Pyramid, or pyramid representation, is a type of multi-scale signal representation developed by the computer vision, image processing and signal processing communities, in which a signal or an image is subject to repeated smoothing and subsampling. Pyramid representation is a predecessor to scale-space representation and multiresolution analysis. == Pyramid generation == There are two main types of pyramids: lowpass and bandpass. A lowpass pyramid is made by smoothing the image with an appropriate smoothing filter and then subsampling the smoothed image, usually by a factor of 2 along each coordinate direction. The resulting image is then subjected to the same procedure, and the cycle is repeated multiple times. Each cycle of this process results in a smaller image with increased smoothing, but with decreased spatial sampling density (that is, decreased image resolution). If illustrated graphically, the entire multi-scale representation will look like a pyramid, with the original image on the bottom and each cycle's resulting smaller image stacked one atop the other. A bandpass pyramid is made by forming the difference between images at adjacent levels in the pyramid and performing image interpolation between adjacent levels of resolution, to enable computation of pixelwise differences. == Pyramid generation kernels == A variety of different smoothing kernels have been proposed for generating pyramids. Among the suggestions that have been given, the binomial kernels arising from the binomial coefficients stand out as a particularly useful and theoretically well-founded class. Thus, given a two-dimensional image, we may apply the (normalized) binomial filter (1/4, 1/2, 1/4) typically twice or more along each spatial dimension and then subsample the image by a factor of two. This operation may then proceed as many times as desired, leading to a compact and efficient multi-scale representation. If motivated by specific requirements, intermediate scale levels may also be generated where the subsampling stage is sometimes left out, leading to an oversampled or hybrid pyramid. With the increasing computational efficiency of CPUs available today, it is in some situations also feasible to use wider supported Gaussian filters as smoothing kernels in the pyramid generation steps. === Gaussian pyramid === In a Gaussian pyramid, subsequent images are weighted down using a Gaussian average (Gaussian blur) and scaled down. Each pixel containing a local average corresponds to a neighborhood pixel on a lower level of the pyramid. This technique is used especially in texture synthesis. === Laplacian pyramid === A Laplacian pyramid is very similar to a Gaussian pyramid but saves the difference image of the blurred versions between each levels. Only the smallest level is not a difference image to enable reconstruction of the high resolution image using the difference images on higher levels. This technique can be used in image compression. === Steerable pyramid === A steerable pyramid, developed by Simoncelli and others, is an implementation of a multi-scale, multi-orientation band-pass filter bank used for applications including image compression, texture synthesis, and object recognition. It can be thought of as an orientation selective version of a Laplacian pyramid, in which a bank of steerable filters are used at each level of the pyramid instead of a single Laplacian or Gaussian filter. == Applications of pyramids == === Alternative representation === In the early days of computer vision, pyramids were used as the main type of multi-scale representation for computing multi-scale image features from real-world image data. More recent techniques include scale-space representation, which has been popular among some researchers due to its theoretical foundation, the ability to decouple the subsampling stage from the multi-scale representation, the more powerful tools for theoretical analysis as well as the ability to compute a representation at any desired scale, thus avoiding the algorithmic problems of relating image representations at different resolution. Nevertheless, pyramids are still frequently used for expressing computationally efficient approximations to scale-space representation. === Detail manipulation === Levels of a Laplacian pyramid can be added to or removed from the original image to amplify or reduce detail at different scales. However, detail manipulation of this form is known to produce halo artifacts in many cases, leading to the development of alternatives such as the bilateral filter. Some image compression file formats use the Adam7 algorithm or some other interlacing technique. These can be seen as a kind of image pyramid. Because those file format store the "large-scale" features first, and fine-grain details later in the file, a particular viewer displaying a small "thumbnail" or on a small screen can quickly download just enough of the image to display it in the available pixels—so one file can support many viewer resolutions, rather than having to store or generate a different file for each resolution.
