Manim Slides

Welcome everyone! Today I will present our work on unified GPU-ready differentiable path tracing for reflection and diffraction sequences.

In recent years, we have observed a number of exciting improvements in radio propagation modeling.

Today, we will focus on the ray tracing techniques.

RT uses path tracing methods based on Fermat's principle to determine the coordinates of each ray path.

Recently, we have differentiable RT has emerged as a powerfull tool for optimization or solving inverse problems, by allowing us to compute the gradient of any output with respect to the input parameters.

In parallel, the use of GPU accelerators is becoming increasingly common.

Even in small scenes, one must trace thousands of paths to capture all the revelant interactions (e.g., reflections, diffractions) that contribute to the received signal. In coverage scenarios, this scale to the number of receiver locations. Using differentiable RT, we could optimize the transmistting antenna location, or perform material calibration. This motivates the need for fast algorithms and GPU acceleration.

So far, we have thus observed a paradigm shift: from...

to...

In practice, such applications pose some implementation challenges:

Challenge bullet

Quick map of the presentation and pacing.

We will now briefly review the main approaches using find ray paths.

Specifically, we are interested in methods that can find a ray path from a given TX--RX pair, and a number of predefined interactions.

State of the art bullet

Qualitative comparison of the different methods in terms of generality and speed: can we have a single method that is fast, accurate, and supports all interactions?

Let's dive into the methodology and problem formulation.

Methodology (1) bullet

Going back to our example geometry, the goal is to find the vector of parameters T that minimizes the path length. Each interaction point is parameterized using the unified formulation, using two base vectors and one reference point. For diffraction, one of the two base vectors is set to zero.

Short parenthesis: we didn't include this in the paper, but our formulation can be easily extended to handle refraction as well,.

Apart bullet

The min. path length problem becomes...

As for most optimization problems, we use an iterative method. Here, we use BFGS, which is a quasi-Newton method that approximates the Hessian using only gradient information, and is known for its robustness and efficiency on a wide range of problems. It works as follows: ...

Methodology (2) bullet

Why not simply use a Newton method or gradient descent, as done by G. Carluccio and M. Albani?

BFGS bullet

To introduce the final component of our approach, it is first important to recall how reverse-mode AD works. Here, we illustrate it on a simple example function with two inputs and two outputs., where each operation is represented as a node in the computational graph.

The compute the gradients, AD first performs a forward pass to compute the function values. Each intermediate variable is stored for later use in the backward pass.

To actually compute the gradients, AD then performs a backward pass, starting from the output gradients and applying the chain rule to compute the gradients for each intermediate variable.

so... (don't say anything, just continue with bullet points)

Implicit differentiation motivation bullet

Use the optimality condition and implicit function theorem to compute gradients without storing all iterations.

Key equations: the optimality condition states the gradient at the solution is zero

Key equations: from the optimality condition, we can derive the implicit gradient formula that only depends on the converged solution, not the entire trajectory.

Let's summarize our main contributions.

Contributions bullet

We will now presents the benchmark setup we used in the results.

Main benchmark figure, split into two panels: reflection-only on the left and diffraction-only on the right. The error formula at the bottom quantifies the average error across all paths and interaction points, comparing our method's predicted interaction points to the ground truth. The number of reflecitions/diffractions (n) is indicated in the badge below, and we will update it from 1 to 5 in subsequent slides to show how performance evolves with more interactions. The dashed vertical line indicates the runtime of the image-based method, which is exact and provides a reference for the best possible execution time.

Draw GD for n=1 on both panels.

Draw CA for n=1 on both panels.

Draw L-BFGS for n=1 on both panels.

Draw ours for n=1 on both panels.

Draw ours-64 for n=1 on both panels.

Update both panels to n=2 while preserving solver ordering and style.

Update both panels to n=3 while preserving solver ordering and style.

Update both panels to n=4 while preserving solver ordering and style.

Update both panels to n=5 while preserving solver ordering and style.

So far, we have shown promising results in terms of speed and improvements regarding a unified path tracing method. However, we have also seen that the accuracy of the methods, especially in the reflection-only case, is still not satisfactory. We are thus actively working on improving the solver formulation and implementation to further close the gap with the image-based method, and we have several ongoing and future research directions in this regard.

Future bullet

Final note on the solver bottleneck and the need for more open implementations to bridge theory and practice.

Closing slide with thanks, and QR codes.