Rodrigo Santa Cruz

Léo Lebrat, Rodrigo Santa Cruz, Remi Chierchia et al.

Wound management poses a significant challenge, particularly for bedridden patients and the elderly. Accurate diagnostic and healing monitoring can significantly benefit from modern image analysis, providing accurate and precise measurements of wounds. Despite several existing techniques, the shortage of expansive and diverse training datasets remains a significant obstacle to constructing machine learning-based frameworks. This paper introduces Syn3DWound, an open-source dataset of high-fidelity simulated wounds with 2D and 3D annotations. We propose baseline methods and a benchmarking framework for automated 3D morphometry analysis and 2D/3D wound segmentation.

Rodrigo Santa Cruz, Léo Lebrat, Darren Fu et al.

The problem of Cortical Surface Reconstruction from magnetic resonance imaging has been traditionally addressed using lengthy pipelines of image processing techniques like FreeSurfer, CAT, or CIVET. These frameworks require very long runtimes deemed unfeasible for real-time applications and unpractical for large-scale studies. Recently, supervised deep learning approaches have been introduced to speed up this task cutting down the reconstruction time from hours to seconds. Using the state-of-the-art CorticalFlow model as a blueprint, this paper proposes three modifications to improve its accuracy and interoperability with existing surface analysis tools, while not sacrificing its fast inference time and low GPU memory consumption. First, we employ a more accurate ODE solver to reduce the diffeomorphic mapping approximation error. Second, we devise a routine to produce smoother template meshes avoiding mesh artifacts caused by sharp edges in CorticalFlow's convex-hull based template. Last, we recast pial surface prediction as the deformation of the predicted white surface leading to a one-to-one mapping between white and pial surface vertices. This mapping is essential to many existing surface analysis tools for cortical morphometry. We name the resulting method CorticalFlow$^{++}$. Using large-scale datasets, we demonstrate the proposed changes provide more geometric accuracy and surface regularity while keeping the reconstruction time and GPU memory requirements almost unchanged.

Remi Chierchia, Leo Lebrat, David Ahmedt-Aristizabal et al.

Managing chronic wounds is a global challenge that can be alleviated by the adoption of automatic systems for clinical wound assessment from consumer-grade videos. While 2D image analysis approaches are insufficient for handling the 3D features of wounds, existing approaches utilizing 3D reconstruction methods have not been thoroughly evaluated. To address this gap, this paper presents a comprehensive study on 3D wound reconstruction from consumer-grade videos. Specifically, we introduce the SALVE dataset, comprising video recordings of realistic wound phantoms captured with different cameras. Using this dataset, we assess the accuracy and precision of state-of-the-art methods for 3D reconstruction, ranging from traditional photogrammetry pipelines to advanced neural rendering approaches. In our experiments, we observe that photogrammetry approaches do not provide smooth surfaces suitable for precise clinical measurements of wounds. Neural rendering approaches show promise in addressing this issue, advancing the use of this technology in wound care practices. We encourage the readers to visit the project page: https://remichierchia.github.io/SALVE/.

Alejandro Ascarate, Leo Lebrat, Rodrigo Santa Cruz et al.

Variational autoencoders (VAE) encode data into lower-dimensional latent vectors before decoding those vectors back to data. Once trained, one can hope to detect out-of-distribution (abnormal) latent vectors, but several issues arise when the latent space is high dimensional. This includes an exponential growth of the hypervolume with the dimension, which severely affects the generative capacity of the VAE. In this paper, we draw insights from high dimensional statistics: in these regimes, the latent vectors of a standard VAE are distributed on the `equators' of a hypersphere, challenging the detection of anomalies. We propose to formulate the latent variables of a VAE using hyperspherical coordinates, which allows compressing the latent vectors towards a given direction on the hypersphere, thereby allowing for a more expressive approximate posterior. We show that this improves both the fully unconditional-OOD and conditional-OOD anomaly detection ability of the VAE, achieving the best performance on the datasets we considered, outperforming existing methods. For the unconditional-OOD and conditional-OOD modalities, respectively, these are: i) detecting unusual landscape from the Mars Rover camera and unusual Galaxies from ground based imagery (complex, real world datasets); ii) standard benchmarks like Cifar10 and subsets of ImageNet as the in-distribution (ID) class.

