arXiv subjects

Qing Guo

Publications and source records attributed to Qing Guo.

At least 163 records · Page 9Linked to original sources

arXiv2022

NPC: Neuron Path Coverage via Characterizing Decision Logic of Deep Neural Networks

Deep learning has recently been widely applied to many applications across different domains, e.g., image classification and audio recognition. However, the quality of Deep Neural Networks (DNNs) still raises concerns in the practical operational environment, which calls for systematic testing, especially in safety-critical scenarios. Inspired by software testing, a number of structural coverage criteria are designed and proposed to measure the test adequacy of DNNs. However, due to the blackbox nature of DNN, the existing structural coverage criteria are difficult to interpret, making it hard to understand the underlying principles of these criteria. The relationship between the structural coverage and the decision logic of DNNs is unknown. Moreover, recent studies have further revealed the non-existence of correlation between the structural coverage and DNN defect detection, which further posts concerns on what a suitable DNN testing criterion should be. In this paper, we propose the interpretable coverage criteria through constructing the decision structure of a DNN. Mirroring the control flow graph of the traditional program, we first extract a decision graph from a DNN based on its interpretation, where a path of the decision graph represents a decision logic of the DNN. Based on the control flow and data flow of the decision graph, we propose two variants of path coverage to measure the adequacy of the test cases in exercising the decision logic. The higher the path coverage, the more diverse decision logic the DNN is expected to be explored. Our large-scale evaluation results demonstrate that: the path in the decision graph is effective in characterizing the decision of the DNN, and the proposed coverage criteria are also sensitive with errors including natural errors and adversarial examples, and strongly correlated with the output impartiality.

cs.LG↗

arXiv2022

Countering Malicious DeepFakes: Survey, Battleground, and Horizon

The creation or manipulation of facial appearance through deep generative approaches, known as DeepFake, have achieved significant progress and promoted a wide range of benign and malicious applications, e.g., visual effect assistance in movie and misinformation generation by faking famous persons. The evil side of this new technique poses another popular study, i.e., DeepFake detection aiming to identify the fake faces from the real ones. With the rapid development of the DeepFake-related studies in the community, both sides have formed the relationship of battleground, pushing the improvements of each other and inspiring new directions, e.g., the evasion of DeepFake detection. Nevertheless, the overview of such battleground and the new direction is unclear and neglected by recent surveys due to the rapid increase of related publications, limiting the in-depth understanding of the tendency and future works. To fill this gap, in this paper, we provide a comprehensive overview and detailed analysis of the research work on the topic of DeepFake generation, DeepFake detection as well as evasion of DeepFake detection, with more than 318 research papers carefully surveyed. We present the taxonomy of various DeepFake generation methods and the categorization of various DeepFake detection methods, and more importantly, we showcase the battleground between the two parties with detailed interactions between the adversaries (DeepFake generation) and the defenders (DeepFake detection). The battleground allows fresh perspective into the latest landscape of the DeepFake research and can provide valuable analysis towards the research challenges and opportunities as well as research trends and future directions. We also elaborately design interactive diagrams (http://www.xujuefei.com/dfsurvey) to allow researchers to explore their own interests on popular DeepFake generators or detectors.

cs.CV↗

arXiv2022

MISF: Multi-level Interactive Siamese Filtering for High-Fidelity Image Inpainting

Although achieving significant progress, existing deep generative inpainting methods are far from real-world applications due to the low generalization across different scenes. As a result, the generated images usually contain artifacts or the filled pixels differ greatly from the ground truth. Image-level predictive filtering is a widely used image restoration technique, predicting suitable kernels adaptively according to different input scenes. Inspired by this inherent advantage, we explore the possibility of addressing image inpainting as a filtering task. To this end, we first study the advantages and challenges of image-level predictive filtering for image inpainting: the method can preserve local structures and avoid artifacts but fails to fill large missing areas. Then, we propose semantic filtering by conducting filtering on the deep feature level, which fills the missing semantic information but fails to recover the details. To address the issues while adopting the respective advantages, we propose a novel filtering technique, i.e., Multilevel Interactive Siamese Filtering (MISF), which contains two branches: kernel prediction branch (KPB) and semantic & image filtering branch (SIFB). These two branches are interactively linked: SIFB provides multi-level features for KPB while KPB predicts dynamic kernels for SIFB. As a result, the final method takes the advantage of effective semantic & image-level filling for high-fidelity inpainting. We validate our method on three challenging datasets, i.e., Dunhuang, Places2, and CelebA. Our method outperforms state-of-the-art baselines on four metrics, i.e., L1, PSNR, SSIM, and LPIPS. Please try the released code and model at https://github.com/tsingqguo/misf.