Clarizen
Clarizen, Inc. is a project management software and collaborative work management company. Clarizen uses a software as a service business model. Clarizen's features include attaching CAD drawings to a project, moving between the project view and design view and an E-mail reporting feature. In May 2014 Clarizen raised $35 million in venture capital investment led by Goldman Sachs. The round brought investment to $90 million. Previous investors, including Benchmark Capital, Carmel Ventures, DAG Ventures, Opus Capital and Vintage Investment Partners participated. In April 2020, Clarizen appointed Matt Zilli as its new CEO, replacing Boaz Chalamish who is appointed as Executive Chairman. In January 2021 Clarizen was acquired by Planview.
Wayve
Wayve Technologies Ltd is a British autonomous driving technology company focused on developing self-driving vehicle systems through end-to-end deep learning. Founded in 2017 by researchers from the University of Cambridge, Wayve’s approach eschews detailed 3D maps and hand-coded rules, in favor of a self-learning “AI driver” that learns from camera data and driving experience. The London-headquartered startup has garnered significant attention and funding for its visually-based method. == History == Wayve was founded in Cambridge, England, on August 21, 2017, by Amar Shah and Alex Kendall, two machine learning PhD students at the University of Cambridge. Shah initially served as CEO while Kendall was CTO, and the pair set out to develop an unconventional self-driving car system using machine learning at every layer of the driving task. In May 2018, Wayve emerged from stealth mode with backing from early-stage investors. At this time the company had around 10 employees, and its advisory investors included Uber’s Chief Scientist, Zoubin Ghahramani, who shared Wayve’s vision of a learning-centric driving AI. In 2019, Wayve achieved a milestone by training a car to drive autonomously on public roads it had never seen before, using only cameras, a basic GPS map, and end-to-end deep learning control. The company moved its base to London and secured a $20 million Series A funding round in November 2019. This investment enabled Wayve to launch a pilot fleet of autonomous electric vehicles in central London for real-world testing. During these trials, Wayve’s cars (such as retrofitted Jaguar I-Pace SUVs) began navigating the complex, narrow streets of London to prove the system’s ability to adapt to challenging urban scenarios. In 2020, co-founder Amar Shah departed the company, and Alex Kendall assumed the role of CEO. The startup joined the Microsoft for Startups: Autonomous Driving program in 2020, leveraging Microsoft Azure’s cloud computing for training its machine learning models at scale. It also committed to testing exclusively on electric vehicles, and a goal to reduce carbon emissions. In 2021, Wayve entered pilot programs with major UK retailers. It launched a 12-month autonomous delivery trial with supermarket chain Asda, and received a £10 million ($13.6 million) investment from online grocer Ocado Group as part of a partnership to develop self-driving grocery delivery vans. Ocado’s backing gave Wayve access to a fleet of delivery vans for data collection and testing on busy London routes (with human safety drivers present) to train its AI in urban traffic. In 2022, after a successful Series B funding round, the company extended road testing beyond the UK to other regions, and, by 2023, in multiple countries. The company had begun operating in the United States and in continental Europe, in preparation for larger commercial deployments. In 2023, Wayve announced a collaboration with Nissan to integrate Wayve’s AI-driven software into its ProPilot ADAS system, slated to launch in fiscal year 2027. Wayve received strategic investment from Uber, in 2024, to jointly develop autonomous ride-hailing services. The two companies plan to trial a fully driverless robotaxi service in London, supported by a UK government program to accelerate commercial self-driving pilots to as early as 2026. To demonstrate the scalability of its technology, Wayve conducted an “AI-500” roadshow project, driving in dozens of cities across Asia, Europe, and North America using the same AI model. By mid-2025, it had completed autonomous driving demos in 90 cities without prior HD mapping. In April 2025, Wayve opened its first Asian research hub in Japan, with investment by SoftBank, to improve its model’s generalization using local driving data. That year, the company conducted driving tests in over 500 cities in Europe, North America and Japan without city-specific programming. In February 2026, Nissan, Uber and Wayve announced their collaboration on robotaxi development, with the aim of launching a pilot programme in Tokyo by