Alejandro Ascarate, Leo Lebrat, Rodrigo Santa Cruz et al.

Many practical anomalies are not merely rare inputs, but violations of semantic constraints: objects co-occur in structured ways, actions imply preconditions, and events satisfy temporal or relational regularities. We study anomaly detection in this setting, where constraints are given as logical rules over learned visual concepts, but real rule violations are rare or absent during training. We propose a neural rule evaluator that compiles each constraint into a directed acyclic graph and learns feature-aware subtree MLP gates for its internal logical operators. Each gate maps child features and edge-level negations to a parent representation and a rule-satisfaction probability, with intermediate supervision obtained from exact Boolean propagation over ground-truth concept labels. The key difficulty is that same-image training data often provide insufficient coverage of informative truth configurations and also allow shortcut solutions. To address this, we introduce chimera training: an operand-level counterfactual construction at the feature level. Instead of mixing input images, we concatenate subtree features from different samples; each operand keeps the hard truth label of the sample it came from, and the chimera target is obtained by applying the node's logical operator to those inherited labels. This supplies supervised logical counterexamples without requiring real anomalous images. Across CLEVRER, OpenImages, and VidOR, the resulting evaluator improves rule-level anomaly AUROC over independent-events and same-image semantic-training baselines, especially for compositional and relational rules. The method yields both scalar anomaly scores and rule-level attributions.

Alejandro Ascarate, Leo Lebrat, Rodrigo Santa Cruz et al.

Within-dataset class-split evaluation is widely used as a proxy for fully unconditional out-of-distribution anomaly detection. We show that this protocol can become ill-posed when the held-out anomaly class overlaps the normal mixture in representation space. In this regime, anomaly scores may collapse toward chance or even invert, and the preferred score direction can depend on the unknown anomaly class. We introduce a simple training-free diagnostic, neighborhood class leakage, and show that it predicts score-direction instability across Fashion-MNIST, CIFAR-10, and Imagenette, in both pixel and VAE latent spaces. Our results suggest that class-split AD benchmarks should be treated as geometry-dependent stress tests rather than unconditional evidence of anomaly-detection ability.

Alejandro Ascarate, Leo Lebrat, Rodrigo Santa Cruz et al.

We study autoencoder and variational-autoencoder latent spaces through the lens of spin-glass theory. The paper has two components. First, we formalize a latent-space spin-glass dictionary: for a fixed decoder, the reconstruction term together with a hyperspherical coordinates prior induces a Hamiltonian on the latent sphere, where latent coordinates play the role of continuous spins and the prior acts as an external magnetic field. This allows us to import operational spin-glass diagnostics -- overlap distributions, susceptibility, and block-spin coarse-graining -- to detect ordered, disordered, and edge-of-stability phases in trained latent representations. Second, we show that deliberately driving the latent system toward the edge-of-stability of the topological trivialization regime has concrete downstream consequences. In generation, hyperspherical compression improves the reconstruction-generation trade-off on CIFAR-10 and CelebA64, yielding lower self-FID while preserving or improving reconstruction. In anomaly detection, the same semi-ordered latent geometry improves both fully unsupervised and conditional OOD detection, including real-world Mars Rover and Galaxy Zoo datasets, as well as CIFAR-10/100 and Imagenette-based OOD benchmarks. We therefore advocate a phase-aware evaluation paradigm for AEs/VAEs, in which spin-glass observables complement standard ML metrics and expose the latent regimes that underlie downstream success or failure in many cases.

Léo Lebrat, Rodrigo Santa Cruz, Frédéric de Gournay et al.

In this paper we introduce CorticalFlow, a new geometric deep-learning model that, given a 3-dimensional image, learns to deform a reference template towards a targeted object. To conserve the template mesh's topological properties, we train our model over a set of diffeomorphic transformations. This new implementation of a flow Ordinary Differential Equation (ODE) framework benefits from a small GPU memory footprint, allowing the generation of surfaces with several hundred thousand vertices. To reduce topological errors introduced by its discrete resolution, we derive numeric conditions which improve the manifoldness of the predicted triangle mesh. To exhibit the utility of CorticalFlow, we demonstrate its performance for the challenging task of brain cortical surface reconstruction. In contrast to current state-of-the-art, CorticalFlow produces superior surfaces while reducing the computation time from nine and a half minutes to one second. More significantly, CorticalFlow enforces the generation of anatomically plausible surfaces; the absence of which has been a major impediment restricting the clinical relevance of such surface reconstruction methods.