cs.CV↗

arXiv2022

Adversarial Rain Attack and Defensive Deraining for DNN Perception

Rain often poses inevitable threats to deep neural network (DNN) based perception systems, and a comprehensive investigation of the potential risks of the rain to DNNs is of great importance. However, it is rather difficult to collect or synthesize rainy images that can represent all rain situations that would possibly occur in the real world. To this end, in this paper, we start from a new perspective and propose to combine two totally different studies, i.e., rainy image synthesis and adversarial attack. We first present an adversarial rain attack, with which we could simulate various rain situations with the guidance of deployed DNNs and reveal the potential threat factors that can be brought by rain. In particular, we design a factor-aware rain generation that synthesizes rain streaks according to the camera exposure process and models the learnable rain factors for adversarial attack. With this generator, we perform the adversarial rain attack against the image classification and object detection. To defend the DNNs from the negative rain effect, we also present a defensive deraining strategy, for which we design an adversarial rain augmentation that uses mixed adversarial rain layers to enhance deraining models for downstream DNN perception. Our large-scale evaluation on various datasets demonstrates that our synthesized rainy images with realistic appearances not only exhibit strong adversarial capability against DNNs, but also boost the deraining models for defensive purposes, building the foundation for further rain-robust perception studies.

cs.CV↗

arXiv2022

Uncertainty-Aware Cascaded Dilation Filtering for High-Efficiency Deraining

Deraining is a significant and fundamental computer vision task, aiming to remove the rain streaks and accumulations in an image or video captured under a rainy day. Existing deraining methods usually make heuristic assumptions of the rain model, which compels them to employ complex optimization or iterative refinement for high recovery quality. This, however, leads to time-consuming methods and affects the effectiveness for addressing rain patterns deviated from from the assumptions. In this paper, we propose a simple yet efficient deraining method by formulating deraining as a predictive filtering problem without complex rain model assumptions. Specifically, we identify spatially-variant predictive filtering (SPFilt) that adaptively predicts proper kernels via a deep network to filter different individual pixels. Since the filtering can be implemented via well-accelerated convolution, our method can be significantly efficient. We further propose the EfDeRain+ that contains three main contributions to address residual rain traces, multi-scale, and diverse rain patterns without harming the efficiency. First, we propose the uncertainty-aware cascaded predictive filtering (UC-PFilt) that can identify the difficulties of reconstructing clean pixels via predicted kernels and remove the residual rain traces effectively. Second, we design the weight-sharing multi-scale dilated filtering (WS-MS-DFilt) to handle multi-scale rain streaks without harming the efficiency. Third, to eliminate the gap across diverse rain patterns, we propose a novel data augmentation method (i.e., RainMix) to train our deep models. By combining all contributions with sophisticated analysis on different variants, our final method outperforms baseline methods on four single-image deraining datasets and one video deraining dataset in terms of both recovery quality and speed.

cs.CV↗

arXiv2021

Existence and local uniqueness of normalized peak solutions for a Schrodinger-Newton system

In this paper, we investigate the existence and local uniqueness of normalized peak solutions for a Schrödinger-Newton system under the assumption that the trapping potential is degenerate and has non-isolated critical points. First we investigate the existence and local uniqueness of normalized single-peak solutions for the Schrödinger-Newton system. Precisely, we give the precise description of the chemical potential $μ$ and the attractive interaction $a$. Then we apply the finite dimensional reduction method to obtain the existence of single-peak solutions. Furthermore, using various local Pohozaev identities, blow-up analysis and the maximum principle, we prove the local uniqueness of single-peak solutions by precise analysis of the concentrated points and the Lagrange multiplier. Finally, we also prove the nonexistence of multi-peak solutions for the Schrödinger-Newton system, which is markedly different from the corresponding Schrödinger equation. The nonlocal term results in this difference. The main difficulties come from the estimates on Lagrange multiplier, the different degenerate rates along different directions at the critical point of $P(x)$ and some complicated estimates involved by the nonlocal term. To our best knowledge, it may be the first time to study the existence and local uniqueness of solutions with prescribed $L^{2}$-norm for the Schrödinger-Newton system.

