An Inverse Procedural Modeling Pipeline for SVBRDF Maps

Material acquisition seeks to recover the reflectance properties of existing surfaces or objects. Our method complements this body of work, since acquired materials are used as input for our pipeline to create procedural representations. While material acquisition has challenged researchers for decades, as discussed in the excellent survey by Guarnera et al. [29], significant progress was achieved in recent years by leveraging deep learning. In particular, different methods were proposed to acquire materials from one [44, 45, 46, 15] or multiple [16, 26, 33, 10, 18] pictures of surfaces or objects. While our approach also targets the creation of a material representation, we do not seek to recover material properties from captured sample(s), but rather to proceduralize an existing material. This difference not only allows our method to benefit from material acquisition and its progress, but also to handle any existing analytical material.

In our method, we leverage texture synthesis, which creates new textures with larger scale or higher resolution from an exemplar. Several surveys provide a comprehensive overview of example-based texture synthesis [61, 2, 51]. Texture synthesis can be classified into three families: patch re-arrangement based, arranging patches available in the original texture to synthesize a new texture [21, 20, 39], statistics based, estimating statistics of the original texture and transferring them to a new texture [34, 23, 24, 28, 25, 35], and neural-network-based, reproducing texture or material appearance using machine learning [58, 9, 63, 54, 37, 36, 48]. We use the second family of methods to reproduce the local variations of our decomposed atomic components’ sub-materials. Alone, these methods can reproduce micro-structures well, but fail for larger scale patterns. They are also limiting in the scope of possible editability. But by making use of progressive decomposition, extending them in a multi-layer fashion, and incorporating them with an optimizable post-processing step, our method can represent complex SVBRDFs as procedural models with arbitrary scale and resolution as well as high-level editability.

Procedural models allow artists to retain editability and produce arbitrary scale and resolution. Specialized procedural models have been proposed but are limited to specific materials such as wood or leather and mostly stationary materials [32]. A more generic option lies in the design software allowing artists to combine procedures to create a new material (see Figure 1). While this procedural representation provides great editability and versatility, it remains time-consuming and challenging, even for expert artists, to reproduce an existing or imagined material.

Recent methods explore inverse procedural modeling of materials, aiming at the reproduction of a given material appearance using a procedural model. Hu et al. [38] and Shi et al. (MATch) [53] both propose different frameworks that leverage existing Substance graph procedural models and infer their parameters to match the appearance of input textures. To do so, Hu et al. rely on neural networks trained for each individual material graph, while Shi et al. (MATch) rely on gradient descent and a differentiable version of the Substance Engine. Both of these methods, however, work on the premise of known pre-existing procedural node graphs, manually created by artists. This approach requires a large, expressive enough dataset of procedural graphs to be available and non-trivial search methods to find the closest ones among hundred of options. As opposed to these methods, our proposed pipeline does not require predefined material graphs and simply relies on a few user scribbles, which typically only require two minutes of interaction.

A key component of our inverse procedural modeling pipeline is the procedural representation of the structure of materials. That is, the global spatial distributions of decomposed sub-materials, which we represent using a set of binary mask maps. To preserve the procedural aspect of our results, we need to represent these masks procedurally. Rosenberger et al. [52] propose a shape synthesis method to generate layered control maps but is limited to unstructured shapes with fractal-like boundaries. Alternatively, we could use an L-system [56], but that approach requires predefined grammars. Rather, we make procedural generation of structured mask maps possible by proposing a by-example inverse Point Process Texture Basis Functions (PPTBF) [30] modeling approach and leveraging random sampling.

Image segmentation aims at separating an image into different regions, based on some specific criteria. In our material proceduralization framework, segmentation is used for partitioning the given material into several sub-materials. Image segmentation has been extensively studied in the past decades [40], with recent methods leveraging deep learning [47]. Close to our challenge are the recent works of Cimpoi et al. and Bell et al. [[8], [14]], which semantically split natural scenes into their different components (such as wood, plastic, etc.). However, these methods focus on the segmentation of complete scene images and use context [8] to better recognize materials, which is not available in our material exemplars.

Most existing segmentation methods do not perform well in our context, as they result in significant error on the boundaries between two sub-materials. We therefore use image matting, originally developed to segment background and foreground with fuzzy boundaries. We found alpha matting to better represent the transition between sub-materials in a SVBRDF and to preserve good quality boundaries, even after thresholding.

