H_{s}$. Therefore, it follows from the continuity of
$\mathcal{F}$ that $\mathcal{F}V_{s}=H_{s}$.

Consequently, for any $f\in H_{s}$ and $g\in G$, we obtain
${\mathcal{F}}^{-1}f\in V_{s},$ $\pi_{s}(g){\mathcal{F}}^{-1}f\in V_{s},$ and
$\mathcal{F}\pi_{s}(g){\mathcal{F}}^{-1}f\in H_{s}.$ Therefore, we have proved
that $\rho_{s}(g)=\mathcal{F}\pi_{s}(g){\mathcal{F}}^{-1}$ is well-defined for
any $g\in G$. Finally, since $\pi_{s}$ is a representation of $G$ on $V_{s}$
then it follows immediately that $\rho_{s}$ also is a well-defined
representation of $G$ on $H_{s}.$ ∎

##  5\. Unitarity of the complementary series

Throughout this section, we will use the notation defined in (3.1)-(3.3) for
the elements of the subgroups of $G$. The main of this section is to establish
the following result.

######  Theorem 5.1.

For $s\in(0,1)$, the operator $\rho_{s}$ defined in (4.2) is a unitary
representation of $G$ on $H_{s}$.

Before embarking on the proof of Theorem 5.1, we shall establish several
preliminary results.

######  Lemma 5.2.

The subgroup $Q$ of upper-triangular subgroup preserves the norm of $H_{s}$
when $\rho_{s}$ is restricted to $Q$.

###### Proof.

Notice that if $q=q(a,c,d)\in Q$ then, by $(\ref{special})$,

(5.1) $(\pi_{s}(q)f)(x)=|a|^{-(s+1)/2}|d|^{(s+1)/2}f(a^{-1}(c+xd)).$

Applying the Fourier transform, we obtain

$\displaystyle(\rho_{s}(q)f)(x)$
$\displaystyle=(\mathcal{F}\pi_{s}(q)\mathcal{F}^{-1}f)(x)$
$\displaystyle=e^{icd^{-1}x}|a|^{(1-s)/2}|d|^{(s-1)/2}f(d^{-1}xa).$

Hence the result follows by a simple calculation. ∎

Define $R(\gamma)=\rho_{s}(\gamma)$ for $\gamma=\gamma(a,d)\in D$. Then, it
follows from the above lemma that $R$ is a unitary representation of $D$ on
$H_{s}$ and

$(R(\gamma)f)(x)=|a|^{(1-s)/2}|d|^{(s-1)/2}f(d^{-1}xa).$

######  Lemma 5.3.

For $s\in\mathbb{C}$ such that $0<\Re(s)<1$, define

(5.2) $\phi_{s}(x)=\int_{-\infty}^{\infty}e^{-ixy}\,(1+y^{2})^{-(s+1)/2}\,dy.$

Then $\phi_{s}\in H_{s}$, and the set
${\mathbf{\Phi}}^{\circ}=\\{R(\gamma)\phi_{s}:\gamma\in D\\}$ spans a dense
subspace of $H_{s}^{+}=\\{f\in H_{s}:f\text{ is even}\\}$.

###### Proof.

Since $0<\Re(s)<1$ then the functions $(1+y^{2})^{-(s+1)/2}$ and
$(1+y^{2})^{-(s+1)}$ are elements of $L^{1}(\mathbb{R})$. Therefore, by the
$L^{1}$ and $L^{2}$ properties of the Fourier transform, it follows that
$\phi_{s}\in L^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$, and

$\displaystyle\langle\phi_{s}|\phi_{s}\rangle_{s}$
$\displaystyle=\int_{-\infty}^{\infty}|\phi_{s}(x)|^{2}|x|^{-s}dx$
$\displaystyle=\int_{|x|\leq
1}|\phi_{s}(x)|^{2}|x|^{-s}dx+\int_{|x|>1}|\phi_{s}(x)|^{2}|x|^{-s}dx$
$\displaystyle\leq\|\phi_{s}\|_{\infty}\int_{|x|\leq
1}|x|^{-s}dx+\int_{|x|>1}|\phi_{s}(x)|^{2}dx$ $\displaystyle\leq
2(1-s)^{-1}\|\phi_{s}\|_{\infty}+\|\phi_{s}\|_{2}^{2}.$

where $\|\phi_{s}\|_{\infty}$ and $\|\phi_{s}\|_{2}$ are the norms of
$\phi_{s}$ in $L^{\infty}(\mathbb{R})$ and $L^{2}(\mathbb{R})$, respectively.
Therefore $\phi_{s}\in H_{s}$, and