math.AP↗

arXiv2021

ArchRepair: Block-Level Architecture-Oriented Repairing for Deep Neural Networks

Over the past few years, deep neural networks (DNNs) have achieved tremendous success and have been continuously applied in many application domains. However, during the practical deployment in the industrial tasks, DNNs are found to be erroneous-prone due to various reasons such as overfitting, lacking robustness to real-world corruptions during practical usage. To address these challenges, many recent attempts have been made to repair DNNs for version updates under practical operational contexts by updating weights (i.e., network parameters) through retraining, fine-tuning, or direct weight fixing at a neural level. In this work, as the first attempt, we initiate to repair DNNs by jointly optimizing the architecture and weights at a higher (i.e., block) level. We first perform empirical studies to investigate the limitation of whole network-level and layer-level repairing, which motivates us to explore a novel repairing direction for DNN repair at the block level. To this end, we first propose adversarial-aware spectrum analysis for vulnerable block localization that considers the neurons' status and weights' gradients in blocks during the forward and backward processes, which enables more accurate candidate block localization for repairing even under a few examples. Then, we further propose the architecture-oriented search-based repairing that relaxes the targeted block to a continuous repairing search space at higher deep feature levels. By jointly optimizing the architecture and weights in that space, we can identify a much better block architecture. We implement our proposed repairing techniques as a tool, named ArchRepair, and conduct extensive experiments to validate the proposed method. The results show that our method can not only repair but also enhance accuracy & robustness, outperforming the state-of-the-art DNN repair techniques.

cs.LG↗

arXiv2021

Benchmarking Shadow Removal for Facial Landmark Detection and Beyond

Facial landmark detection is a very fundamental and significant vision task with many important applications. In practice, facial landmark detection can be affected by a lot of natural degradations. One of the most common and important degradations is the shadow caused by light source blocking. While many advanced shadow removal methods have been proposed to recover the image quality in recent years, their effects to facial landmark detection are not well studied. For example, it remains unclear whether shadow removal could enhance the robustness of facial landmark detection to diverse shadow patterns or not. In this work, for the first attempt, we construct a novel benchmark to link two independent but related tasks (i.e., shadow removal and facial landmark detection). In particular, the proposed benchmark covers diverse face shadows with different intensities, sizes, shapes, and locations. Moreover, to mine hard shadow patterns against facial landmark detection, we propose a novel method (i.e., adversarial shadow attack), which allows us to construct a challenging subset of the benchmark for a comprehensive analysis. With the constructed benchmark, we conduct extensive analysis on three state-of-the-art shadow removal methods and three landmark detectors. The observation of this work motivates us to design a novel detection-aware shadow removal framework, which empowers shadow removal to achieve higher restoration quality and enhance the shadow robustness of deployed facial landmark detectors.

cs.CV↗

arXiv2021

FakeLocator: Robust Localization of GAN-Based Face Manipulations

Full face synthesis and partial face manipulation by virtue of the generative adversarial networks (GANs) and its variants have raised wide public concerns. In the multi-media forensics area, detecting and ultimately locating the image forgery has become an imperative task. In this work, we investigate the architecture of existing GAN-based face manipulation methods and observe that the imperfection of upsampling methods therewithin could be served as an important asset for GAN-synthesized fake image detection and forgery localization. Based on this basic observation, we have proposed a novel approach, termed FakeLocator, to obtain high localization accuracy, at full resolution, on manipulated facial images. To the best of our knowledge, this is the very first attempt to solve the GAN-based fake localization problem with a gray-scale fakeness map that preserves more information of fake regions. To improve the universality of FakeLocator across multifarious facial attributes, we introduce an attention mechanism to guide the training of the model. To improve the universality of FakeLocator across different DeepFake methods, we propose partial data augmentation and single sample clustering on the training images. Experimental results on popular FaceForensics++, DFFD datasets and seven different state-of-the-art GAN-based face generation methods have shown the effectiveness of our method. Compared with the baselines, our method performs better on various metrics. Moreover, the proposed method is robust against various real-world facial image degradations such as JPEG compression, low-resolution, noise, and blur.

