Mastering 3D Spaces: A Comprehensive Guide to Coordinate System Conversions in OpenCV, COLMAP, PyTorch3D, and OpenGL

Photo by CHUTTERSNAP on Unsplash.

Introduction

Navigating through the world of 3D frameworks like OpenCV, COLMAP, PyTorch3D, and OpenGL can be daunting due to their different coordinate systems. This article simplifies these complexities by comparing their coordinate systems and providing practical guides for converting between them. Whether you’re a developer, researcher, or enthusiast, you’ll find essential insights and tools to seamlessly work across these diverse 3D environments. Let’s explore and understand these coordinate systems in a clear and straightforward way.

Understanding Coordinate Systems

Bridging the gap from 2D imaging to 3D modeling necessitates a solid grasp of coordinate systems, which provide a structural framework for object orientation in a three-dimensional context. Such systems are defined by three principal axes:

• X-Axis: This axis determines lateral placement, extending from left to right.
• Y-Axis: It controls vertical positioning, which can be upwards or downwards.
• Z-Axis: The axis of depth, it projects forward or backward from the point of view.

The origin, where these axes intersect, is the pivotal reference that anchors all spatial measurements.

Aligning objects correctly across various frameworks such as OpenCV, COLMAP, PyTorch3D, and OpenGL is contingent on the orientation of these axes. A vertical object aligned with a positive Y-axis in one system may be inverted if the Y-axis is interpreted as negative in another, leading to a mirrored or upside-down model.

Understanding the unique axis conventions of each framework is imperative to ensure models are represented accurately and transformed correctly when transitioning between platforms. This consistency is key to seamless integration and manipulation of 3D models.

Below is an illustrative image of the 3D coordinate system:

This diagram delineates the standard orientation of the X, Y, and Z axes. It’s important to note that different 3D tools might alter these orientations, so verifying the specific conventions of each framework is crucial for precise model manipulation.

Overview of Major 3D Frameworks

Understanding the unique characteristics of coordinate systems across various 3D frameworks is key to effectively translating and manipulating models. Below, we provide a snapshot of the coordinate systems and typical applications for each major framework.

OpenCV

Pinhole camera model illustrating 3D to 2D projection in OpenCV, with the world coordinates (Xw, Yw, Zw) being mapped to image plane coordinates (u, v).

• Coordinate System Overview: OpenCV’s camera calibration routines assume a pinhole camera model and utilize a coordinate system where the X-axis extends to the right and the Y-axis down, following the usual image coordinate space. The Z-axis points out of the camera, towards the scene.
• Common Use Cases: OpenCV is widely used for a variety of computer vision tasks such as motion analysis, augmented reality, gesture recognition, and automated inspection. Its camera calibration capabilities are vital for applications requiring precise measurements of object sizes and positions in 3D space, such as robotics and advanced driver assistance systems (ADAS).

COLMAP

• Coordinate System Overview: In COLMAP, the camera coordinate system is set up with the X-axis pointing to the right, consistent with the image plane. However, the Y-axis points downward (to the bottom), diverging from some conventional graphics systems. The Z-axis extends to the front, as seen from the camera’s perspective. It’s akin to reaching out into the photo, moving deeper into the image.
• Common Use Cases: COLMAP is extensively used for photogrammetry applications, including 3D reconstruction from unordered image collections and Structure from Motion (SFM) tasks. Its unique coordinate system plays a critical role in accurately modeling and reconstructing three-dimensional environments from two-dimensional image data. This makes it a preferred choice in fields like archaeology, architecture, and virtual reality, where precise spatial measurements and reconstructions are key.

PyTorch3D

Note that z is going pointing directly into the page

• Coordinate System Overview: In PyTorch3D, the camera coordinate system is set up with the X-axis pointing to the left, the Y-axis up, and the Z-axis points into the screen.
• Common Use Cases: It is designed for deep learning with 3D data, including tasks like 3D shape prediction, rendering, and processing.

OpenGL

Think of your screen being the center of the 3 axes and the positive z-axis going through your screen towards you.

