Categories

## Neural Implicit Boundary Representations

By SGI Fellow Xinwen Ding and Ahmed Elhag

During the third and fourth week of SGI, Xinwen Ding, Ahmed A. A. Elhag and Miles Silberling-Cook (week 3 only) worked under the guidance of Benjamin Jones and Prof. Adriana Schulz to explore methods that can define a continuous relaxation of CAD geometry.

## Background

In general, there are two ways to represent shapes: explicit representations and implicit representations. Explicit representations are easier to model and allow local differentiable parameterizations. CAD geometry, stored in an explicit form called parametric boundary representations (B-reps), is one example, while triangle mesh is another typical example.

However, just as each triangle facet in a triangle mesh has its independent parameterization, it is hard to represent a surface using one single function under an explicit representation. We call this property discrete at the global scale. This discreteness forces us to catch continuous changes using discontinuous shape parameterization and results in weirdness and artifacts. For example, explicit representations can be incompatible with some gradient-based methods and some neural network techniques on a non-local scale.

One possible fix to this issue is to use implicit shape representations, such as signed distance field (SDF). SDFs are global functions that are continuously differentiable almost everywhere in the domain, which addresses the issues caused by explicit representations. Motivated by this, we want to play the same trick by defining a continuous relaxation of CAD geometry.

## Problem Description

To define this continuous relaxation of CAD geometry, we need to find a continuous relaxation of the boundary element type. Consider a simple case where the CAD data define a geometry that only contains two types; lines and circles. While it is natural that we map lines to 0 and circles to 1,  there is no type defined in the CAD geometry as the pre-image of (0,1). So, we want to define these intermediate states between lines and circles.

The task contains two parts. First, we need to learn the SDF and thus obtain the implicit shape representation of the CAD geometry. As an alignment to the input data type, we want to convert the type-blended geometry to CAD data. So next, we want to convert the SDF back to valid boundary representation by recovering the parameters of the elements we encoded in the SDF and then blending their element type.

## The Method

To make it easier for the reconstruction task, we decided to learn multiple SDFs, one for each type of geometry. According to these learned SDFs, we can step into the process of recovering the geometries based on their types. Now, let us consider a concrete example. If we have a CAD shape that consists of two types of geometries, say lines and circles, we need to learn two SDFs: one for edges (part of circles) and another for arcs (part of circles). With these learned SDFs, We hope to recover all the lines that appear in the input shape from the line SDF, and a similar expectation applies to the circle SDF.

Before jumping into detailed implementations, we want to acknowledge Miles Silberling-Cook for bringing up the multi-SDF idea. Due to the time limitation at SGI, we only tested this method for edges in 2D. We start with the CAD data defining a shape in Figure 1. All the results we show later are based on this geometry.

Learned SDF

Our goal is to learn a function that maps a coordinate of a query point $$(x,y) \in \mathbb{R^2}$$ to a signed distance in $$\mathbb{R}$$ from $$(x,y)$$ to the surface. The output of this function is positive if $$(x,y)$$ is outside the surface, negative if $$(x,y)$$ is enclosed by the suface, and zero if $$(x,y)$$ lies on the surface. Thus, our neural network is defined as $$f: \mathbb{R} ^2 \to \mathbb{R}$$. For Figure 1, we learned two neural networks, the first network maps $$(x,y)$$ to its distance from the line edge, and the second network maps this point to its distance for the circle edge.  For this task, we use a Decoder network (multi-layer perceptron, MLP), and optimize it using a gradient descent until convergence. Our dataset was created from a grid with appropriate dimensions, as these are our 2D points. Then, for each point in the grid, we calculate its distance from the line edge and the circle edge.

We compare the image from the learned SDF and the ground truth in Figure 2. It clearly shows that we can overfit and learn both the two networks for line and circle edges. Figure 2: The network learned the SDF with respect to the edges (first row) and arcs (second row) of the input geometry displayed in Figure 1. We compare the learned result (left column) with the ground truth (right column).

Reconstruction

After obtaining the learned line SDF model, we need to analytically recover the edges and arcs. To define an edge, we need nothing but a starting point, a direction, and a length. So, we begin the recovery by randomly seeding thousands of points and assigning each point a random direction. Furthermore, we can only accept those points with their associated values in SDF to be close to zero (see Figure 3), which enhances the success rate of finding an edge as a part of the shape boundary. Figure 3: we iteratively generate points until 6000 of them are accepted (i.e. SDF value small enough). The accepted points are plotted in red.

