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Owing to the diverse scales and varying distributions of sparse matrices arising from practical problems, a multitude of choices are present in the design and implementation of sparse matrix-vector multiplication (SpMV). Researchers have proposed many machine learning-based optimization methods for SpMV. However, these efforts only support one area of sparse matrix format selection, SpMV algorithm selection, or parameter configuration, and rarely consider a large amount of time overhead associated with feature extraction, model inference, and compression format conversion. This paper introduces a machine learning-based cascaded prediction method for SpMV computations that spans various computing stages and hierarchies. Besides, an asynchronous and concurrent computing model has been designed and implemented for runtime model prediction and iterative algorithm solving on heterogeneous computing platforms. It not only offers comprehensive support for the iterative algorithm-solving process leveraging machine learning technology, but also effectively mitigates the preprocessing overheads. Experimental results demonstrate that the cascaded prediction introduced in this paper accelerates SpMV by 1.33x on average, and the iterative algorithm, enhanced by cascaded prediction and asynchronous execution, optimizes by 2.55x on average., Comment: 12 pages, 9 figures, 7 tables

Iterative solvers are frequently used in scientific applications and engineering computations. However, the memory-bound Sparse Matrix-Vector (SpMV) kernel computation hinders the efficiency of iterative algorithms. As modern hardware increasingly supports low-precision computation, the mixed-precision optimization of iterative algorithms has garnered widespread attention. Nevertheless, existing mixed-precision methods pose challenges, including format conversion overhead, tight coupling between storage and computation representation, and the need to store multiple precision copies of data. This paper proposes a floating-point representation based on the group-shared exponent and segmented storage of the mantissa, enabling higher bit utilization of the representation vector and fast switches between different precisions without needing multiple data copies. Furthermore, a stepped mixed-precision iterative algorithm is proposed. Our experimental results demonstrate that, compared with existing floating-point formats, our approach significantly improves iterative algorithms' performance and convergence residuals., Comment: 13 pages, 9 figures

In this paper, we investigate the task of general conversational image retrieval on open-domain images. The objective is to search for images based on interactive conversations between humans and computers. To advance this task, we curate a dataset called ChatSearch. This dataset includes a multi-round multimodal conversational context query for each target image, thereby requiring the retrieval system to find the accurate image from database. Simultaneously, we propose a generative retrieval model named ChatSearcher, which is trained end-to-end to accept/produce interleaved image-text inputs/outputs. ChatSearcher exhibits strong capability in reasoning with multimodal context and can leverage world knowledge to yield visual retrieval results. It demonstrates superior performance on the ChatSearch dataset and also achieves competitive results on other image retrieval tasks and visual conversation tasks. We anticipate that this work will inspire further research on interactive multimodal retrieval systems. Our dataset will be available at https://github.com/joez17/ChatSearch.

Fine-tuning Large Language Models (LLMs) has proven effective for a variety of downstream tasks. However, as LLMs grow in size, the memory demands for backpropagation become increasingly prohibitive. Zeroth-order (ZO) optimization methods offer a memory-efficient alternative by using forward passes to estimate gradients, but the variance of gradient estimates typically scales linearly with the model's parameter dimension$\unicode{x2013}$a significant issue for LLMs. In this paper, we propose the random Subspace Zeroth-order (SubZero) optimization to address the challenges posed by LLMs' high dimensionality. We introduce a low-rank perturbation tailored for LLMs that significantly reduces memory consumption while improving training performance. Additionally, we prove that our gradient estimation closely approximates the backpropagation gradient, exhibits lower variance than traditional ZO methods, and ensures convergence when combined with SGD. Experimental results show that SubZero enhances fine-tuning performance and achieves faster convergence compared to standard ZO approaches like MeZO across various language modeling tasks.