late 2026. Wayve also formed a strategic alliance with Mercedes-Benz and Stellantis on personal vehicle and robotaxi applications. == Financing and investors == Wayve has been backed by a mix of venture capital (VC) firms, corporate investors, and individuals. Its initial seed funding came from funds such as Compound (NYC) and Firstminute Capital (London), as well as Cambridge-based angel investors, in 2018. Academic Pieter Abbeel and Uber’s chief scientist, Zoubin Ghahramani, were early backers. In November 2019, Wayve raised a $20 million Series A led by Eclipse Ventures, with participation from Balderton Capital and other prior investors. The Series A financing was used to fund the company’s first autonomous trials in London, and marked the first time a European self-driving car startup had secured a U.S. VC as lead investor. In October 2021, Ocado Group invested £10 million (approximately $13.6 million) in Wayve as a strategic partner in autonomous grocery delivery. This brought Wayve’s total funding to around $60 million at that time. The Series B round followed in January 2022, when Wayve announced $200 million in new funding led by Eclipse Ventures, with D1 Capital Partners, Moore Strategic Ventures, and Linse Capital. Balderton, Microsoft and Virgin Group joined as strategic backers. Baillie Gifford and Compound also participated; Ocado increased its stake as a strategic investor; and Meta AI head Yann LeCun and Richard Branson also became investors. Wayve’s Series C in May 2024 closed a $1.05 billion, led by Japan’s SoftBank Group. The funding round was the largest-ever for a UK AI company, and included new investor Nvidia, and returning investors Microsoft and Eclipse Ventures, among others. Uber also joined as a stratgic partner and a stakeholder. The Series C round increased Wayve’s total funding raised to about $1.3 billion to date from investors including SoftBank, Microsoft and Nvidia, and lifted Wayve’s valuation into “unicorn” status. In February 2026, Wayve announced a $1.2 billion Series D funding round; later that month, the company reported that $1.5 billion had been raised from, primarily, Mercedes-Benz, Stellantis, Nissan, and existing backers Uber, Microsoft and Nvidia, increasing Wayve's overall valuation to $8.6 billion. == Technology == Wayve’s self-driving approach centers on end-to-end deep learning and a vision-based AI system. Unlike conventional autonomous vehicles that depend on high-definition maps, hand-coded rules, and arrays of expensive lidar sensors, Wayve’s platform learns to drive predominantly using camera data and machine learning algorithms. The company refers to its AI-driven driving software as an “Embodied AI” or AI Driver, emphasizing that the system learns from experience (both real and simulated) to handle complex or novel situations rather than following pre-programmed instructions, not unlike Tesla's approach. The Wayve hardware-agnostic autonomy stack consists of a suite of video cameras, with basic automotive sensors, mounted on the vehicle, and paired with onboard compute units that are powered by GPUs to run the AI models. This vision-only philosophy is similar to Tesla’s Autopilot/FSDB model, but Wayve’s solution is vehicle-agnostic and mapless. Wayve’s strategy is to provide its driving AI as an OEM-ready platform; it plans to license or embed its technology into vehicles made by established automakers rather than build its own cars. Wayve’s development vehicles currently use Nvidia’s Orin system-on-chip as the onboard computer for running the AI model, but CEO Kendall has noted that the software can run on “whatever GPU [an automaker] already has in their vehicles” Wayve has built a cloud infrastructure, largely on Microsoft Azure, to process petabytes of this data, and uses simulation tools (known internally as the “Wayve Infinity” simulator) to synthetically generate and practice rare or dangerous scenarios for the AI to learn from. == Corporate affairs == Wayve is a privately held company headquartered in London, England, with its primary research and development office in the Kings Cross area of London. The company was initially incorporated as Wayve Technologies Ltd in the UK. Wayve has also established a presence in the U.S., in Silicon Valley); in Canada, with a research hub in Vancouver; in Yokohama, Japan; in Leonberg, Germany; and in Herzliya, Israel. The Leadership team includes research scientists and engineers with backgrounds in computer vision, robotics, and automotive systems. President Erez Dagan was hired in 2024, following two decades at Mobileye; chief scientist Jamie Shotton is formerly of Microsoft Research; CEO Alex Kendall, originally from New Zealand with a PhD in computer vision from Cambridge, took over as CEO in 2020 after the departure of his co-founder Amar Shah.