cs.CV↗

arXiv2021

Learning to Adversarially Blur Visual Object Tracking

Motion blur caused by the moving of the object or camera during the exposure can be a key challenge for visual object tracking, affecting tracking accuracy significantly. In this work, we explore the robustness of visual object trackers against motion blur from a new angle, i.e., adversarial blur attack (ABA). Our main objective is to online transfer input frames to their natural motion-blurred counterparts while misleading the state-of-the-art trackers during the tracking process. To this end, we first design the motion blur synthesizing method for visual tracking based on the generation principle of motion blur, considering the motion information and the light accumulation process. With this synthetic method, we propose optimization-based ABA (OP-ABA) by iteratively optimizing an adversarial objective function against the tracking w.r.t. the motion and light accumulation parameters. The OP-ABA is able to produce natural adversarial examples but the iteration can cause heavy time cost, making it unsuitable for attacking real-time trackers. To alleviate this issue, we further propose one-step ABA (OS-ABA) where we design and train a joint adversarial motion and accumulation predictive network (JAMANet) with the guidance of OP-ABA, which is able to efficiently estimate the adversarial motion and accumulation parameters in a one-step way. The experiments on four popular datasets (e.g., OTB100, VOT2018, UAV123, and LaSOT) demonstrate that our methods are able to cause significant accuracy drops on four state-of-the-art trackers with high transferability. Please find the source code at \url{https://github.com/tsingqguo/ABA}.

cs.CV↗

arXiv2021

AdvFilter: Predictive Perturbation-aware Filtering against Adversarial Attack via Multi-domain Learning

High-level representation-guided pixel denoising and adversarial training are independent solutions to enhance the robustness of CNNs against adversarial attacks by pre-processing input data and re-training models, respectively. Most recently, adversarial training techniques have been widely studied and improved while the pixel denoising-based method is getting less attractive. However, it is still questionable whether there exists a more advanced pixel denoising-based method and whether the combination of the two solutions benefits each other. To this end, we first comprehensively investigate two kinds of pixel denoising methods for adversarial robustness enhancement (i.e., existing additive-based and unexplored filtering-based methods) under the loss functions of image-level and semantic-level, respectively, showing that pixel-wise filtering can obtain much higher image quality (e.g., higher PSNR) as well as higher robustness (e.g., higher accuracy on adversarial examples) than existing pixel-wise additive-based method. However, we also observe that the robustness results of the filtering-based method rely on the perturbation amplitude of adversarial examples used for training. To address this problem, we propose predictive perturbation-aware & pixel-wise filtering}, where dual-perturbation filtering and an uncertainty-aware fusion module are designed and employed to automatically perceive the perturbation amplitude during the training and testing process. The method is termed as AdvFilter. Moreover, we combine adversarial pixel denoising methods with three adversarial training-based methods, hinting that considering data and models jointly is able to achieve more robust CNNs. The experiments conduct on NeurIPS-2017DEV, SVHN and CIFAR10 datasets and show advantages over enhancing CNNs' robustness, high generalization to different models and noise levels.

cs.CV↗

arXiv2021

JPGNet: Joint Predictive Filtering and Generative Network for Image Inpainting

Image inpainting aims to restore the missing regions of corrupted images and make the recovery result identical to the originally complete image, which is different from the common generative task emphasizing the naturalness or realism of generated images. Nevertheless, existing works usually regard it as a pure generation problem and employ cutting-edge deep generative techniques to address it. The generative networks can fill the main missing parts with realistic contents but usually distort the local structures or introduce obvious artifacts. In this paper, for the first time, we formulate image inpainting as a mix of two problems, predictive filtering and deep generation. Predictive filtering is good at preserving local structures and removing artifacts but falls short to complete the large missing regions. The deep generative network can fill the numerous missing pixels based on the understanding of the whole scene but hardly restores the details identical to the original ones. To make use of their respective advantages, we propose the joint predictive filtering and generative network (JPGNet) that contains three branches: predictive filtering & uncertainty network (PFUNet), deep generative network, and uncertainty-aware fusion network (UAFNet). The PFUNet can adaptively predict pixel-wise kernels for filtering-based inpainting according to the input image and output an uncertainty map. This map indicates the pixels should be processed by filtering or generative networks, which is further fed to the UAFNet for a smart combination between filtering and generative results. Note that, our method as a novel inpainting framework can benefit any existing generation-based methods. We validate our method on three public datasets, Dunhuang, Places2, and CelebA, and demonstrate that our method can enhance three state-of-the-art generative methods significantly with slightly extra time costs.