Effective activation functions introduce non-linear transformations, providing neural networks with stronger fitting capa-bilities, which help them better adapt to real data distributions. Huawei Noah's Lab believes that dynamic activation functions are more suitable than static activation functions for enhancing the non-linear capabilities of neural networks. Tsinghua University's related research also suggests using dynamically adjusted activation functions. Building on the ideas of using fine-tuned activation functions from Tsinghua University and Huawei Noah's Lab, we propose a series-based learnable ac-tivation function called LSLU (Learnable Series Linear Units). This method simplifies deep learning networks while im-proving accuracy. This method introduces learnable parameters {\theta} and {\omega} to control the activation function, adapting it to the current layer's training stage and improving the model's generalization. The principle is to increase non-linearity in each activation layer, boosting the network's overall non-linearity. We evaluate LSLU's performance on CIFAR10, CIFAR100, and specific task datasets (e.g., Silkworm), validating its effectiveness. The convergence behavior of the learnable parameters {\theta} and {\omega}, as well as their effects on generalization, are analyzed. Our empirical results show that LSLU enhances the general-ization ability of the original model in various tasks while speeding up training. In VanillaNet training, parameter {\theta} initially decreases, then increases before stabilizing, while {\omega} shows an opposite trend. Ultimately, LSLU achieves a 3.17% accuracy improvement on CIFAR100 for VanillaNet (Table 3). Codes are available at https://github.com/vontran2021/Learnable-series-linear-units-LSLU.

As an emerging vision sensor, the event camera has gained popularity in various vision tasks such as optical flow estimation, stereo matching, and depth estimation due to its high-speed, sparse, and asynchronous event streams. Unlike traditional approaches that use specialized architectures for each specific task, we propose a unified framework, EventMatch, that reformulates these tasks as an event-based dense correspondence matching problem, allowing them to be solved with a single model by directly comparing feature similarities. By utilizing a shared feature similarities module, which integrates knowledge from other event flows via temporal or spatial interactions, and distinct task heads, our network can concurrently perform optical flow estimation from temporal inputs (e.g., two segments of event streams in the temporal domain) and stereo matching from spatial inputs (e.g., two segments of event streams from different viewpoints in the spatial domain). Moreover, we further demonstrate that our unified model inherently supports cross-task transfer since the architecture and parameters are shared across tasks. Without the need for retraining on each task, our model can effectively handle both optical flow and disparity estimation simultaneously. The experiment conducted on the DSEC benchmark demonstrates that our model exhibits superior performance in both optical flow and disparity estimation tasks, outperforming existing state-of-the-art methods. Our unified approach not only advances event-based models but also opens new possibilities for cross-task transfer and inter-task fusion in both spatial and temporal dimensions. Our code will be available later.

Event-based video reconstruction has garnered increasing attention due to its advantages, such as high dynamic range and rapid motion capture capabilities. However, current methods often prioritize the extraction of temporal information from continuous event flow, leading to an overemphasis on low-frequency texture features in the scene, resulting in over-smoothing and blurry artifacts. Addressing this challenge necessitates the integration of conditional information, encompassing temporal features, low-frequency texture, and high-frequency events, to guide the Denoising Diffusion Probabilistic Model (DDPM) in producing accurate and natural outputs. To tackle this issue, we introduce a novel approach, the Temporal Residual Guided Diffusion Framework, which effectively leverages both temporal and frequency-based event priors. Our framework incorporates three key conditioning modules: a pre-trained low-frequency intensity estimation module, a temporal recurrent encoder module, and an attention-based high-frequency prior enhancement module. In order to capture temporal scene variations from the events at the current moment, we employ a temporal-domain residual image as the target for the diffusion model. Through the combination of these three conditioning paths and the temporal residual framework, our framework excels in reconstructing high-quality videos from event flow, mitigating issues such as artifacts and over-smoothing commonly observed in previous approaches. Extensive experiments conducted on multiple benchmark datasets validate the superior performance of our framework compared to prior event-based reconstruction methods., Comment: Accepted by ECCV2024

Existing learning-based denoising methods typically train models to generalize the image prior from large-scale datasets, suffering from the variability in noise distributions encountered in real-world scenarios. In this work, we propose a new perspective on the denoising challenge by highlighting the distinct separation between noise and image priors. This insight forms the basis for our development of conditional optimization framework, designed to overcome the constraints of traditional denoising framework. To this end, we introduce a Locally Noise Prior Estimation (LoNPE) algorithm, which accurately estimates the noise prior directly from a single raw noisy image. This estimation acts as an explicit prior representation of the camera sensor's imaging environment, distinct from the image prior of scenes. Additionally, we design an auxiliary learnable LoNPE network tailored for practical application to sRGB noisy images. Leveraging the estimated noise prior, we present a novel Conditional Denoising Transformer (Condformer), by incorporating the noise prior into a conditional self-attention mechanism. This integration allows the Condformer to segment the optimization process into multiple explicit subspaces, significantly enhancing the model's generalization and flexibility. Extensive experimental evaluations on both synthetic and real-world datasets, demonstrate that the proposed method achieves superior performance over current state-of-the-art methods. The source code is available at https://github.com/YuanfeiHuang/Condformer., Comment: Under Review