Remi Chierchia, Léo Lebrat, David Ahmedt-Aristizabal et al.

Wound care is often challenged by the economic and logistical burdens that consistently afflict patients and hospitals worldwide. In recent decades, healthcare professionals have sought support from computer vision and machine learning algorithms. In particular, wound segmentation has gained interest due to its ability to provide professionals with fast, automatic tissue assessment from standard RGB images. Some approaches have extended segmentation to 3D, enabling more complete and precise healing progress tracking. However, inferring multi-view consistent 3D structures from 2D images remains a challenge. In this paper, we evaluate WoundNeRF, a NeRF SDF-based method for estimating robust wound segmentations from automatically generated annotations. We demonstrate the potential of this paradigm in recovering accurate segmentations by comparing it against state-of-the-art Vision Transformer networks and conventional rasterisation-based algorithms. The code will be released to facilitate further development in this promising paradigm.

Remi Chierchia, Léo Lebrat, David Ahmedt-Aristizabal et al.

Neural Surface Reconstruction has become a standard methodology for indoor 3D reconstruction, with Signed Distance Functions (SDFs) proving particularly effective for representing scene geometry. A variety of applications require a detailed understanding of the scene context, driving the need for object-level semantic signals. While recent methods successfully integrate semantic labels, they often inherit the slow training time and limited scalability of multi-SDF learning. In this paper, we introduce FSTM, a unified approach for learning geometry and semantics through a two-step process: a geometry warm-up using RGB inputs and geometric cues, followed by semantic field estimation. By first optimising geometry without semantic supervision, we observe substantial improvements compared to the standard joint optimisation. Rather than relying on specialised modules or complex multi-SDF designs, FSTM shows that a streamlined formulation is sufficient to achieve strong geometric and semantic reconstructions. Experiments on both synthetic and real-world indoor datasets show that our method outperforms multi-SDF approaches. It trains 2.3x faster on Replica, improves robustness to real-world imperfections on ScanNet++, and achieves higher recall by recovering the surfaces of more objects in the scene. The code will be made available at https://remichierchia.github.io/FSTM.

Wenhui Xiao, Remi Chierchia, Rodrigo Santa Cruz et al.

Neural Radiance Fields (NeRFs) have remodeled 3D scene representation since release. NeRFs can effectively reconstruct complex 3D scenes from 2D images, advancing different fields and applications such as scene understanding, 3D content generation, and robotics. Despite significant research progress, a thorough review of recent innovations, applications, and challenges is lacking. This survey compiles key theoretical advancements and alternative representations and investigates emerging challenges. It further explores applications on reconstruction, highlights NeRFs' impact on computer vision and robotics, and reviews essential datasets and toolkits. By identifying gaps in the literature, this survey discusses open challenges and offers directions for future research.

Wenhui Xiao, Ethan Goan, Rodrigo Santa Cruz et al.

Using accurate depth priors in 3D Gaussian Splatting helps mitigate artifacts caused by sparse training data and textureless surfaces. However, acquiring accurate depth maps requires specialized acquisition systems. Foundation monocular depth estimation models offer a cost-effective alternative, but they suffer from scale ambiguity, multi-view inconsistency, and local geometric inaccuracies, which can degrade rendering performance when applied naively. This paper addresses the challenge of reliably leveraging monocular depth priors for Gaussian Splatting (GS) rendering enhancement. To this end, we introduce a training framework integrating scale-ambiguous and noisy depth priors into geometric supervision. We highlight the importance of learning from weakly aligned depth variations. We introduce a method to isolate ill-posed geometry for selective monocular depth regularization, restricting the propagation of depth inaccuracies into well-reconstructed 3D structures. Extensive experiments across diverse datasets show consistent improvements in geometric accuracy, leading to more faithful depth estimation and higher rendering quality across different GS variants and monocular depth backbones tested.

Ines Vati, Pierrick Bourgeat, Rodrigo Santa Cruz et al.