Many methods for image matting suggest affinity-based solutions, solving a Laplacian clustering problem [43, 12, 4]. Recent work [13, 62] attempts to learn alpha maps directly using deep neural networks, but is limited to two-layer (foreground and background) alpha matting due to available training data. Aksoy et al. [3] proposed an improved affinity-based method by combining deep semantic features with affinity Laplacian. While it supports multiple layers, it is geared toward natural scene images and requires user to provide the number of layers. Our decomposition approach of materials is similar to the one adopted by Lawrence et al. [41]: We aim to decompose the input spatially varying material maps into regions of similar sub-materials. The approach by Lawrence et al. was further developed by AppWand [49], AppProp [5], and Material Matting [42], which introduce segmenting measured or analytical material images to make them easy to edit. However, in practice, these approaches seem to handle limited normal variations. More importantly, they do not target the creation of fully procedural representations. They treat the materials as SVBRDF images, limiting the editing of results to pixelwise (rather than parametric) modifications of the segmentation and/or uniform (rather than multi-scale) material editing. Beyond the editing limits, these methods do not provide for the generation of textures of arbitrary spatial extent.

We choose to build upon KNN matting [12] by proposing a spectrum-sensitive affinity-based image matting method. KNN matting supports multiple layers and enables user control to conveniently define layers of interest, which are useful features for our material segmentation purpose. The aforementioned material segmentation approaches [49, 5, 42] could produce interesting segmentation results, but we chose KNN matting because of its generality, the availability of its implementation, and its efficient runtime.

We first consider how to decompose a set of SVBRDF maps into sub-materials. As classical segmentation [40] does not allow for smooth boundaries, we use alpha matting [12]. Inspired by KNN matting, our algorithm allows users to draw a few strokes to conveniently indicate different regions of interest.

As an affinity-based algorithm, KNN matting [12] relies on a feature vector \(X(i)\) for each pixel \(i\) of the image to compute an affinity matrix. For SVBRDF maps, traditionally used features are albedo color (r, g, b), height (h), roughness (\(\alpha\)), and position (x,y):We propose an additional feature, based on the noise spectrum, enforcing the statistical uniformity expected from decomposed sub-materials. To distinguish between the differences in local noise, we take the noise Fourier spectrum into account. Notice that although position features \((x, y)\) are taken into account, we do not enforce spatial continuity of the regions, because we sample multiple non-local neighborhoods with different weights of \((x, y)\), to explore nonlocality (as done in the original KNN matting)

We estimate the local spectrum at each pixel \(i\) using Welch’s method [60] and reduce its dimensionality to 3 using Principal Component Analysis (PCA). We therefore compute the affinity matrix and matting using a feature vector \(X(i)\) composed of albedo color, height, roughness, position, and our spectrum estimation.

Alpha matting results in multiple alpha maps that we process to generate binary mask maps. Each pixel in the image is assigned to the binary for which it has the highest alpha map value. The thresholded alpha-matting better represents the transition between sub-materials than direct segmentation methods. Figure 4 shows a comparison between the decomposition results with and without spectrum features on example textures. In these two images, color and position features do not provide enough hints to separate two layers, because they are similar in color but differ in noise spectra. More decomposition results on material maps with user scribbles can be found in our supplemental document.

Once a material decomposed, we provide the option to further process each sub-material using the same matting algorithm with the newly generated mask map(s) as an additional constraint.

As an alternative to a progressive decomposition into sub-materials, we provide a lightweight solution for instance-based decomposition. This is particularly useful for repeating materials such as a tile wall composed of different types of tiles (the input shown in Figure 5). Rather than manually segmenting each different tile, we frame this as an instance detection and clustering problem, using—in this example—the mask map of segmented tiles to extract each instances of tile. Different instances correspond to disconnected regions in the mask map segmented from the previous layer in the decomposed material tree. We then scale each instance to the same size based on its bounding box. For each instance, we estimate its color histogram and local spectrum as features and build a feature matrix for agglomerative clustering, as shown in Figure 5. The clustering result allows us to extract the different types of tiles, their frequency, and their sub-material to assign them with a similar frequency in the procedural model.

