Determining camera parameters?

Question

Given a picture taken by a simple digital that contains an image of a rectangle of known dimensions. How can I - to some degree of accuracy - determine the parameters of this camera? I am mostly interested in Pan-, Tilt- and Swing angles. Optionally distance to the rectangle would be nice. I acknowledge the fact that the focal length and resolution of the camera should be known, but it can be assumed that they are available (for example through exif metadata). Are there any simple algorithms for this problems?

Answer 1

To determine the camera parameters (pan, tilt, and swing angles) from an image of a rectangle with known dimensions, follow these steps:

### 1. **Extract Relevant Information**

- **Known Data**: Dimensions of the rectangle in real life, focal length of the camera, and image resolution.

- **Measured Data**: The dimensions of the rectangle in the image (in pixels).

### 2. **Calculate the Distance to the Rectangle**

Using the known dimensions of the rectangle and its image size, you can calculate the distance from the camera to the rectangle.

- **Formula**:

\[

\text{Distance} = \frac{\text{Real Rectangle Height} \times \text{Focal Length}}{\text{Image Rectangle Height in Pixels}}

\]

Adjust this formula if you have both height and width measurements.

### 3. **Determine Camera Parameters**

**Pan and Tilt Angles**:

- **Pan Angle (Rotation around the vertical axis)**:

\[

\text{Pan Angle} = \text{atan2}\left(\frac{\text{Rectangle Center X (in Pixels)} - \text{Image Center X}}{\text{Focal Length}}, \text{Distance}\right)

\]

- **Tilt Angle (Rotation around the horizontal axis)**:

\[

\text{Tilt Angle} = \text{atan2}\left(\frac{\text{Rectangle Center Y (in Pixels)} - \text{Image Center Y}}{\text{Focal Length}}, \text{Distance}\right)

\]

In these formulas:

- The `atan2` function calculates the angle considering the x and y offsets and distance.

- `Rectangle Center X` and `Rectangle Center Y` are the coordinates of the center of the rectangle in the image.

- `Image Center X` and `Image Center Y` are the coordinates of the image center.

**Swing Angle (Rotation around the camera’s optical axis)**:

This is more complex to determine directly from a single image, as it typically requires additional information about the camera’s orientation or multiple images. If you assume the camera’s optical axis is aligned with the image plane, the swing angle may be considered minimal or zero.

### 4. **Implementing in Practice**

- **Extract Rectangle Measurements**: Use image processing tools to find the rectangle’s dimensions and center in pixels.

- **Calculate Parameters**: Use the provided formulas to compute the angles.

### 5. **Optional Distance Calculation**

For a more accurate distance measurement:

- Use the known dimensions of the rectangle and the image size to infer how far the object is. Adjust the focal length accordingly.

### Example Workflow

1. **Measure**: Find the width and height of the rectangle in the image and in real life.

2. **Calculate Distance**: Use the formula to find the distance.

3. **Compute Angles**: Use the pan and tilt formulas to calculate the angles based on the rectangle’s position in the image.

By following these steps, you can estimate the camera parameters with reasonable accuracy given that the focal length and image resolution are known.