We introduce neural cortical maps, a continuous and compact neural representation for cortical feature maps, as an alternative to traditional discrete structures such as grids and meshes. It can learn from meshes of arbitrary size and provide learnt features at any resolution. Neural cortical maps enable efficient optimization on the sphere and achieve runtimes up to 30 times faster than classic barycentric interpolation (for the same number of iterations). As a proof of concept, we investigate rigid registration of cortical surfaces and propose NC-Reg, a novel iterative algorithm that involves the use of neural cortical feature maps, gradient descent optimization and a simulated annealing strategy. Through ablation studies and subject-to-template experiments, our method demonstrates sub-degree accuracy ($<1^\circ$ from the global optimum), and serves as a promising robust pre-alignment strategy, which is critical in clinical settings.

Lena Harkämper, Leo Lebrat, David Ahmedt-Aristizabal et al.

Chronic wound monitoring and management require accurate and efficient wound measurement methods. This paper presents a fast, non-invasive 3D wound measurement algorithm based on RGB-D imaging. The method combines RGB-D odometry with B-spline surface reconstruction to generate detailed 3D wound meshes, enabling automatic computation of clinically relevant wound measurements such as perimeter, surface area, and dimensions. We evaluated our system on realistic silicone wound phantoms and measured sub-millimetre 3D reconstruction accuracy compared with high-resolution ground-truth scans. The extracted measurements demonstrated low variability across repeated captures and strong agreement with manual assessments. The proposed pipeline also outperformed a state-of-the-art object-centric RGB-D reconstruction method while maintaining runtimes suitable for real-time clinical deployment. Our approach offers a promising tool for automated wound assessment in both clinical and remote healthcare settings.

Xun Li, Rodrigo Santa Cruz, Mingze Xi et al.

To enable robots to comprehend high-level human instructions and perform complex tasks, a key challenge lies in achieving comprehensive scene understanding: interpreting and interacting with the 3D environment in a meaningful way. This requires a smart map that fuses accurate geometric structure with rich, human-understandable semantics. To address this, we introduce the 3D Queryable Scene Representation (3D QSR), a novel framework built on multimedia data that unifies three complementary 3D representations: (1) 3D-consistent novel view rendering and segmentation from panoptic reconstruction, (2) precise geometry from 3D point clouds, and (3) structured, scalable organization via 3D scene graphs. Built on an object-centric design, the framework integrates with large vision-language models to enable semantic queryability by linking multimodal object embeddings, and supporting object-level retrieval of geometric, visual, and semantic information. The retrieved data are then loaded into a robotic task planner for downstream execution. We evaluate our approach through simulated robotic task planning scenarios in Unity, guided by abstract language instructions and using the indoor public dataset Replica. Furthermore, we apply it in a digital duplicate of a real wet lab environment to test QSR-supported robotic task planning for emergency response. The results demonstrate the framework's ability to facilitate scene understanding and integrate spatial and semantic reasoning, effectively translating high-level human instructions into precise robotic task planning in complex 3D environments.

Remi Chierchia, Rodrigo Santa Cruz, Léo Lebrat et al.

Managing chronic wounds remains a major healthcare challenge, with clinical assessment often relying on subjective and time-consuming manual documentation methods. Although 2D digital videometry frameworks aided the measurement process, these approaches struggle with perspective distortion, a limited field of view, and an inability to capture wound depth, especially in anatomically complex or curved regions. To overcome these limitations, we present Wound3DAssist, a practical framework for 3D wound assessment using monocular consumer-grade videos. Our framework generates accurate 3D models from short handheld smartphone video recordings, enabling non-contact, automatic measurements that are view-independent and robust to camera motion. We integrate 3D reconstruction, wound segmentation, tissue classification, and periwound analysis into a modular workflow. We evaluate Wound3DAssist across digital models with known geometry, silicone phantoms, and real patients. Results show that the framework supports high-quality wound bed visualization, millimeter-level accuracy, and reliable tissue composition analysis. Full assessments are completed in under 20 minutes, demonstrating feasibility for real-world clinical use.

Alejandro Ascarate, Leo Lebrat, Rodrigo Santa Cruz et al.

Variational autoencoders (VAE) encode data into lower-dimensional latent vectors before decoding those vectors back to data. Once trained, decoding a random latent vector from the prior usually does not produce meaningful data, at least when the latent space has more than a dozen dimensions. In this paper, we investigate this issue by drawing insight from high dimensional statistics: in these regimes, the latent vectors of a standard VAE are by construction distributed uniformly on a hypersphere. We propose to formulate the latent variables of a VAE using hyperspherical coordinates, which allows compressing the latent vectors towards an island on the hypersphere, thereby reducing the latent sparsity and we show that this improves the generation ability of the VAE. We propose a new parameterization of the latent space with limited computational overhead.

