analyze whether this research proposal is coherent, novel and logical. consider whether is actually viable topic for masters thesis.

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\title{Dimensionality Reduction for Efficient 3-DoF Orientation Estimation: A General Framework}
\author{Gilad Freidkin}
\date{17.09.2025}

\begin{document}

\maketitle

\section{Introduction}
This research proposal introduces a novel framework for enhancing the efficiency and accuracy of 3-DoF (Degrees of Freedom) orientation estimation from 2D RGB images. Using rotation-invariant functions, we reduce the dimensionality of this problem, potentially leading to significant improvements in computational efficiency and memory utilization across various orientation detection methods.

The core innovation of our approach lies in its ability to transform a complex 3D optimization problem into a series of lower-dimensional optimizations. This not only promises to enhance the performance of existing methods, but also opens up new possibilities for more accurate and resource-efficient orientation estimation techniques.

\section{Background and Existing Approaches}
3-DoF orientation estimation from 2D images is a crucial task in computer vision, with applications that span robotics, autonomous systems, and industrial automation. The challenge lies in accurately inferring the full 3D orientation of an object given only its 2D projection in an image, a problem complicated by the inherent information loss in the projection process. Current approaches to this problem can be broadly categorized into three main types:

\begin{enumerate}
    \item \textbf{Template-Matching Methods:} These methods compare the input image against a pre-computed set of features (templates) with known ground-truth orientations. Typically, these features are extracted from rendered views of a CAD model of the object at different orientations. An example of this approach is presented in \cite{sundermeyer2018implicit}, where a deep learning model is used to encode orientation information in its latent space. Figure \ref{template-matching-image} depicts the said structure. 

    \item \textbf{Refinement-Based Methods:} These approaches start with an initial orientation estimate and iteratively refine it, usually using NN (neural-networks). Unlike template-matching methods where computations are performed offline, refinement-based methods often involve online rendering and comparison. Examples include DeepIM \cite{li2018deepim} and RNNPose \cite{wen2020rnnpose}.

    \item \textbf{Direct-Search Approaches:} These methods perform a direct search over the orientation space, comparing a similarity metric between the input image and candidate renders. In a recent academic project supervised by Dr. Sarah Keren, we explored a direct-search method using Black-Box optimization algorithms.
\end{enumerate}

\begin{figure*}
    \centering
    \includegraphics[width=1.0\linewidth]{images/test_pipeline.pdf}
    \caption{Image from \cite{sundermeyer2018implicit}. A general diagram of the template-based method proposed in the paper. The input image latent space is compared to an offline template-bank of latent spaces, calculated from different orientations of the object. }
    \label{template-matching-image}
\end{figure*}

Each of these approaches faces significant challenges in computational complexity, memory requirements, and dealing with the non-smoothness of similarity functions in orientation space. Especially challenging are scenarios with real-time orientation detection requirements, for which only the most highly optimized algorithms are applicable. 

\section{Proposed Framework}

This framework introduces two key ideas to address these challenges: a canonization function and a rotation-invariant similarity function.

\subsection{Canonization Function}
We define a canonization function that returns a canonical representation of an image with respect to rotations around the central pixel. Basically, this function normalizes the rotation of the input image. 

Let $rotate(x, \theta)$ be a function that rotates image $x$ clockwise by $\theta$ degrees around its center.
Formally, for any input image $x$ and angle $\theta \in [0, 2\pi]$ we demand:

\begin{equation*}
    canonize(rotate(x, \theta)) = canonize(x)
\end{equation*}

This process effectively reduces the problem from three dimensions to two by eliminating the need to search across one axis of rotation. A potential implementation of this function could use image moments to determine a consistent orientation for any input image.

\subsection{Rotation-Invariant Similarity Function}
We propose a similarity function that compares images in a rotation-invariant manner. Formally, for any input image $x$, CAD model rendering $y$, and angle $\theta \in [0, 2\pi]$:

\begin{equation*}
    sim(x, rotate(y, \theta)) = sim(x, y)
\end{equation*}

This function allows for consistent comparison of the input image with candidate render images regardless of their rotation around the central axis, hence removing one-degree of freedom of the object's orientation. One way to implement this could be by transforming the images into polar coordinates, calculating the shift in the angle dimension using phase cross-correlation, correcting for this shift, and then applying a standard image similarity measure like SSIM or MSE \cite{Reddy1996AnFT}, \cite{scikit-image}.

\section{Application to Existing Methods}
Our framework can be applied to enhance various existing orientation detection methods.
Throughout this section, we describe the orientation space using the Euler angle representation of rage $\Theta = [0, 2\pi] \times [0, 2\pi] \times [0, \pi]$. W.L.G we assume that the $rotate$ function affects only the first dimension of the Euler-angle representation. 

\subsection{Template-Matching Methods}
In the original approach, templates are created from orientations sampled in the entire 3D space $\Theta$. Assuming $N$ samples per dimension, the original template bank size is of size $O(N^3)$. By applying the canonization function to both input images and templates:

\begin{itemize}
    \item Every two templates, resulting from images $y, y'$ with orientations differing only at the first dimension, can be replaced by a single template of $canonize(y)$. Hence, the search space is reduced from 3D to 2D.
    \item Allows to decrease the feature bank size to $O(N^2)$ without losing any coverage.
    \item Subsequent 1D dynamic search is required at the remaining dimension.
\end{itemize}

An alternative approach is to use rotation-invariant template calculation function, resulting in the same benefits as described above, while also removing the need of canonization altogether.