$\displaystyle\phi_{s}(x)$
$\displaystyle=\int_{-\infty}^{\infty}e^{-ixy}\,(1+y^{2})^{-(s+1)/2}\,dy$
$\displaystyle=\frac{1}{\Gamma((1+s)/2)}\int_{-\infty}^{\infty}e^{-ixy}\left[\int_{0}^{\infty}\xi^{(s-1)/2}e^{-\xi(1+y^{2})}d\xi\right]dy.$

As these integrals are absolutely convergent, we now apply Fubini’s theorem to
reverse the order of integration; then the inner integral with respect to $y$
is seen to be the Fourier transform of the Gaussian; on evaluating that
integral we obtain

$\displaystyle\phi_{s}(x)$
$\displaystyle=\frac{\sqrt{\pi}}{\Gamma((1+s)/2)}\int_{0}^{\infty}\xi^{(s/2)-1}e^{-\xi}e^{-x^{2}/4\xi}d\xi$
(5.3)
$\displaystyle=\frac{\sqrt{\pi}}{\Gamma((1+s)/2)}\,|x/2|^{s/2}\int_{0}^{\infty}\xi^{(s/2)-1}\exp(-|x|(\xi+\xi^{-1})/2)d\xi.$

In order to prove that $\Phi^{\circ}$ spans a dense subspace in $H_{s}^{+}$,
it suffices to show that if $f\in H_{s}$ is even and is such that $\langle
R(\gamma(a,1))\phi_{s}|f\rangle_{s}=0$ for any $a\in\mathbb{R}^{\times}$ then
$f=0$, almost everywhere. It follows from the condition $\langle
R(\gamma(a,1))\phi_{s}|f\rangle=0$ that

$\int_{-\infty}^{\infty}|a|^{(1-s)/2}\phi_{s}(xa)\overline{f(x)}|x|^{-s}dx=0$

and that

$\int_{-\infty}^{\infty}\phi_{s}(xa)\overline{f(x)}|x|^{-s}dx=0$

for all $a\in\mathbb{R}^{\times}$. From (5), we have

$\phi_{s}(xa)=\frac{\sqrt{\pi}}{\Gamma((1+s)/2)}|xa/2|^{s/2}\int_{0}^{\infty}\xi^{(s/2)-1}\exp(-|xa|(\xi+\xi^{-1})/2)d\xi,$

and replacing $\xi$ by $\xi/|x|$ in the latter integral, we obtain

$\phi_{s}(xa)=\frac{\sqrt{\pi}}{\Gamma((1+s)/2)}|a|^{s}\int_{0}^{\infty}\xi^{\frac{s}{2}-1}\exp\Big{(}-|a|\xi-\frac{x^{2}}{4\xi}\Big{)}d\xi.$

Therefore,

$\int_{-\infty}^{\infty}\overline{f(x)}|x|^{-s}\left[\int_{0}^{\infty}\xi^{\frac{s}{2}-1}\exp\Big{(}-|a|\xi-\frac{x^{2}}{4\xi}\Big{)}d\xi\right]dx=0$

for all $a\in\mathbb{R}^{\times}$. Again applying Fubini’s theorem to
interchange the order of integration, we obtain

$\int_{0}^{\infty}\xi^{(s/2)-1}\left[\int_{-\infty}^{\infty}\overline{f(x)}|x|^{-s}\exp(-x^{2}/4\xi)dx\right]e^{-|a|\xi}d\xi=0.$