• Coordinate System Overview: OpenGL uses a right-handed system, with the X-axis pointing to the right, the Y-axis pointing up, and the Z-axis pointing out of the screen.
• Common Use Cases: OpenGL is widely used for rendering 2D and 3D vector graphics and is a core technology in many games, simulations, and graphical user interfaces.

Each of these frameworks has been developed with specific goals and applications in mind, which has influenced the design choices behind their coordinate systems. As a developer or 3D artist, it’s crucial to understand these differences to ensure that assets and data are accurately represented when transitioning between frameworks.

Navigating 3D Spaces: A Comparative Analysis of Axis Orientations Across Different Frameworks

In this comparative analysis, we explore the axis orientations in OpenCV, COLMAP, PyTorch3D, and OpenGL. The table below highlights the consistent direction of the X-axis in most frameworks, while showcasing the distinct Y and Z axis orientations. These differences are crucial in the realm of 3D modeling and visualization, directly influencing how objects are represented and manipulated across these platforms. For professionals in 3D spaces, understanding these variations ensures accurate model representations and smooth transitions between different software environments.

Comparing Axis Orientations: A Quick Reference to Coordinate Systems in OpenCV, COLMAP, PyTorch3D, and OpenGL.

Conversion Guides Between Frameworks

Transitioning between OpenCV, COLMAP, PyTorch3D, and OpenGL involves understanding and manipulating their coordinate systems. Let’s explore practical guides enhanced by Python functions for flipping axes, making these conversions efficient and accurate.

Understanding Rotation and Translation in 3D Frameworks

Before diving into conversions, let’s briefly touch upon rotation matrices and translation vectors. In 3D frameworks, rotation matrices define the orientation of an object, while translation vectors specify its position in space. For example, in COLMAP, the rotation matrix and translation vector dictate how a 3D point in the world is transformed to the camera’s coordinate system.

Applying Conversions: From COLMAP to PyTorch3D

Here are the Python functions designed to adjust rotation matrices and translation vectors for coordinate system conversions:

def flip_rotation_axes(rotation_matrix, flip_x=False, flip_y=False, flip_z=False):
    """
    Flip the specified axes of a 3x3 rotation matrix.

    Args:
    rotation_matrix (np.array): The original 3x3 rotation matrix.
    flip_x (bool): Whether to flip the X-axis.
    flip_y (bool): Whether to flip the Y-axis.
    flip_z (bool): Whether to flip the Z-axis.

    Returns:
    np.array: The rotation matrix after flipping the specified axes.
    """
    flipped_matrix = rotation_matrix.copy()

    if flip_x:
        flipped_matrix[1:3, :] = -flipped_matrix[1:3, :]

    if flip_y:
        flipped_matrix[[0, 2], :] = -flipped_matrix[[0, 2], :]

    if flip_z:
        flipped_matrix[:, [0, 1]] = -flipped_matrix[:, [0, 1]]

    return flipped_matrix

The flip_rotation_axes function flips each axis of a 3x3 rotation matrix by negating specific elements: flipping the X-axis inverts the second and third rows, flipping the Y-axis inverts the first and third rows, and flipping the Z-axis inverts the first two columns.

def flip_translation_vector(translation_vector, flip_x=False, flip_y=False, flip_z=False):
    """
    Flip the specified axes of a translation vector.

    Args:
    translation_vector (np.array): The original translation vector.
    flip_x (bool): Whether to flip along the X-axis.
    flip_y (bool): Whether to flip along the Y-axis.
    flip_z (bool): Whether to flip along the Z-axis.

    Returns:
    np.array: The translation vector after flipping the specified axes.
    """
    flipped_vector = translation_vector.copy()

    if flip_x:
        flipped_vector[0] = -flipped_vector[0]

    if flip_y:
        flipped_vector[1] = -flipped_vector[1]

    if flip_z:
        flipped_vector[2] = -flipped_vector[2]

    return flipped_vector

The flip_translation_vector function inverts selected axes of a translation vector. It selectively negates the components of the vector along the X, Y, or Z axes based on the provided boolean arguments, effectively mirroring the vector's position along the specified axes. This is useful for adapting spatial data to different coordinate systems in 3D environments.