In this section, we describe the conversion of the segmented sub-material into procedural models. Each leaf node in our tree is a sub-material while the intermediate nodes store the mask maps. Each sub-material is represented by a set of masked SVBRDF maps. We use a multi-layer procedural noise model to proceduralize their texture appearance. As our images are masked and incomplete, the spectrum of the entire image is unavailable, preventing the direct fitting of procedural noise models using noise synthesis methods, e.g., Galerne et al. [23], Galerne et al. [24], Gilet et al. [28], and Heitz and Neyret [35]. While Guingo et al. [31] propose estimating local spectra in the valid regions only, their approach cannot capture the global variation of the noise textures, e.g., when the scale of the randomness is comparable to the scale of the full image. We leverage a similar sliding window approach but propose a multi-layer procedural model—shown in Algorithm 1—to deal with incomplete images. Given an input masked image, we decompose it into several noise layers using a progressive filtering strategy. For each layer, we filter our input image with a Gaussian kernel. The kernel size of the filter becomes larger as the number of layers grows, aiming at capturing larger scale spatial variation. For each filtered layer, we estimate its local spectrum and convert it to a procedural noise model similar to Guingo et al. [31]. Finally, our algorithm uses the mean value of filtered image \(I\) as the base color \(C\) and extracts the final noise layer \(N \leftarrow I-C\).

This last layer, however, represents the lowest frequency of the input image, preventing the use of sliding window spectrum estimation. As each sub-material is masked, the full image spectrum is unavailable. We therefore inpaint the missing data [57, 7]. To reduce artifacts introduced by this step, we apply a set of Gabor noises as a basis to approximate the power spectrum of the inpainted image [24]. This last step and our use of multi-layer noise allows our method to produce a fully procedural representation of the sub-material’s properties. Figure 6 shows an example of our multi-layer decomposition on a masked colored texture image, where each layer captures different levels of details from an input image and the last layer shows the global variation of the noise image.

Using the procedural noise \(p_i\) fitted to each layer \(i\) and the base color \(C\), we reconstruct the original texture as \(\sum p_i + C, i=1,2,\ldots ,n\) where \(n\) is the number of noise layers. We further improve the results of this reconstruction for SVBRDF maps by introducing a differentiable rendering-based optimization scheme, described in Section 5.3. Our approach provides a fully procedural representation and can reproduce fine-grained details and yield a smoother noise texture than single-layer methods (see Figure 7). Furthermore, our multi-layer approach gives users more control over level of detail.

While we represent the sub-material local variations using our multi-layer noise model, we use mask maps to represent the global spatial distributions of the sub-materials. Proceduralizing these mask maps allows us to reach a full procedural representation of the material. With it, we can easily edit, resample, and extend the resolution of the spatial distribution. Similar to sub-material proceduralization, mask proceduralization is done recursively during traversal of the material tree. We model mask maps by two methods (1) Point Process Texture Basis Functions (PPTBFs) [30] to model decomposed binary mask maps by matting algorithm (Section 4.1); (2) Random Sampling to model decomposed instances (Section 4.2).

Finally, using our generated procedural noise maps and masks tree, we compose them into an output SVBRDF to reproduce the appearance of the original SVBRDF inputs. To relate this approach to a classical Substance Designer [1] pipeline, our procedural noise maps and binary masks function as generators nodes, which do not rely on existing artist designed graphs.

To better match the original SVBRDF inputs, we add optimizable operators to control the appearance of procedural noise and binary maps. Given a noise map \(I\), we modify its appearance by \(G(I*\alpha +\delta , \sigma)\), where \(G\) is a Gaussian filter, \(\alpha\) controls the intensity, \(\delta\) biases the noise, and \(\sigma\) is the standard deviation of the Gaussian filter. For a binary mask map \(M\) that represents the distribution of sub-materials, we model the smooth transition between the boundaries of sub-materials by \(G(M, \sigma)\). The parameters \(\alpha\), \(\delta\), and \(\sigma\) are optimizable for each Gaussian filter per noise or mask map. In each leaf node of our tree, the noise maps are multiplied by their corresponding mask maps and linearly combined to reconstruct the SVBRDF maps for a procedural sub-material. Procedural Sub-materials from different layers are recursively computed and aggregated in a bottom-up fashion to build the final output SVBRDF maps.

These optimizable operators, together with our reconstructed noise maps and binary maps, build a small optimizable material graph. We optimize this material graph, using a differentiable rendering-based optimization routine. The reconstructed SVBRDF maps and the input SVBRDF maps are rendered using a Cook-Torrance, GGX shading model under randomly sampled lighting configurations. Considering that the structure of the procedural SVBRDF maps might not align perfectly with the original inputs, we define the loss as a combination of Style loss and SSIM loss. Style loss is defined as the \(L_1\) difference between Gram matrices computed over VGG [55] features of the renderings, similar to the style transfer literature [27]. SSIM loss is computed by the \(L_1\) difference between the structural similarity (SSIM) indices [64] of the renderings. The full loss is written aswhere \(GM\) is the Gram matrix and SSIM is an operator to compute structure similarity index; \(I_i\) and \(I^*_i\) are input/procedural albedo map, normal map (computed from height map), roughness map, and their renderings; \(\beta\) balances the weight between the style term and the SSIM term. In Figure 10, we show an example with and without this optimization process. Often in real data local texture appearance cannot be well represented by Gaussian noise models, resulting in fitting errors and artifacts when generating procedural models (third row in Figure 10). This is particularly problematic for normal/height map modeling. Our optimization helps refine the parameters of our procedural graph, significantly improving the results, even when the exemplar violates our assumption of locally uniform appearance such as in Figure 10.

Our final procedural representation of material maps is the equivalent of small material graphs with (i) procedural noise maps, (ii) procedural mask maps, (iii) optimizable operators, similar to Figure 1 . In Shi et al. [53]’s terminology, each procedural noise and mask model is a generator, while optimizable operators serve as filters. As a fully procedural model, editable parameters of our small material graph are (i) parameters of our generators, e.g., noise models and mask models; (ii) parameters in the filter nodes. Editing can be done on individual maps or jointly on all of them. The procedural model can then generate the material at arbitrary resolution and scale.

Although our pipeline was designed to take material maps as input, it can also work on natural and flash images, benefiting from existing material acquisition methods [15, 16, 17, 26, 33]. Providing a set of captured images, we first apply a material acquisition method to generate material maps and use the generated SVBRDF maps with our method. Figure 12 shows examples of picture(s) as input. We convert the input natural image(s) to SVBRDF maps using state-of-the-art material acquisition methods [16, 17, 33]. As opposed to “artist-designed” SVBRDF maps or manually post-processed SVBRDF maps, automatically generated SVBRDF maps are not perfectly clean. Different maps can be noisy and exhibit shading and color variations due to incomplete lighting removal. Normal maps in particular can represent strong height variations even within a single sub-material. This makes it challenging to recover an exact match to the input image. Furthermore, irregularities often occur in acquired SVBRDF, which should be regular. We show in Figure 12 that our pipeline is capable of matching acquired SVBRDF overall appearance, and also of enforcing better regularity.

We compare our results to previous work in inverse procedural modeling and in texture synthesis. Since we take the output of a material acquisition method as input to our pipeline, we do not compare to material acquisition methods.

To characterize the limitations of our work, we describe the space of possible materials we attempt to model. We consider materials as collections of elements. The space of different element collections has four dimensions defining the complexity of the proceduralization of a material, illustrated in Figure 19. The first axis is the number of different material element types in the texture. Both the types of elements and their spatial distribution relative to one another need to be modeled. The second axis is the nature of the spatial distribution, whether the elements on the texture are sparsely or densely positioned, with overlap for example. Dense spatial distribution are harder to segment using masks. The third axis is complexity of the contour of the spatial mask. The more complex, the more difficult it is for PPTBF or any procedural approach to represent them faithfully. The fourth axis is the texture within each element, whether it can easily be represented as noise or is more structured and semantically meaningful.

Our general pipeline that segments SVBRDF and then estimates parameters for each element type applies to the complete space we have described. However, our current implementation is limited to the less complex end of each of the last three axes. Our segmentation method is limited in that it cannot appropriately segment dense overlapping material. As stated, PPTBF performs well with irregular simple contours but not with complex contours. Finally, for the fourth axis, we have not developed a method to proceduralize semantically meaningful elements such as the bunny.

Our results show that our pipeline is able to proceduralize different types of materials with multiple SVBRDF maps, e.g., albedo, normal, and roughness. In Figure 20, we show failure cases that result from the general limitations we just described. The combination of complex contours and a semantically meaningful structure results in the failure shown in the top row of Figure 20. The complex contours form a “flower” shape that is not preserved by the PPTBF mask. The presence of a sub-material that is itself composed of a densely packed set of elements results in the failure shown in the bottom row of Figure 20. Neither a noise pattern nor our segmentation method can capture the arrangement of the small pebbles forming the fill material between the square elements. Finer segmentation could produce better results, but the overlapping partial shapes would still be difficult to match using PPTBF. In some cases, such as the second example of Figure 16, the retrieved procedural mask will have small differences in the contours compared to the original. Adding a loss more sensitive to contours could improve such cases.

Finally, as our procedural material model is built upon multi-scale Gaussian noise models and Gaussian filters, it poorly reproduces highly structured variations, e.g., extremely strong and directional normal variations seen as Figure 21. In these cases, the segmentation of the height map is also ambiguous, leading to less faithful procedural reconstruction.