Then, we need to tell which lines are more likely to be the ones that define the boundary of our CAD shape and reject the ones that are unlikely to be on the boundary. To guarantee a fair selection, we need to fix the length of the randomly generated edges and pick the ones whose line integral of the learned line SDF is small enough. Moreover, to save more time, we approximate the integral by a finite sum, where we sum up the SDF value assumed by a fixed number of sample points along every edge. Stopping here, we have a pool of edge boundary candidates. We visualize them in terms of their starting points and direction using a quiver plot in Figure 4.

In the next step, we want to extend the candidate edges as long as possible, as our goal is to reconstruct the whole boundary. The extension ends once the SDF value of some extended point exceeds some threshold. After the extension, we cluster the extended edges using the mean shift algorithm. We adopt this clustering algorithm since it does not need to pre-determine the number of clusters. As shown in Figure 5, the algorithm successfully predicts the correct number of edge clusters after carefully tuning the parameters. Figure 5: Clustered edges. Each color represents one cluster. After carefully tuning parameters, the optimal number of clusters found by the mean shift algorithm reflects the actual number of edges in the geometry.

Finally, we want to extract the lines that best define the shape boundary. As we set a threshold in the extension process, we simply need to choose the longest edge from each cluster and name it a boundary edge. The three boundary edges in our example, one in each color, appear in Figure 6.

## Results

To sum up, during the project’s two-week active period, we managed to complete the following items:

• We set up a neural network to learn multiple SDFs. The model learns the SDF for edge and arc components on the boundary of a 2D input shape.
• We developed and implemented a sequence of procedures to reconstruct the lines from the trained line SDF model.

## Future Work

Even though we showed the results we achieved during the two weeks, there are more things to improve in the future. First of all, we need to reconstruct the arcs in 2D and ensure the whole procedure to be successful in more complicated 2D geometries. Second, we would like to generalize the whole process to 3D. More importantly, we are interested in establishing a way to smoothly and continuously characterize the shape transfer after the reconstruction. Finally, we need to transfer the continuous shape representation back to a CAD geometry.

Categories

## Text-Guided Shape Assembly

By Tiago De Souza Fernandes, Bryan Dumond, Daniel Scrivener and Vivien van Veldhuizen

If you have been keeping up with some of the recent developments in artificial intelligence, you might have seen AI models that can generate images from a text-based description, such as CLIP or DALL·E. These models are able to generate completely new images from just a single text-prompt (see for example this twitter account). Models such as CLIP and DALL·E operate by embedding text and images in the same vector space, which makes it possible to frame the image synthesis as an optimization problem, ie: find  an image that, when processed by a neural network, yields an embedding similar to a corresponding text embedding. A problem with this setup, however, is that it is severely underconstrained: there are many such images and some of them are not very interesting from the perspective of a human. To circumvent this issue, previous research has focused on incorporating priors into this optimization.

Over the last few weeks, our team, under the supervision of Matheus Gadelha from Adobe Research, looked into a specific instance of such constraints: namely, what would happen if the images could only be formed through a simple assembly of geometric primitives (spheres, cuboids, planes, etc)?

## An Overview of the Optimization Problem

The ability to unite text and images in the same vector space through neural networks like CLIP lends itself to an intuitive formulation of the problem we’d like to solve: What image comprising a set of geometric primitives is most similar to a given text prompt?

We’ll represent CLIP’s mapping between images/text and vectors as a function, $\phi$. Since CLIP represents text and images alike as n-dimensional vectors, we can measure their similarity in terms of the angle between the two vectors. Assuming that the vectors it generates are normalized, the cosine of the angle between them is simply the dot product $$\langle \phi(Image), \phi(Text) \rangle$$. Minimizing this quantity with respect to the image generated from our geometric primitive of choice should yield a locally optimal solution: something that more closely approximates the desired text prompt than its neighbors. Two sample images — a golden retriever and a 2006 Toyota Camry — alongside their respective descriptions according to CLIP. The cosine similarity of the first image relative to the prompt “a golden retriever” is 0.2690, compared to 0.1586 for the second image.

In order to get a sense of where the solution might exist relative to our starting point, we’d like all operations involved in the process of computing this similarity score to be differentiable. That way, we’ll be able to follow the gradient of the function and use it to update our image at each step.
To achieve our goals within a limited two-week timeframe, we desire rasterizers built on simple but powerful frameworks that would allow for as rapid iteration as possible. As such, PyTorch-based differentiable renderers like diffvg (2D) and Nvdiffrast (3D) provide the machinery to relate image embeddings with the parameters used to draw our primitives to the screen: we’ll be making extensive use of both throughout this project.