Now, let’s apply our Python functions to a COLMAP to PyTorch3D conversion:

from collections import OrderedDict
from read_write_model import read_model, qvec2rotmat

colmap_path = "your_colmap_dataset_dir"
cameras, images, points_3D = read_model(path=f"{colmap_path}/sparse", ext=".bin")
images = OrderedDict(sorted(images.items()))
print(f"Number of images: {len(images)}")

# Choose your required image number from the available images
sample_image = 2 
colmap_image = images[sample_image]

# Extract rotation matrix 
R_colmap = qvec2rotmat(colmap_image.qvec)

# Converting COLMAP to PyTorch3D format
R_pt3d = flip_rotation_axes(R_colmap, flip_x=True, flip_y=True, flip_z=False)
R_pt3d = R_pt3d.T

# Extract translation vector
T_colmap = colmap_image.tvec

# Converting COLMAP to PyTorch3D format
T_pt3d = flip_translation_vector(T_colmap, flip_x=True, flip_y=True, flip_z=False)

The above code snippet demonstrates how to convert rotation matrices and translation vectors from COLMAP to PyTorch3D format. It starts by loading the COLMAP model using read_model and qvec2rotmat functions from COLMAP's official GitHub repository. The rotation matrix (R_colmap) is extracted from the COLMAP image and then converted to PyTorch3D's format using flip_rotation_axes. This function inverts the axes as required, and the matrix is transposed to conform to PyTorch3D's convention of using row vectors for matrix multiplication. Similarly, the translation vector (T_colmap) is extracted and converted using flip_translation_vector. This code is a practical example of adapting COLMAP's data for use in PyTorch3D, showcasing the necessary transformations for coordinate system compatibility.

Challenges and Considerations

When converting between different 3D frameworks, several challenges and considerations come into play:

1. Framework-Specific Requirements: Each framework has unique requirements. Reading official documentation is crucial to understand specifics like vector input formats for matrix multiplication, and whether the framework expects batch-shaped or non-batched inputs.
2. Scaling Differences: Be aware of different scaling conventions used in frameworks. A model’s scale in one framework might not directly translate to another, necessitating adjustments for consistent representation.
3. Data Loss and Accuracy: Conversion processes can lead to loss of precision or detail. Ensuring accuracy in transformation is vital, as small errors can significantly impact the 3D representation.
4. Testing and Validation: Rigorous testing is essential. Regularly validate conversions to ensure they meet the requirements of the target framework.
5. Orientation and Handedness: Pay attention to the orientation and handedness of the coordinate systems. Misalignment here can lead to flipped or inverted models.

Understanding and addressing these challenges is key to successful integration and manipulation of 3D data across diverse platforms.

Conclusion

Navigating the diverse coordinate systems of OpenCV, COLMAP, PyTorch3D, and OpenGL can be complex, but it’s a journey worth taking for anyone in the field of 3D modeling and computer vision. Understanding these differences is key to accurately transforming and integrating 3D data across various platforms. This guide has aimed to simplify these complexities, providing practical insights for conversions and highlighting the importance of attention to detail. Embrace these challenges as opportunities to deepen your understanding, and remember, the right conversions can unlock new dimensions in your 3D projects.

References and Further Reading

To deepen your understanding of 3D coordinate systems and their conversions across various frameworks, the following resources are invaluable:

1. OpenCV Documentation: A comprehensive guide for OpenCV’s functionalities and coordinate system conventions. OpenCV Documentation
2. COLMAP Documentation: Detailed information on COLMAP’s structure and coordinate system. COLMAP Documentation
3. PyTorch3D Documentation: For insights into PyTorch3D’s rendering and coordinate system. PyTorch3D Documentation
4. OpenGL Reference Pages: A useful resource for understanding OpenGL’s coordinate system and rendering techniques. OpenGL Documentation

These resources not only provide specific information about each framework’s coordinate system but also offer broader insights into 3D computer vision and modeling techniques.