Wenhui Xiao, Rodrigo Santa Cruz, David Ahmedt-Aristizabal et al.

Neural Rendering representations have significantly contributed to the field of 3D computer vision. Given their potential, considerable efforts have been invested to improve their performance. Nonetheless, the essential question of selecting training views is yet to be thoroughly investigated. This key aspect plays a vital role in achieving high-quality results and aligns with the well-known tenet of deep learning: "garbage in, garbage out". In this paper, we first illustrate the importance of view selection by demonstrating how a simple rotation of the test views within the most pervasive NeRF dataset can lead to consequential shifts in the performance rankings of state-of-the-art techniques. To address this challenge, we introduce a unified framework for view selection methods and devise a thorough benchmark to assess its impact. Significant improvements can be achieved without leveraging error or uncertainty estimation but focusing on uniform view coverage of the reconstructed object, resulting in a training-free approach. Using this technique, we show that high-quality renderings can be achieved faster by using fewer views. We conduct extensive experiments on both synthetic datasets and realistic data to demonstrate the effectiveness of our proposed method compared with random, conventional error-based, and uncertainty-guided view selection.

Rongkai Ma, Leo Lebrat, Rodrigo Santa Cruz et al.

Neural radiance fields (NeRFs) have exhibited potential in synthesizing high-fidelity views of 3D scenes but the standard training paradigm of NeRF presupposes an equal importance for each image in the training set. This assumption poses a significant challenge for rendering specific views presenting intricate geometries, thereby resulting in suboptimal performance. In this paper, we take a closer look at the implications of the current training paradigm and redesign this for more superior rendering quality by NeRFs. Dividing input views into multiple groups based on their visual similarities and training individual models on each of these groups enables each model to specialize on specific regions without sacrificing speed or efficiency. Subsequently, the knowledge of these specialized models is aggregated into a single entity via a teacher-student distillation paradigm, enabling spatial efficiency for online render-ing. Empirically, we evaluate our novel training framework on two publicly available datasets, namely NeRF synthetic and Tanks&Temples. Our evaluation demonstrates that our DaC training pipeline enhances the rendering quality of a state-of-the-art baseline model while exhibiting convergence to a superior minimum.

Léo Lebrat, Rodrigo Santa Cruz, Clinton Fookes et al.

Recent advances in geometric deep-learning introduce complex computational challenges for evaluating the distance between meshes. From a mesh model, point clouds are necessary along with a robust distance metric to assess surface quality or as part of the loss function for training models. Current methods often rely on a uniform random mesh discretization, which yields irregular sampling and noisy distance estimation. In this paper we introduce MongeNet, a fast and optimal transport based sampler that allows for an accurate discretization of a mesh with better approximation properties. We compare our method to the ubiquitous random uniform sampling and show that the approximation error is almost half with a very small computational overhead.

Rodrigo Santa Cruz, Leo Lebrat, Pierrick Bourgeat et al.

The study of neurodegenerative diseases relies on the reconstruction and analysis of the brain cortex from magnetic resonance imaging (MRI). Traditional frameworks for this task like FreeSurfer demand lengthy runtimes, while its accelerated variant FastSurfer still relies on a voxel-wise segmentation which is limited by its resolution to capture narrow continuous objects as cortical surfaces. Having these limitations in mind, we propose DeepCSR, a 3D deep learning framework for cortical surface reconstruction from MRI. Towards this end, we train a neural network model with hypercolumn features to predict implicit surface representations for points in a brain template space. After training, the cortical surface at a desired level of detail is obtained by evaluating surface representations at specific coordinates, and subsequently applying a topology correction algorithm and an isosurface extraction method. Thanks to the continuous nature of this approach and the efficacy of its hypercolumn features scheme, DeepCSR efficiently reconstructs cortical surfaces at high resolution capturing fine details in the cortical folding. Moreover, DeepCSR is as accurate, more precise, and faster than the widely used FreeSurfer toolbox and its deep learning powered variant FastSurfer on reconstructing cortical surfaces from MRI which should facilitate large-scale medical studies and new healthcare applications.