## 2D Shape Optimization

We started by looking at the 2D case, taking this paper as our basis. In this paper, the authors propose CLIPDraw: an algorithm that creates drawings from textual input. The authors use a pre-trained CLIP language-image encoder as a metric for maximizing the similarity between the given description and a generated drawing. This idea is similar to what our project hopes to accomplish, with the main difference being that CLIPDraw operates over vector strokes. We tried a couple of methods to make the algorithm more geometrically aware, the first of which is some data augmentations.

By default, the optimizer has too much creative leeway to interpret which images match our text prompt, leading to some borderline incomprehensible results. Here’s an example of a result that CLIP identifies as a “golden retriever” when left to its own devices:

Here, the optimizer gets the general color palette right while missing out completely on the geometric features of the image. Fortunately, we can force the optimizer to approach the problem in a more comprehensive manner by “augmenting” (transforming) the output at each step. In our case, we applied four random perspective & crop transformations (as do the authors of CLIPDraw) and a grayscale transformation at each step. This forces the optimizer to imitate human perception in the sense that it must identify the same object when viewed under different ambient conditions.

Fortunately, the introduction of data augmentations produced near-instant improvements in our results! Here is a taste of what the neural network can generate using triangles:

### Generating Meshes

In addition to manipulating individual shapes, we also tried to generate 2D triangular meshes using CLIP and diffvg. Since diffvg doesn’t provide automatic support for meshes, we circumvented this problem using our own implementation where individual triangles are connected to form a mesh. We started our algorithm with a simple uniform randomly-colored triangulation:

Simply following each step in the  gradient direction could change their positions individually, destroying the mesh structure. So, at each iteration, we merge together vertices that would have been pushed away from each other. We also prevent triangles from “flipping”, or changing orientation in such a way that would produce an intersection with another existing triangle. Two of the results can be seen below:

Throughout this process, we introduce and subsequently undo a lot of changes, making the process inefficient and the convergence slow. Another nice approach is to modify only the colors of the triangles rather than their positions and shapes, allowing us to color the mesh structure like a pixel grid:

### Non-differentiable Operations and the Evolutionary Approach

Up until this point, we’ve only considered gradient descent as a means of solving our central optimization problem. In reality, certain operations involved in this pipeline are not always guaranteed to be differentiable, especially when it comes to rendering. Furthermore, gradient descent optimization narrows the range of images that we’re able to explore. Once a locally optimal solution is found, the optimizer tends to “settle” on it without experimenting further — often to the detriment of the result.

On the other end of the “random-deterministic” spectrum are evolutionary models, which work by introducing random changes at each step. Changes that improve the result are preserved through future steps, whereas superfluous or detrimental changes are discarded. Unlike gradient descent optimization, evolutionary approaches are not guaranteed to improve the result at each step, which makes them considerably slower. However, by exploring a wider set of possible changes to the images, rather than just the changes introduced by gradient descent, we gain the ability to explore more images.

Though we did not tune our evolutionary model to the same extent as our gradient descent optimizer, we were able to produce a version of the program that performs simple tasks such as optimizing with respect to the overall image color.

## 3D Shape Optimization

Moving into the third dimension not only gives us a whole new set of geometric primitives to play with, but also introduces fascinating ideas for image augmentations, such as changing the position of the virtual camera. While Nvdiffrast provides a powerful interface for rendering in 3D with automatic differentiation, we quickly discovered that we’d need to implement our own geometric framework before we could test its capabilities.

Nvdiffrast’s renderer is very elegant in its design. It needs only a set of triangles and their indices in addition to a set of vertex colors in order to render a scene. Our first task was to define a set of geometric primitives, as Nvdiffrast doesn’t provide out-of-the-box support for anything but triangles. For anyone familiar with OpenGL, creating an elementary shape such as a sphere is very similar to setting up a VBO/EBO. We set to work creating classes for a sphere, a cylinder, and a cube. Random configuration of geometric primitives — a cube, a cylinder, and a sphere — rendered with Nvdiffrast. One example of an input to our algorithm.

Because the input to Nvdiffrast is one contiguous set of triangles, we also had to design data structures to mark the boundaries between discrete shapes in our triangle list. We did not want the optimizer to operate erratically on individual triangles, potentially breaking up the connectivity of the underlying shape: to this end, we devised a system by which shapes could only be manipulated by a series of standard linear transformations. Specifically, we allowed the optimizer to rotate, scale, and translate shapes at will. We also optimized according to vertex colors as in our previous 2D implementation.

With more time, it would have been great to experiment with new augmentations and learning rates for these parameters: however, setting up a complex environment like Nvdiffrast takes more time than one might expect, and so we have only begun to explore different results at the writing of this blog post. Some features that show promising outcomes are color gradient optimization, as well as the general positioning of shapes: