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Camera Calibration Matrix Decomposition

Decompose a $3 \times 3$ upper-triangular camera intrinsic matrix $K$ into its individual parameters.

The intrinsic matrix $K$ has the form:

$K = \begin{pmatrix} f_x & s & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{pmatrix}$

where:

$f_x, f_y$ are the focal lengths (in pixels) along the x and y axes
$c_x, c_y$ is the principal point (where the optical axis meets the image plane)
$s$ is the skew coefficient (usually 0 for modern cameras)

Given such a matrix, extract and return these parameters as a dictionary with keys 'fx', 'fy', 'cx', 'cy', and 'skew'.

This decomposition is a fundamental step in camera calibration and is used extensively in 3D reconstruction, augmented reality, and visual SLAM.

Example:

Input:

K = [[800.0, 0.0, 320.0],
     [0.0, 800.0, 240.0],
     [0.0, 0.0, 1.0]]

Output:

{'fx': 800.0, 'fy': 800.0, 'cx': 320.0, 'cy': 240.0, 'skew': 0.0}

Reasoning:

The given intrinsic matrix $K$ is a $3 \times 3$ upper-triangular matrix with the form: $K = \begin{pmatrix} f_x & s & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{pmatrix}$
We directly extract the values from the input matrix $K$ : $f_x = 800.0$ , $f_y = 800.0$ , $c_x = 320.0$ , $c_y = 240.0$ , and $s = 0.0$ .
These extracted values are then used to create a dictionary with the corresponding keys: 'fx', 'fy', 'cx', 'cy', and 'skew'.
The resulting dictionary is the output, containing the individual parameters of the camera intrinsic matrix: {'fx': 800.0, 'fy': 800.0, 'cx': 320.0, 'cy': 240.0, 'skew': 0.0}.

Constraints:

Input: A 3x3 upper-triangular intrinsic matrix K as a list of lists
K[2][2] is always 1
Return a dictionary with keys 'fx', 'fy', 'cx', 'cy', 'skew'
Values should be floats

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Background Knowledge

The camera intrinsic matrix $K$ is a fundamental concept in computer vision, representing the internal characteristics of a camera. It maps 3D world points to 2D image points, taking into account the camera's focal length, principal point, and skew. The matrix $K$ is typically represented as a $3 \times 3$ upper-triangular matrix, with the form: $K = \begin{pmatrix} f_x & s & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{pmatrix}$ where $f_x$ and $f_y$ are the focal lengths, $c_x$ and $c_y$ are the coordinates of the principal point, and $s$ is the skew coefficient. Understanding the structure and components of this matrix is crucial for tasks like camera calibration, 3D reconstruction, and image processing.

The camera intrinsic matrix $K$ can be thought of as a transformation matrix that converts 3D world coordinates to 2D image coordinates. The focal lengths $f_x$ and $f_y$ determine the scale of the image, while the principal point $(c_x, c_y)$ represents the location of the optical axis on the image plane. The skew coefficient $s$ accounts for any non-perpendicularity between the image sensor's rows and columns. In modern cameras, $s$ is often assumed to be 0, simplifying the matrix structure.

To decompose the camera intrinsic matrix $K$ , one needs to understand the relationships between its components and how they contribute to the overall transformation. This involves identifying the individual parameters ( $f_x$ , $f_y$ , $c_x$ , $c_y$ , and $s$ ) and extracting them from the given matrix $K$ . This process requires a solid grasp of linear algebra and matrix operations, as well as an understanding of the camera's geometric properties.

Algorithm/Approach

The general approach to decomposing the camera intrinsic matrix $K$ involves analyzing its structure and identifying the individual parameters. This can be achieved through a combination of matrix operations and geometric insights. The algorithm typically involves:

Identifying the focal lengths $f_x$ and $f_y$ from the diagonal elements of $K$
Extracting the principal point coordinates $(c_x, c_y)$ from the off-diagonal elements
Determining the skew coefficient $s$ from the remaining elements
Combining these parameters into a dictionary or data structure for output

Step-by-Step Strategy

To implement the solution, follow these steps:

Import necessary libraries: Load the required libraries for matrix operations and data structures.
Define the input matrix: Represent the given camera intrinsic matrix $K$ as a $3 \times 3$ matrix.
Extract focal lengths: Identify $f_x$ and $f_y$ from the diagonal elements of $K$ .
Extract principal point coordinates: Determine $c_x$ and $c_y$ from the off-diagonal elements of $K$ .
Extract skew coefficient: Find the value of $s$ from the remaining elements of $K$ .
Create output dictionary: Combine the extracted parameters into a dictionary with the required keys.

Common Pitfalls

When implementing the solution, watch out for:

Incorrectly indexing the matrix elements
Failing to handle potential edge cases (e.g., zero or negative values)
Not validating the input matrix structure and dimensions
Incorrectly combining the extracted parameters into the output dictionary

Time & Space Complexity

The expected time complexity for this problem is O(1), as it involves a fixed number of operations to extract the parameters from the input matrix. The space complexity is also O(1), as the output dictionary has a fixed number of elements. However, the actual complexity may vary depending on the specific implementation and library functions used.

solution.py

Python 3.13

Test Results

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Run code to see test results.