cs.CV↗

arXiv2021

Pasadena: Perceptually Aware and Stealthy Adversarial Denoise Attack

Image denoising can remove natural noise that widely exists in images captured by multimedia devices due to low-quality imaging sensors, unstable image transmission processes, or low light conditions. Recent works also find that image denoising benefits the high-level vision tasks, e.g., image classification. In this work, we try to challenge this common sense and explore a totally new problem, i.e., whether the image denoising can be given the capability of fooling the state-of-the-art deep neural networks (DNNs) while enhancing the image quality. To this end, we initiate the very first attempt to study this problem from the perspective of adversarial attack and propose the adversarial denoise attack. More specifically, our main contributions are three-fold: First, we identify a new task that stealthily embeds attacks inside the image denoising module widely deployed in multimedia devices as an image post-processing operation to simultaneously enhance the visual image quality and fool DNNs. Second, we formulate this new task as a kernel prediction problem for image filtering and propose the adversarial-denoising kernel prediction that can produce adversarial-noiseless kernels for effective denoising and adversarial attacking simultaneously. Third, we implement an adaptive perceptual region localization to identify semantic-related vulnerability regions with which the attack can be more effective while not doing too much harm to the denoising. We name the proposed method as Pasadena (Perceptually Aware and Stealthy Adversarial DENoise Attack) and validate our method on the NeurIPS'17 adversarial competition dataset, CVPR2021-AIC-VI: unrestricted adversarial attacks on ImageNet,etc. The comprehensive evaluation and analysis demonstrate that our method not only realizes denoising but also achieves a significantly higher success rate and transferability over state-of-the-art attacks.

cs.CV↗

arXiv2021

CarveNet: Carving Point-Block for Complex 3D Shape Completion

3D point cloud completion is very challenging because it heavily relies on the accurate understanding of the complex 3D shapes (e.g., high-curvature, concave/convex, and hollowed-out 3D shapes) and the unknown & diverse patterns of the partially available point clouds. In this paper, we propose a novel solution,i.e., Point-block Carving (PC), for completing the complex 3D point cloud completion. Given the partial point cloud as the guidance, we carve a3D block that contains the uniformly distributed 3D points, yielding the entire point cloud. To achieve PC, we propose a new network architecture, i.e., CarveNet. This network conducts the exclusive convolution on each point of the block, where the convolutional kernels are trained on the 3D shape data. CarveNet determines which point should be carved, for effectively recovering the details of the complete shapes. Furthermore, we propose a sensor-aware method for data augmentation,i.e., SensorAug, for training CarveNet on richer patterns of partial point clouds, thus enhancing the completion power of the network. The extensive evaluations on the ShapeNet and KITTI datasets demonstrate the generality of our approach on the partial point clouds with diverse patterns. On these datasets, CarveNet successfully outperforms the state-of-the-art methods.

cs.CV↗

arXiv2021

Simpler, Faster, Stronger: Breaking The log-K Curse On Contrastive Learners With FlatNCE

InfoNCE-based contrastive representation learners, such as SimCLR, have been tremendously successful in recent years. However, these contrastive schemes are notoriously resource demanding, as their effectiveness breaks down with small-batch training (i.e., the log-K curse, whereas K is the batch-size). In this work, we reveal mathematically why contrastive learners fail in the small-batch-size regime, and present a novel simple, non-trivial contrastive objective named FlatNCE, which fixes this issue. Unlike InfoNCE, our FlatNCE no longer explicitly appeals to a discriminative classification goal for contrastive learning. Theoretically, we show FlatNCE is the mathematical dual formulation of InfoNCE, thus bridging the classical literature on energy modeling; and empirically, we demonstrate that, with minimal modification of code, FlatNCE enables immediate performance boost independent of the subject-matter engineering efforts. The significance of this work is furthered by the powerful generalization of contrastive learning techniques, and the introduction of new tools to monitor and diagnose contrastive training. We substantiate our claims with empirical evidence on CIFAR10, ImageNet, and other datasets, where FlatNCE consistently outperforms InfoNCE.

