Transforming Shapes with Polynomial Projection: A New Take on Non-Linear Mapping, Powered by PyTorch

In the vast landscape of computer vision and geometric transformations, finding an efficient way to map shapes non-linearly — without the typical grind of iterative calculations or complex neural networks — feels like discovering a shortcut that works. Imagine if you could morph a circle into a rectangle, align distorted medical images, or create data-augmented versions of objects, all with one simple method. Polynomial projection is a robust, one-step transformation method that’s as versatile as elegant.

What Makes Polynomial Projection Unique?

Let’s break down what makes this method special. At its core, polynomial projection takes the coordinates of a shape, expands them with polynomial terms (like x², y³), and matches them to target points. The beauty of this approach lies in its training. We simply don’t need to train the model that transforms the points. We do the training by solving x to y mapping with exact solution, meaning you get immediate results with precision.

Built with PyTorch, this method not only makes it simple to implement but also runs efficiently on GPUs, handling large datasets and complex shapes in real-time. Instead of training a neural network or setting up radial basis functions, polynomial projection does it all in one go. You provide the initial points (say, a circle), and the target points (perhaps a rectangle), and let the projection transform the shape. It’s a convenient method for anyone working with shape transformations, especially when efficiency matters.

Real-World Applications

The power of this transformation method stretches across several domains:

1. Shape Morphing for Animation: Morphing a circle into a star or a square in smooth steps? Easy. This method generates each frame with fluid transitions, perfect for animations that need non-linear, natural transformations.
2. Distortion Correction in Imaging: In fields like photography and satellite imaging, lens distortion often warps an image. Polynomial projection lets you reverse these effects quickly by mapping distorted points back to an undistorted grid, saving time and preserving detail.
3. Medical Image Alignment: Aligning complex structures like organs in medical images often calls for non-linear mapping. Polynomial projection aligns points between different scans without requiring iterative adjustments, simplifying workflows in diagnostics and treatment planning.
4. Data Augmentation in AI: Imagine training an AI model to recognize objects from varied perspectives. With polynomial projection, you can easily create non-linear deformations, generating augmented data that enhances model robustness, whether it’s for autonomous driving, augmented reality, or facial recognition.

Why Polynomial Projection with PyTorch?

This method is precise, fast, and GPU-ready. Whether you’re aligning point clouds, correcting image distortions, or transforming shapes for animation, PyTorch’s polynomial projection offers a shortcut to efficiency with exact results. It’s perfect for anyone looking to implement shape transformations in practical applications — without spending hours on computation.

Ready to dive in? here is the code:

Future Use Cases:

This method could be useful in cryptography. If we want to send a message, we can just convert it to x, and y coordinates, calculate the transformation model with the proposed method, and save the transformation model as a key.

We could use it for fitting noisy data to a requested shape by calculating the transformation of the noisy data to the shape of interest.

I believe that there are more applications of this. Let me know if there is anything that comes to your mind in the comments.

Cheers,