Rodrigo Santa Cruz, Léo Lebrat, Pierrick Bourgeat et al.

Brain morphometry from magnetic resonance imaging (MRI) is a consolidated biomarker for many neurodegenerative diseases. Recent advances in this domain indicate that deep convolutional neural networks can infer morphometric measurements within a few seconds. Nevertheless, the accuracy of the devised model for insightful bio-markers (mean curvature and thickness) remains unsatisfactory. In this paper, we propose a more accurate and efficient neural network model for brain morphometry named HerstonNet. More specifically, we develop a 3D ResNet-based neural network to learn rich features directly from MRI, design a multi-scale regression scheme by predicting morphometric measures at feature maps of different resolutions, and leverage a robust optimization method to avoid poor quality minima and reduce the prediction variance. As a result, HerstonNet improves the existing approach by 24.30% in terms of intraclass correlation coefficient (agreement measure) to FreeSurfer silver-standards while maintaining a competitive run-time.

Rodrigo Santa Cruz, Anoop Cherian, Basura Fernando et al.

This paper presents a framework to recognize temporal compositions of atomic actions in videos. Specifically, we propose to express temporal compositions of actions as semantic regular expressions and derive an inference framework using probabilistic automata to recognize complex actions as satisfying these expressions on the input video features. Our approach is different from existing works that either predict long-range complex activities as unordered sets of atomic actions, or retrieve videos using natural language sentences. Instead, the proposed approach allows recognizing complex fine-grained activities using only pretrained action classifiers, without requiring any additional data, annotations or neural network training. To evaluate the potential of our approach, we provide experiments on synthetic datasets and challenging real action recognition datasets, such as MultiTHUMOS and Charades. We conclude that the proposed approach can extend state-of-the-art primitive action classifiers to vastly more complex activities without large performance degradation.

Rodrigo Santa Cruz, Basura Fernando, Anoop Cherian et al.

The world is fundamentally compositional, so it is natural to think of visual recognition as the recognition of basic visually primitives that are composed according to well-defined rules. This strategy allows us to recognize unseen complex concepts from simple visual primitives. However, the current trend in visual recognition follows a data greedy approach where huge amounts of data are required to learn models for any desired visual concept. In this paper, we build on the compositionality principle and develop an "algebra" to compose classifiers for complex visual concepts. To this end, we learn neural network modules to perform boolean algebra operations on simple visual classifiers. Since these modules form a complete functional set, a classifier for any complex visual concept defined as a boolean expression of primitives can be obtained by recursively applying the learned modules, even if we do not have a single training sample. As our experiments show, using such a framework, we can compose classifiers for complex visual concepts outperforming standard baselines on two well-known visual recognition benchmarks. Finally, we present a qualitative analysis of our method and its properties.

Rodrigo Santa Cruz, Basura Fernando, Anoop Cherian et al.

We present a principled approach to uncover the structure of visual data by solving a novel deep learning task coined visual permutation learning. The goal of this task is to find the permutation that recovers the structure of data from shuffled versions of it. In the case of natural images, this task boils down to recovering the original image from patches shuffled by an unknown permutation matrix. Unfortunately, permutation matrices are discrete, thereby posing difficulties for gradient-based methods. To this end, we resort to a continuous approximation of these matrices using doubly-stochastic matrices which we generate from standard CNN predictions using Sinkhorn iterations. Unrolling these iterations in a Sinkhorn network layer, we propose DeepPermNet, an end-to-end CNN model for this task. The utility of DeepPermNet is demonstrated on two challenging computer vision problems, namely, (i) relative attributes learning and (ii) self-supervised representation learning. Our results show state-of-the-art performance on the Public Figures and OSR benchmarks for (i) and on the classification and segmentation tasks on the PASCAL VOC dataset for (ii).

Stephen Gould, Basura Fernando, Anoop Cherian et al.

Some recent works in machine learning and computer vision involve the solution of a bi-level optimization problem. Here the solution of a parameterized lower-level problem binds variables that appear in the objective of an upper-level problem. The lower-level problem typically appears as an argmin or argmax optimization problem. Many techniques have been proposed to solve bi-level optimization problems, including gradient descent, which is popular with current end-to-end learning approaches. In this technical report we collect some results on differentiating argmin and argmax optimization problems with and without constraints and provide some insightful motivating examples.