Deep priors have emerged as potent methods in hyperspectral image (HSI) reconstruction. While most methods emphasize space-domain learning using image space priors like non-local similarity, frequency-domain learning using image frequency priors remains neglected, limiting the reconstruction capability of networks. In this paper, we first propose a Hyperspectral Frequency Correlation (HFC) prior rooted in in-depth statistical frequency analyses of existent HSI datasets. Leveraging the HFC prior, we subsequently establish the frequency domain learning composed of a Spectral-wise self-Attention of Frequency (SAF) and a Spectral-spatial Interaction of Frequency (SIF) targeting low-frequency and high-frequency components, respectively. The outputs of SAF and SIF are adaptively merged by a learnable gating filter, thus achieving a thorough exploitation of image frequency priors. Integrating the frequency domain learning and the existing space domain learning, we finally develop the Correlation-driven Mixing Domains Transformer (CMDT) for HSI reconstruction. Extensive experiments highlight that our method surpasses various state-of-the-art (SOTA) methods in reconstruction quality and computational efficiency., Comment: 14 pages, 11 figures

Second-order optimizers, maintaining a matrix termed a preconditioner, are superior to first-order optimizers in both theory and practice. The states forming the preconditioner and its inverse root restrict the maximum size of models trained by second-order optimizers. To address this, compressing 32-bit optimizer states to lower bitwidths has shown promise in reducing memory usage. However, current approaches only pertain to first-order optimizers. In this paper, we propose the first 4-bit second-order optimizers, exemplified by 4-bit Shampoo, maintaining performance similar to that of 32-bit ones. We show that quantizing the eigenvector matrix of the preconditioner in 4-bit Shampoo is remarkably better than quantizing the preconditioner itself both theoretically and experimentally. By rectifying the orthogonality of the quantized eigenvector matrix, we enhance the approximation of the preconditioner's eigenvector matrix, which also benefits the computation of its inverse 4-th root. Besides, we find that linear square quantization slightly outperforms dynamic tree quantization when quantizing second-order optimizer states. Evaluation on various networks for image classification and natural language modeling demonstrates that our 4-bit Shampoo achieves comparable performance to its 32-bit counterpart while being more memory-efficient.

Accurately estimating scene lighting is critical for applications such as mixed reality. Existing works estimate illumination by generating illumination maps or regressing illumination parameters. However, the method of generating illumination maps has poor generalization performance and parametric models such as Spherical Harmonic (SH) and Spherical Gaussian (SG) fall short in capturing high-frequency or low-frequency components. This paper presents MixLight, a joint model that utilizes the complementary characteristics of SH and SG to achieve a more complete illumination representation, which uses SH and SG to capture low-frequency ambient and high-frequency light sources respectively. In addition, a special spherical light source sparsemax (SLSparsemax) module that refers to the position and brightness relationship between spherical light sources is designed to improve their sparsity, which is significant but omitted by prior works. Extensive experiments demonstrate that MixLight surpasses state-of-the-art (SOTA) methods on multiple metrics. In addition, experiments on Web Dataset also show that MixLight as a parametric method has better generalization performance than non-parametric methods.

Sparse matrix-vector multiplication (SpMV) is a crucial computing kernel with widespread applications in iterative algorithms. Over the past decades, research on SpMV optimization has made remarkable strides, giving rise to various optimization contributions. However, the comprehensive and systematic literature survey that introduces, analyzes, discusses, and summarizes the advancements of SpMV in recent years is currently lacking. Aiming to fill this gap, this paper compares existing techniques and analyzes their strengths and weaknesses. We begin by highlighting two representative applications of SpMV, then conduct an in-depth overview of the important techniques that optimize SpMV on modern architectures, which we specifically classify as classic, auto-tuning, machine learning, and mixed-precision-based optimization. We also elaborate on the hardware-based architectures, including CPU, GPU, FPGA, processing in Memory, heterogeneous, and distributed platforms. We present a comprehensive experimental evaluation that compares the performance of state-of-the-art SpMV implementations. Based on our findings, we identify several challenges and point out future research directions. This survey is intended to provide researchers with a comprehensive understanding of SpMV optimization on modern architectures and provide guidance for future work., Comment: 34 pages, 18 figures, 16 tables

