https://github.com/bchao1/homography
2D projection geometry with homography.
https://github.com/bchao1/homography
computer-graphics computer-vision homography image-processing projection-geometry
Last synced: 4 months ago
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2D projection geometry with homography.
- Host: GitHub
- URL: https://github.com/bchao1/homography
- Owner: bchao1
- Created: 2020-08-10T14:06:36.000Z (about 5 years ago)
- Default Branch: master
- Last Pushed: 2020-08-11T14:28:52.000Z (about 5 years ago)
- Last Synced: 2025-03-22T23:27:53.794Z (7 months ago)
- Topics: computer-graphics, computer-vision, homography, image-processing, projection-geometry
- Language: Python
- Homepage:
- Size: 3.47 MB
- Stars: 2
- Watchers: 2
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# homography
The meme square with projection geometry.
## Usage
The main algorithm is in `homography/main.py`. To process the images, you need to specify custom parameters in `homography/config.yaml` file:
- `canvas_image`: file path of the background canvas image
- `project_images`: list of file paths of the images to project onto the canvas
- `output_image`: file path of the output image
- `coors`: a list of quadrilateral corners in canvas coordinates. Should be same length as `project_images`. Note that in the config file, `x` is defined as the vertical axis, while `y` is the horizontal axis.
```
This pixel is (x, y) = (5, 2).
__ __ __ __ __ __ __ y (width)
|__|__|__|__|__|__
|__|__|__|__|__|__
|__|__|__|__|__|__
|__|__|__|__|__|__
|__|██|__|__|__|__
|__|__|__|__|__|__
|
x (height)
```
Referring to `homography/config.yaml`, the four corners of image `1.jpg`, starting from top-left in clockwise order, will be projected to the first entry of `coors`, which are four canvas coordinates:
```yaml
coors:
- top_left:
x: 296.3452162756598
y: 795.149285190616
top_right:
x: 49.88489736070392
y: 1194.9957844574778
bottom_right:
x: 346.23295454545456
y: 1197.9741568914956
bottom_left:
x: 460.90029325513194
y: 799.6168438416423
```
After modifying `config.yaml`, simply run:
```
python3 main.py
```
in `homography/`. You can also try out the demo `config.yaml` provided, using images in `data/`.
## How this works
### Homography
Projecting a 2D image onto another image (the "canvas") involves computing the tranformation between two homogeneous coordinate systems. We refer to the coordinate system of the image to project as the **source coordinates**, and the coordinate system of the canvas as the **target coordinates**.
Suppose a point `(u_x, u_y, 1)` in the source coordinate is projected to a point in `(v_x, v_y, 1)` in target coordinates, we can represent the projection as a **homography** **H**, which is a 3*3 matrix.
Since H is *invariant to scaling*, we usually constrain **h_33 = 1** so that **H** has 8 degree of freedom (DoF) rather than 9. Another way to enforce 8 DoF is constraining the squared sum of all elements in **H** to be 1 (which turns out to be a more robust constraint).
### Solving H
To solve **H**, we need 8 equations since **H** has 8 DoF. This is equivalent to 4 pairs of mappings between source and target coordinates(4 points define a homography). We can simply choose the four corners of the source image and its expected transformed target coordinates to construct our equations.
For each homogeneous coordinate mapping `(u_x, u_y, 1) -> (v_x, v_y, 1)`, we can write down two equations:
Writing in matrix multiplication form:
Since we have 4 pairs of coordinate mappings, we can stack all 2 * 9 matrices into a single 8 * 9 matrix **A**, and solve the homogeneous system:
```
Ah = 0, subject to ||h|| = 1
```
We solve **h** by performing [singular value decomposition](https://en.wikipedia.org/wiki/Singular_value_decomposition) on **A**T**A**. **h** would be the eigenvector corresponding to the smallest eigenvalue.