\subsection{Refinement-Based Methods}
In traditional refinement approaches, at each iteration, a CAD model $y$ is rendered in some orientation $(\theta_1, \theta_2, \theta_3) \in \Theta$ and compared to the input image $x$. With our framework:

\begin{itemize}
    \item We can compare $canonize(x)$ and $canonize(y)$, effectively removing a dimension from the refinement procedure.
    \item Instead of refining the whole 3D orientation $(\theta_1, \theta_2, \theta_3)$, we only need to refine the $\theta_2, \theta_3$ components. A subsequent refinement of the first component $\theta_1$ is required. 
\end{itemize}

Alternatively, rotation-invariant NN \cite{mo2022riccnnrotationinvariantcoordinateconvolutional} \cite{worrall2017harmonicnetworksdeeptranslation} could be explored to refine orientation coordinates in the first two dimensions, removing the need for canonization altogether since the invariance is built-in to the architecture of the network.

\subsection{Direct-Search Approaches}
In the original direct search method, we optimize the similarity in the 3D orientation space $\Theta$. Using our rotation-invariant similarity function $sim$:

\begin{itemize}
    \item We can reduce the search space to $[0, 2\pi] \times [0, \pi]$. For each orientation $(0, \theta_2, \theta_3)$ we render the matching CAD model $y$ and compare it to the input image $x$ using $sim(x, y)$.
    \item This results in reduction from a 3D to a 2D search space. A consequent 1D search is required to find the $\theta_1$ component.

\end{itemize}


\section{Novelty and Potential Impact}
Our framework offers several novel contributions that distinguish it from existing approaches in the field of 3-DoF orientation estimation.

\begin{enumerate}
    \item \textbf{Systematic Dimensionality Reduction:} We provide a general approach to reduce the 3D orientation problem to lower-dimensional optimizations.  Unlike previous methods that work directly in the 3D space, our approach systematically decomposes the problem into lower-dimensional sub-problems. This decomposition is achieved through the novel use of rotation-invariant functions, a concept not previously applied in this context. In many cases solving two consecutive lower-dimension problems is more efficient than solving a higher dimensional optimization task.

    \item \textbf{Efficiency Improvements:} We offer the potential for exponential reductions in template bank sizes and significant computational efficiency gains across different methods. For template-matching methods, our approach reduces the template bank size from $O(N^3)$ to $O(N^2)$. In refinement-based methods, we reduce the search space dimensionality, potentially leading to faster convergence and reduction in the total number of renders. For direct-search approaches, the dimensionality reduction results in convergence in exponentially smaller amount of steps for classical optimization algorithms.

    \item \textbf{Improved Accuracy:} By allowing for larger template banks or more refined searches within the same computational budget, our approach may lead to improved accuracy in orientation estimation. The reduced computational complexity allows for more comprehensive searches or larger template banks, potentially improving accuracy without increasing computational cost.

    \item \textbf{Generalized Approach:} Our method is applicable to a wide range of existing orientation detection techniques. The framework acts as a meta-algorithm, enhancing various existing methods rather than replacing them. It provides a unified way to think about and improve orientation estimation across different paradigms. Additionally, many 6DoF pose estimation algorithms separate the orientation detection and spatial position estimation into two different parts. Hence, the described technique could be applied in these cases also. 

\end{enumerate}

\section{Research Challenges and Methodology}


Despite the promise of the proposed framework, several challenges must be addressed for successful implementation. Developing a computationally efficient canonization function that can be applied in real time without introducing prohibitive computational overhead remains a critical goal. Similarly, ensuring that the similarity function can operate efficiently without compromising accuracy presents its own difficulties. 

A proposed evaluation methodology includes integrating this framework into well-established algorithms and comparing them with their original implementation. Key performance metrics will include computational efficiency and accuracy. Improving existing algorithms, or even achieving state-of-the-art performance or accuracy metrics would count as success. 

\section{Feasibility and Preliminary Results}
The proposed research is feasible for a master's thesis due to several factors:

\begin{enumerate}
    \item \textbf{Preliminary implementations:} Canonization function using image moments and similarity function using polar coordinate transformation have been developed.
    \item \textbf{Successful integration:} The similarity function has been successfully integrated into a direct-orientation search algorithm.
    \item \textbf{Built on existing work:} The research builds upon well-established approaches while introducing significant novel elements.
\end{enumerate}

Initial, non-optimized integration of the similarity function into the direct-search algorithm has shown competitive results with the original implementation. Although it merely serves as proof-of-concept for the feasibility of our approach, we acknowledge that the actual implementations of the canonization and alignment functions may change drastically as we optimize for performance and accuracy.

\section{Conclusion}
This research proposes a general framework to enhance 3-DoF orientation detection methods using rotation-invariant functions. By reducing the dimensionality of the problem, we aim to significantly improve the efficiency and potentially the accuracy of existing approaches. The framework's broad applicability and potential for substantial performance improvements make it a promising direction for advancing the field of 3D orientation estimation.

The combination of theoretical innovation, practical implementations, and a potential wide-ranging impact makes this an exciting and valuable area of research. Success in this project could lead to more efficient and accurate orientation estimation techniques, with broad implications for fields such as robotics and computer vision.

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