Spike cameras, leveraging spike-based integration sampling and high temporal resolution, offer distinct advantages over standard cameras. However, existing approaches reliant on spike cameras often assume optimal illumination, a condition frequently unmet in real-world scenarios. To address this, we introduce SpikeNeRF, the first work that derives a NeRF-based volumetric scene representation from spike camera data. Our approach leverages NeRF's multi-view consistency to establish robust self-supervision, effectively eliminating erroneous measurements and uncovering coherent structures within exceedingly noisy input amidst diverse real-world illumination scenarios. The framework comprises two core elements: a spike generation model incorporating an integrate-and-fire neuron layer and parameters accounting for non-idealities, such as threshold variation, and a spike rendering loss capable of generalizing across varying illumination conditions. We describe how to effectively optimize neural radiance fields to render photorealistic novel views from the novel continuous spike stream, demonstrating advantages over other vision sensors in certain scenes. Empirical evaluations conducted on both real and novel realistically simulated sequences affirm the efficacy of our methodology. The dataset and source code are released at https://github.com/BIT-Vision/SpikeNeRF., Comment: Accepted by CVPR 2024

As a bio-inspired vision sensor, the spike camera emulates the operational principles of the fovea, a compact retinal region, by employing spike discharges to encode the accumulation of per-pixel luminance intensity. Leveraging its high temporal resolution and bio-inspired neuromorphic design, the spike camera holds significant promise for advancing computer vision applications. Saliency detection mimics the behavior of human beings and captures the most salient region from the scenes. In this paper, we investigate the visual saliency in the continuous spike stream for the first time. To effectively process the binary spike stream, we propose a Recurrent Spiking Transformer (RST) framework, which is based on a full spiking neural network. Our framework enables the extraction of spatio-temporal features from the continuous spatio-temporal spike stream while maintaining low power consumption. To facilitate the training and validation of our proposed model, we build a comprehensive real-world spike-based visual saliency dataset, enriched with numerous light conditions. Extensive experiments demonstrate the superior performance of our Recurrent Spiking Transformer framework in comparison to other spike neural network-based methods. Our framework exhibits a substantial margin of improvement in capturing and highlighting visual saliency in the spike stream, which not only provides a new perspective for spike-based saliency segmentation but also shows a new paradigm for full SNN-based transformer models. The code and dataset are available at \url{https://github.com/BIT-Vision/SVS}., Comment: Accepted by AAAI 2024

Using a variety of methods developed in the theory of finite-dimensional quasi-Hopf algebras, we classify all finite-dimensional coradically graded pointed coquasi-Hopf algebras over abelian groups. As a consequence, we partially confirm the generation conjecture of pointed finite tensor categories due to Etingof, Gelaki, Nikshych and Ostrik., Comment: 30 pages. arXiv admin note: text overlap with arXiv:1710.07917

The fusion of a low-spatial-resolution hyperspectral image (LR-HSI) with a high-spatial-resolution multispectral image (HR-MSI) has garnered increasing research interest. However, most fusion methods solely focus on the fusion algorithm itself and overlook the degradation models, which results in unsatisfactory performance in practical scenarios. To fill this gap, we propose physics-inspired degradation models (PIDM) to model the degradation of LR-HSI and HR-MSI, which comprises a spatial degradation network (SpaDN) and a spectral degradation network (SpeDN). SpaDN and SpeDN are designed based on two insights. First, we employ spatial warping and spectral modulation operations to simulate lens aberrations, thereby introducing non-uniformity into the spatial and spectral degradation processes. Second, we utilize asymmetric downsampling and parallel downsampling operations to separately reduce the spatial and spectral resolutions of the images, thus ensuring the matching of spatial and spectral degradation processes with specific physical characteristics. Once SpaDN and SpeDN are established, we adopt a self-supervised training strategy to optimize the network parameters and provide a plug-and-play solution for fusion methods. Comprehensive experiments demonstrate that our proposed PIDM can boost the fusion performance of existing fusion methods in practical scenarios.