As the latter integral is a Laplace transform, it follows that

$\int_{-\infty}^{\infty}\overline{f(x)}|x|^{-s}\exp(-x^{2}/4\xi)dx=0$

$\xi$-almost everywhere. Since $f$ is even, it follows that

$\int_{0}^{\infty}\overline{f(x)}|x|^{-s}\exp(-x^{2}/4\xi)dx$

almost everywhere in $\xi>0$; equivalently,

$\int_{0}^{\infty}\overline{f(\sqrt{x})}|x|^{-(s+1)/2}\exp(-x/4\xi)dx=0$

almost everywhere in $\xi>0$. Hence, the Laplace transform of
$\overline{f(\sqrt{x})}|x|^{-(s+1)/2}$ is zero almost everywhere on
$\mathbb{R}^{\times}$. Therefore $\overline{f(\sqrt{x})}=0$ for almost every
$x>0$, which implies that $f(x)=0$,  a.e.  $x>0$. Because $f$ is even, we
deduce that $f(x)=0$ a.e. This completes the proof of the Lemma. ∎

######  Lemma 5.4.

For $0<\Re(s)<1$, define

(5.4) $\psi_{s}(x)=\int_{-\infty}^{\infty}ye^{-ixy}(1+y^{2})^{-(s+1)/2}dy$

Then $\psi_{s}\in H_{s}$, and the set
${\mathbf{\Psi}}^{\circ}=\left\\{R(\gamma)\psi_{s}:\gamma\in D\right\\}$ spans
a dense subspace of $H_{s}^{-}=\left\\{f\in H_{s}:f\text{ is odd}\right\\}$.

The proof of this result is similar to that of Lemma 5.3.

Proof of Theorem 5.1. By Lemma 5.2 it follows that $\rho_{s}$ is unitary when
restricted to $Q$. Also, by Lemma 3, we need to prove that $\rho_{s}(p)$ is a
unitary operator on $H_{s}$.

Denoting $\rho_{s}(p)$ by $W$, it is a straightforward calculation to verify
the following properties for $W$:

(i) $W^{2}=I$, the identity operator of $H_{s}$.

(ii) $WR(\gamma)=R(\gamma^{-1})W$ for $\gamma=\gamma(a,d)\in D$.

(iii) $W\phi_{s}=\phi_{s}$.

It is also a simple calculation to verify that
$\int_{-\infty}^{\infty}\phi_{s}(x)\overline{(R(\gamma)\phi_{s})(x)}|x|^{-s}$
is real for any $\gamma\in D$. Hence by Proposition 2.1 and Lemma 5.3, $W$ is
a generalized Watson transform of $H_{s}^{+}$ with respect to the unitary
representations $R(\gamma)$ and $R(\gamma^{-1})$. Therefore, $W$ is unitary on
$H_{s}^{+}$.

Similarly, from Proposition 2.1 and Lemma 5.4, it follows that $W$ is unitary
on $H_{s}^{-}$. Consequently, $W$ is unitary on $H_{s}=H_{s}^{+}\oplus
H_{s}^{-}$. This completes the proof of the unitarity of the complementary
series of $G$. ∎

## Acknowledgements

I wish to express my deep gratitude to my research advisors, the late
Professor Kenneth I. Gross (University of Vermont) and the late Professor Ray
A. Kunze (University of Georgia), who introduced me to the subject of group
representations and suggested that I study the generalized Watson transforms.
I also thank Donald Richards (Penn State University) for his continual
patience and encouragement. Their help and support have been invaluable.

## References

  * [1] V. Bargmann, Representations of the Lorentz group, Ann. Math., 48 (1947), 568–640.
  * [2] A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, Princeton, NJ, 1986.
  * [3] H. R. Pitt, A note on bilinear forms, J. London Math. Soc., 11 (1936), 174–180.
  * [4] E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc., 83 (1956), 482–492.
  * [5] Q. Zheng, Generalized Watson transforms I: General theory, Proc. Amer. Math. Soc., 128 (2000), 2777–2787.
  * [6] Q. Zheng, Generalized Watson transforms III: Hankel transforms on symmetric cones, in preparation.