Computer Vision Project 2024/2025 - Homotopy Continuation

This repository provides a comprehensive framework for exploring the use of gradient descent and homotopy continuation techniques in image rectification. The goal is to evaluate whether these methods offer greater robustness compared to traditional numerical and symbolic solvers, and to determine which of the two performs more effectively.

Documentation

A detailed description of the project is available in the report.

Sample videos demonstrating image rectification via gradient descent can be found here.

Authors - HomoTopi

• Filippo Balzarini
• Paolo Ginefra
• Martina Missana

Overview

Gradient Descent

The project leverages the usage of Adam's optimization method to perform image rectification.

Homotopy Continuation

The project leverages the HomotopyContinuation.jl package for solving systems of polynomial equations using the Homotopy Continuation approach.

Simulator

To test the application of homotopy continuation techniques to image rectification it is used a simulator with the following pipeline:

• Scene Generator:

  Given a scene definition it generates synthetic scenes warping three circles and creating an homography matrix.

• Rectifier:

  Abstract class to reconstruct the scene using different approaches:

  • Homotopy Rectifier
  • Numeric Rectifier
  • Gradient Descent Rectifier

  It reconstruct the image computing an homography matrix.

• Losser:

  Computes reconstruction errors and distortions using various metrics:

  • Circle Loss:

    uses the eccentricity of the reconstructed conics.

  • Angle Distortion Loss:

    uses the cosine of the angle between two perpendicular lines.

  • Circular Points Loss:

    uses the distance of the reconstructed circular points from the true ones.

  • Frobenious Norm Loss:

    uses Frobenius norm of the difference between two homography matrices.

  • Lines at the infinity Loss:

    uses the angle between the lines at infinity.

  • Reconstruction Error Loss:

    uses the L2 norm of the difference between the warped points using the true and the computed homographies.

Installation

Python Dependencies

1. Create a virtual environment:

   python -m venv venv
   source venv/bin/activate

2. Install dependencies:

   pip install -r requirements.txt

3. Install the package in development mode:

   pip install -e .

Julia Dependencies

1. Install Julia from JuliaLang.org

2. Install the HomotopyContinuation package:

   using Pkg
   Pkg.add("HomotopyContinuation")

MATLAB Dependencies

1. Install MATLAB from MathWorks

Docker Setup and usage

1. Build and start the services: docker-compose up -d

2. Example Julia API call:

   curl -X POST http://localhost:8081/rectify \
     -H "Content-Type: application/json" \
     -d '{"conics": [
           [1.0, 0.0, 1.0, 0.0, 0.0, -1.0],
           [1.0, 0.0, 0.0, 0.0, 0.0, -1.0],
           [0.0, 0.0, 1.0, 0.0, 0.0, -1.0]
         ]}'
         ```
   

3. Example Conics Intersection API call:

   curl -X POST http://localhost:8082/intersect \
     -H "Content-Type: application/json" \
     -d '{"conics": [
           [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, -1.0]],
           [[3.0, 0.0, 0.0], [0.0, 0.0, -0.5], [0.0, -0.5, -2.0]]
         ]}'
       ```
   

4. Check services health:

   curl http://localhost:8081/health
   curl http://localhost:8082/health 

Optuna

To test different experiment parameters Optuna is used. To visualize the Optuna dashboard

optuna-dashboard sqlite:///db.sqlite3

Testing

To run the tests for the Python package, use the following command:

pytest