Multi-modal large language models(MLLMs) have achieved remarkable progress and demonstrated powerful knowledge comprehension and reasoning abilities. However, the mastery of domain-specific knowledge, which is essential for evaluating the intelligence of MLLMs, continues to be a challenge. Current multi-modal benchmarks for domain-specific knowledge concentrate on multiple-choice questions and are predominantly available in English, which imposes limitations on the comprehensiveness of the evaluation. To this end, we introduce CMMU, a novel benchmark for multi-modal and multi-type question understanding and reasoning in Chinese. CMMU consists of 3,603 questions in 7 subjects, covering knowledge from primary to high school. The questions can be categorized into 3 types: multiple-choice, multiple-response, and fill-in-the-blank, bringing greater challenges to MLLMs. In addition, we propose an evaluation strategy called Positional Error Variance for assessing multiple-choice questions. The strategy aims to perform a quantitative analysis of position bias. We evaluate seven open-source MLLMs along with GPT4-V, Gemini-Pro, and Qwen-VL-Plus. The results demonstrate that CMMU poses a significant challenge to the recent MLLMs. The data and code are available at https://github.com/FlagOpen/CMMU.

Depth completion, aiming to predict dense depth maps from sparse depth measurements, plays a crucial role in many computer vision related applications. Deep learning approaches have demonstrated overwhelming success in this task. However, high-precision depth completion without relying on the ground-truth data, which are usually costly, still remains challenging. The reason lies on the ignorance of 3D structural information in most previous unsupervised solutions, causing inaccurate spatial propagation and mixed-depth problems. To alleviate the above challenges, this paper explores the utilization of 3D perceptual features and multi-view geometry consistency to devise a high-precision self-supervised depth completion method. Firstly, a 3D perceptual spatial propagation algorithm is constructed with a point cloud representation and an attention weighting mechanism to capture more reasonable and favorable neighboring features during the iterative depth propagation process. Secondly, the multi-view geometric constraints between adjacent views are explicitly incorporated to guide the optimization of the whole depth completion model in a self-supervised manner. Extensive experiments on benchmark datasets of NYU-Depthv2 and VOID demonstrate that the proposed model achieves the state-of-the-art depth completion performance compared with other unsupervised methods, and competitive performance compared with previous supervised methods., Comment: 13 pages, 7 figures

Dual-Camera Compressed Hyperspectral Imaging (DCCHI) offers the capability to reconstruct 3D Hyperspectral Image (HSI) by fusing compressive and Panchromatic (PAN) image, which has shown great potential for snapshot hyperspectral imaging in practice. In this paper, we introduce a novel DCCHI reconstruction network, the Intra-Inter Similarity Exploiting Transformer (In2SET). Our key insight is to make full use of the PAN image to assist the reconstruction. To this end, we propose using the intra-similarity within the PAN image as a proxy for approximating the intra-similarity in the original HSI, thereby offering an enhanced content prior for more accurate HSI reconstruction. Furthermore, we aim to align the features from the underlying HSI with those of the PAN image, maintaining semantic consistency and introducing new contextual information for the reconstruction process. By integrating In2SET into a PAN-guided unrolling framework, our method substantially enhances the spatial-spectral fidelity and detail of the reconstructed images, providing a more comprehensive and accurate depiction of the scene. Extensive experiments conducted on both real and simulated datasets demonstrate that our approach consistently outperforms existing state-of-the-art methods in terms of reconstruction quality and computational complexity. Code will be released., Comment: CVPR 2024

We study the computational complexity of a criterion and an algorithm for diagonalization of multivariate homogeneous polynomials, that is, expressing them as sums of powers of independent linear forms. They are based on Harrison's center theory and only require solving linear and quadratic systems of equations. Detailed descriptions and computational complexity of each step of the algorithm are provided. The complexity analysis focuses on the impacts of problem sizes, including the number of variables and the degree of given polynomials. We show that this algorithm runs in polynomial time and validate it through numerical experiments. Other diagonalization algorithms are reviewed and compared in terms of complexity.