stat.ML↗

arXiv2021

AVA: Adversarial Vignetting Attack against Visual Recognition

Vignetting is an inherited imaging phenomenon within almost all optical systems, showing as a radial intensity darkening toward the corners of an image. Since it is a common effect for photography and usually appears as a slight intensity variation, people usually regard it as a part of a photo and would not even want to post-process it. Due to this natural advantage, in this work, we study vignetting from a new viewpoint, i.e., adversarial vignetting attack (AVA), which aims to embed intentionally misleading information into vignetting and produce a natural adversarial example without noise patterns. This example can fool the state-of-the-art deep convolutional neural networks (CNNs) but is imperceptible to humans. To this end, we first propose the radial-isotropic adversarial vignetting attack (RI-AVA) based on the physical model of vignetting, where the physical parameters (e.g., illumination factor and focal length) are tuned through the guidance of target CNN models. To achieve higher transferability across different CNNs, we further propose radial-anisotropic adversarial vignetting attack (RA-AVA) by allowing the effective regions of vignetting to be radial-anisotropic and shape-free. Moreover, we propose the geometry-aware level-set optimization method to solve the adversarial vignetting regions and physical parameters jointly. We validate the proposed methods on three popular datasets, i.e., DEV, CIFAR10, and Tiny ImageNet, by attacking four CNNs, e.g., ResNet50, EfficientNet-B0, DenseNet121, and MobileNet-V2, demonstrating the advantages of our methods over baseline methods on both transferability and image quality.

eess.IV↗

arXiv2021

Let There be Light: Improved Traffic Surveillance via Detail Preserving Night-to-Day Transfer

In recent years, image and video surveillance have made considerable progresses to the Intelligent Transportation Systems (ITS) with the help of deep Convolutional Neural Networks (CNNs). As one of the state-of-the-art perception approaches, detecting the interested objects in each frame of video surveillance is widely desired by ITS. Currently, object detection shows remarkable efficiency and reliability in standard scenarios such as daytime scenes with favorable illumination conditions. However, in face of adverse conditions such as the nighttime, object detection loses its accuracy significantly. One of the main causes of the problem is the lack of sufficient annotated detection datasets of nighttime scenes. In this paper, we propose a framework to alleviate the accuracy decline when object detection is taken to adverse conditions by using image translation method. We propose to utilize style translation based StyleMix method to acquire pairs of day time image and nighttime image as training data for following nighttime to daytime image translation. To alleviate the detail corruptions caused by Generative Adversarial Networks (GANs), we propose to utilize Kernel Prediction Network (KPN) based method to refine the nighttime to daytime image translation. The KPN network is trained with object detection task together to adapt the trained daytime model to nighttime vehicle detection directly. Experiments on vehicle detection verified the accuracy and effectiveness of the proposed approach.

cs.CV↗

arXiv2021

Concentrated solutions to fractional Schrödinger equations with prescribed $L^2$-norm

We investigate the existence and local uniqueness of normalized $k$-peak solutions for the fractional Schrödinger equations with attractive interactions with a class of degenerated trapping potential with non-isolated critical points. Precisely, applying the finite dimensional reduction method, we first obtain the existence of $k$-peak concentrated solutions and especially describe the relationship between the chemical potential $μ$ and the attractive interaction $a$. Second, after precise analysis of the concentrated points and the Lagrange multiplier, we prove the local uniqueness of the $k$-peak solutions with prescribed $L^2$-norm, by use of the local Pohozaev identities, the blow-up analysis and the maximum principle associated to the nonlocal operator $(-Δ)^s$. To our best knowledge, there is few results on the excited normalized solutions of the fractional Schrödinger equations before this present work. The main difficulty lies in the non-local property of the operator $(-Δ)^s$. First, it makes the standard comparison argument in the ODE theory invalid to use in our analysis. Second, because of the algebraic decay involving the approximate solutions, the estimates, on the Lagrange multiplier for example, would become more subtle. Moreover,when studying the corresponding harmonic extension problem, several local Pohozaev identities are constructed and we have to estimate several kinds of integrals that never appear in the classic local Schrödinger problems. In addition, throughout our discussion, we need to distinct the different cases of $p$, which are called respectively that the mass-subcritical, the mass-critical, and the mass-supercritical case, due to the mass-constraint condition. Another difficulty comes from the influence of the different degenerate rates along different directions at the critical points of the potential.

math.AP↗