Recently, the mainstream practice for training low-light raw image denoising methods has shifted towards employing synthetic data. Noise modeling, which focuses on characterizing the noise distribution of real-world sensors, profoundly influences the effectiveness and practicality of synthetic data. Currently, physics-based noise modeling struggles to characterize the entire real noise distribution, while learning-based noise modeling impractically depends on paired real data. In this paper, we propose a novel strategy: learning the noise model from dark frames instead of paired real data, to break down the data dependency. Based on this strategy, we introduce an efficient physics-guided noise neural proxy (PNNP) to approximate the real-world sensor noise model. Specifically, we integrate physical priors into neural proxies and introduce three efficient techniques: physics-guided noise decoupling (PND), physics-guided proxy model (PPM), and differentiable distribution loss (DDL). PND decouples the dark frame into different components and handles different levels of noise flexibly, which reduces the complexity of noise modeling. PPM incorporates physical priors to constrain the generated noise, which promotes the accuracy of noise modeling. DDL provides explicit and reliable supervision for noise distribution, which promotes the precision of noise modeling. PNNP exhibits powerful potential in characterizing the real noise distribution. Extensive experiments on public datasets demonstrate superior performance in practical low-light raw image denoising. The code will be available at \url{https://github.com/fenghansen/PNNP}., Comment: Under Review

Spike camera is a new type of bio-inspired vision sensor that records light intensity in the form of a spike array with high temporal resolution (20,000 Hz). This new paradigm of vision sensor offers significant advantages for many vision tasks such as high speed image reconstruction. However, existing spike-based approaches typically assume that the scenes are with sufficient light intensity, which is usually unavailable in many real-world scenarios such as rainy days or dusk scenes. To unlock more spike-based application scenarios, we propose a Recurrent Spike-based Image Restoration (RSIR) network, which is the first work towards restoring clear images from spike arrays under general illumination. Specifically, to accurately describe the noise distribution under different illuminations, we build a physical-based spike noise model according to the sampling process of the spike camera. Based on the noise model, we design our RSIR network which consists of an adaptive spike transformation module, a recurrent temporal feature fusion module, and a frequency-based spike denoising module. Our RSIR can process the spike array in a recursive manner to ensure that the spike temporal information is well utilized. In the training process, we generate the simulated spike data based on our noise model to train our network. Extensive experiments on real-world datasets with different illuminations demonstrate the effectiveness of the proposed network. The code and dataset are released at https://github.com/BIT-Vision/RSIR., Comment: Accepted by ACM MM 2023

The recent neural surface reconstruction by volume rendering approaches have made much progress by achieving impressive surface reconstruction quality, but are still limited to dense and highly accurate posed views. To overcome such drawbacks, this paper pays special attention on the consistent surface reconstruction from sparse views with noisy camera poses. Unlike previous approaches, the key difference of this paper is to exploit the multi-view constraints directly from the explicit geometry of the neural surface, which can be used as effective regularization to jointly learn the neural surface and refine the camera poses. To build effective multi-view constraints, we introduce a fast differentiable on-surface intersection to generate on-surface points, and propose view-consistent losses based on such differentiable points to regularize the neural surface learning. Based on this point, we propose a jointly learning strategy for neural surface and camera poses, named SC-NeuS, to perform geometry-consistent surface reconstruction in an end-to-end manner. With extensive evaluation on public datasets, our SC-NeuS can achieve consistently better surface reconstruction results with fine-grained details than previous state-of-the-art neural surface reconstruction approaches, especially from sparse and noisy camera views.

Image denoising is a fundamental problem in computational photography, where achieving high perception with low distortion is highly demanding. Current methods either struggle with perceptual quality or suffer from significant distortion. Recently, the emerging diffusion model has achieved state-of-the-art performance in various tasks and demonstrates great potential for image denoising. However, stimulating diffusion models for image denoising is not straightforward and requires solving several critical problems. For one thing, the input inconsistency hinders the connection between diffusion models and image denoising. For another, the content inconsistency between the generated image and the desired denoised image introduces distortion. To tackle these problems, we present a novel strategy called the Diffusion Model for Image Denoising (DMID) by understanding and rethinking the diffusion model from a denoising perspective. Our DMID strategy includes an adaptive embedding method that embeds the noisy image into a pre-trained unconditional diffusion model and an adaptive ensembling method that reduces distortion in the denoised image. Our DMID strategy achieves state-of-the-art performance on both distortion-based and perception-based metrics, for both Gaussian and real-world image denoising.The code is available at https://github.com/Li-Tong-621/DMID., Comment: 18 pages,15 figures