Interim Measures for the Management of Generative AI Services
The Interim Measures for the Management of Generative AI Services (Chinese: 生成式人工智能服务管理暂行办法; pinyin: Shēngchéng shì réngōng zhìnéng fúwù guǎnlǐ zànxíng bànfǎ) are a set of regulations governing public-facing generative artificial intelligence services in China. Issued on 10 July 2023 and effective from 15 August 2023, they were China's first binding regulation specifically targeting generative AI. They have been described as among the earliest such regulations adopted by any country. The measures were jointly issued by the Cyberspace Administration of China (CAC) and six other national bodies: the National Development and Reform Commission, the Ministry of Education, the Ministry of Science and Technology, the Ministry of Industry and Information Technology, the Ministry of Public Security, and the National Radio and Television Administration. Among the measures' most prominent requirements is that generative AI services must uphold Core Socialist Values and must not generate content that could subvert state power, harm national security, or undermine social stability. The measures also require providers of public-facing generative AI services to undergo security assessments and register their algorithms with the CAC. As of December 2025, 748 generative AI services had completed the filing process at the national level. == Background == The Interim Measures build on two earlier sets of regulations targeting specific algorithm applications. The Administrative Provisions on Algorithm Recommendation for Internet Information Services, effective from March 2022, established China's algorithm registry and required providers of recommendation algorithms with "public opinion properties or social mobilization capabilities" to file with the CAC and undergo security assessments. The Administrative Provisions on Deep Synthesis of Internet Information Services, effective from January 2023, extended similar requirements to algorithms used for generating synthetic media such as deepfakes. In April 2023, the CAC released a draft of the generative AI regulation for public comment. The draft included several requirements that attracted attention, including that generated content should "embody Core Socialist Values" and that training data should be "true and accurate". The public consultation period ran until May 2023. The final version, published in July 2023, was substantially revised from the draft. According to an analysis by the Future of Privacy Forum, changes appeared to reflect feedback from industry stakeholders including Baidu, Xiaomi, SenseTime, and others, as well as input from government-affiliated research institutes. The final measures adopted a more permissive tone, with the CAC describing its approach as "inclusive and prudent" (包容审慎) and emphasising "classified and graded" (分类分级) supervision. == Scope == The measures apply to services that use generative AI technology to provide text, images, audio, video, or other content to the public within mainland China (Article 2). They do not apply to organisations that develop or use generative AI internally without offering services to the domestic public, such as industry associations, enterprises, and research institutions. Overseas providers whose services are accessible to users in China are also subject to the measures. == Key provisions == === Content requirements === Article 4 sets out the core content obligations. Providers and users of generative AI services must uphold the Core Socialist Values. The measures prohibit generating content that incites subversion of national sovereignty or the socialist system, endangers national security or the nation's image, incites separatism, promotes terrorism or extremism, promotes ethnic hatred or discrimination, or contains violence, obscenity, or false information prohibited by law. These content prohibitions largely mirror those in Article 12 of the Cybersecurity Law and in prior regulations governing online content. Article 4 also requires that models be designed and trained to avoid discrimination, that services respect intellectual property rights, and that providers take effective measures to improve the transparency and accuracy of generated content. === Training data and labelling === Article 7 requires providers to ensure that training data is of high quality and legitimately sourced, and that it does not infringe upon intellectual property rights. Where personal information is used, consent must be obtained. The final version of this provision removed language from the draft that would have held providers responsible for the "legitimacy" of all pretraining data, replacing it with a requirement to "employ effective measures to improve the quality of training data". Article 8 requires providers to establish labelling rules for training data and to conduct quality assessments of data annotations. Article 12 requires that generated images, videos, and other synthetic content be labelled as AI-generated. === User rights and privacy === Article 11 requires providers to protect user privacy, to minimise the collection and retention of personal data, and to refrain from unlawfully sharing user information. Users have the right to request review, correction, or deletion of their personal information. Article 10 requires providers to take measures to prevent excessive dependence on or addiction to generative AI services by minors. === Security assessment and algorithm filing === Article 17 requires that providers of generative AI services with "public opinion properties or the capacity for social mobilization" (具有舆论属性或者社会动员能力) carry out security assessments and complete algorithm filing procedures in accordance with the Administrative Provisions on Algorithm Recommendation for Internet Information Services. == Implementation == === Algorithm filing process === In practice, the filing requirements under the Interim Measures have developed into a two-tier process. The first tier is the standard algorithm filing (算法备案) under the pre-existing Algorithm Recommendation Provisions, which involves submitting information about an algorithm's design, purpose, and data sources to the CAC. This process is primarily a registration mechanism. For public-facing generative AI products, there is an additional, more rigorous process commonly referred to as the "large model filing" (大模型备案). This involves submitting a security self-assessment report, data annotation rules, a keyword blocking list, and evaluation test question sets. The process includes technical testing at the provincial level, followed by review at the national CAC level. The algorithm filing targets specific algorithms, while the large model filing evaluates the broader system architecture, training data, model parameters, and potential social impact. The CAC publishes lists of generative AI services that have successfully completed the filing process. The first such list was published on 2 April 2024. According to the CAC's year-end announcements, 302 generative AI services had completed national-level filing by the end of 2024 (of which 238 were new that year), alongside 105 applications that completed local-level registration. By the end of 2025, the cumulative total had risen to 748 national-level filings and 435 local-level registrations. === Content compliance and testing === According to the Carnegie Endowment, the CAC has conducted compliance audits of generative AI services with a particular focus on ensuring appropriate responses to queries about politically sensitive topics. The large model filing process requires providers to pass both provincial-level and national-level technical testing before their services can be made available to the public. On 1 March 2024, the National Technical Committee 260 on Cybersecurity (TC260) published TC260-003, the Basic Security Requirements for Generative AI Services (生成式人工智能服务安全基本要求), a technical standard that provides detailed guidance on the security assessments required under the Interim Measures. The standard covers requirements for training data safety, model security, and content safety evaluation, and is used as a reference for the filing process. == Analysis == === Relationship to broader Chinese internet regulation === The content requirements in the Interim Measures extend China's existing framework for online information control to generative AI. Legal scholars have noted that the "Core Socialist Values" provision and the specific content prohibitions are consistent with longstanding requirements imposed on internet platforms under the Cybersecurity Law and related regulations. The Asia Society Policy Institute has described the Chinese government's highest regulatory priority in this area as retaining control of information, noting that content-related obligations receive stricter enforcement than other provisions. === Nature of the filing system === The character of the filing system has been debated by scholars. Angela Huyue Zh
Deep learning speech synthesis
Deep learning speech synthesis refers to the application of deep learning models to generate natural-sounding human speech from written text (text-to-speech) or spectrum (vocoder). Deep neural networks are trained using large amounts of recorded speech and, in the case of a text-to-speech system, the associated labels and/or input text. == Formulation == Given an input text or some sequence of linguistic units Y {\displaystyle Y} , the target speech X {\displaystyle X} can be derived by X = arg max P ( X | Y , θ ) {\displaystyle X=\arg \max P(X|Y,\theta )} where θ {\displaystyle \theta } is the set of model parameters. Typically, the input text will first be passed to an acoustic feature generator, then the acoustic features are passed to the neural vocoder. For the acoustic feature generator, the loss function is typically L1 loss (Mean Absolute Error, MAE) or L2 loss (Mean Square Error, MSE). These loss functions impose a constraint that the output acoustic feature distributions must be Gaussian or Laplacian. In practice, since the human voice band ranges from approximately 300 to 4000 Hz, the loss function will be designed to have more penalty on this range: l o s s = α loss human + ( 1 − α ) loss other {\displaystyle loss=\alpha {\text{loss}}_{\text{human}}+(1-\alpha ){\text{loss}}_{\text{other}}} where loss human {\displaystyle {\text{loss}}_{\text{human}}} is the loss from human voice band and α {\displaystyle \alpha } is a scalar, typically around 0.5. The acoustic feature is typically a spectrogram or Mel scale. These features capture the time-frequency relation of the speech signal, and thus are sufficient to generate intelligent outputs. The Mel-frequency cepstrum feature used in the speech recognition task is not suitable for speech synthesis, as it reduces too much information. == History == In September 2016, DeepMind released WaveNet, which demonstrated that deep learning-based models are capable of modeling raw waveforms and generating speech from acoustic features like spectrograms or mel-spectrograms. Although WaveNet was initially considered to be computationally expensive and slow to be used in consumer products at the time, a year after its release, DeepMind unveiled a modified version of WaveNet known as "Parallel WaveNet," a production model 1,000 faster than the original. This was followed by Google AI's Tacotron 2 in 2018, which demonstrated that neural networks could produce highly natural speech synthesis but required substantial training data—typically tens of hours of audio—to achieve acceptable quality. Tacotron 2 used an autoencoder architecture with attention mechanisms to convert input text into mel-spectrograms, which were then converted to waveforms using a separate neural vocoder. When trained on smaller datasets, such as 2 hours of speech, the output quality degraded while still being able to maintain intelligible speech, and with just 24 minutes of training data, Tacotron 2 failed to produce intelligible speech. In 2019, Microsoft Research introduced FastSpeech, which addressed speed limitations in autoregressive models like Tacotron 2. FastSpeech utilized a non-autoregressive architecture that enabled parallel sequence generation, significantly reducing inference time while maintaining audio quality. Its feedforward transformer network with length regulation allowed for one-shot prediction of the full mel-spectrogram sequence, avoiding the sequential dependencies that bottlenecked previous approaches. The same year saw the release of HiFi-GAN, a generative adversarial network (GAN)-based vocoder that improved the efficiency of waveform generation while producing high-fidelity speech. In 2020, the release of Glow-TTS introduced a flow-based approach that allowed for fast inference and voice style transfer capabilities. In March 2020, the free text-to-speech website 15.ai was launched. 15.ai gained widespread international attention in early 2021 for its ability to synthesize emotionally expressive speech of fictional characters from popular media with minimal amount of data. The creator of 15.ai (known pseudonymously as 15) stated that 15 seconds of training data is sufficient to perfectly clone a person's voice (hence its name, "15.ai"), a significant reduction from the previously known data requirement of tens of hours. 15.ai is credited as the first platform to popularize AI voice cloning in memes and content creation. 15.ai used a multi-speaker model that enabled simultaneous training of multiple voices and emotions, implemented sentiment analysis using DeepMoji, and supported precise pronunciation control via ARPABET. The 15-second data efficiency benchmark was later corroborated by OpenAI in 2024. == Semi-supervised learning == Currently, self-supervised learning has gained much attention through better use of unlabelled data. Research has shown that, with the aid of self-supervised loss, the need for paired data decreases. == Zero-shot speaker adaptation == Zero-shot speaker adaptation is promising because a single model can generate speech with various speaker styles and characteristic. In June 2018, Google proposed to use pre-trained speaker verification models as speaker encoders to extract speaker embeddings. The speaker encoders then become part of the neural text-to-speech models, so that it can determine the style and characteristics of the output speech. This procedure has shown the community that it is possible to use only a single model to generate speech with multiple styles. == Neural vocoder == In deep learning-based speech synthesis, neural vocoders play an important role in generating high-quality speech from acoustic features. The WaveNet model proposed in 2016 achieves excellent performance on speech quality. Wavenet factorised the joint probability of a waveform x = { x 1 , . . . , x T } {\displaystyle \mathbf {x} =\{x_{1},...,x_{T}\}} as a product of conditional probabilities as follows p θ ( x ) = ∏ t = 1 T p ( x t | x 1 , . . . , x t − 1 ) {\displaystyle p_{\theta }(\mathbf {x} )=\prod _{t=1}^{T}p(x_{t}|x_{1},...,x_{t-1})} where θ {\displaystyle \theta } is the model parameter including many dilated convolution layers. Thus, each audio sample x t {\displaystyle x_{t}} is conditioned on the samples at all previous timesteps. However, the auto-regressive nature of WaveNet makes the inference process dramatically slow. To solve this problem, Parallel WaveNet was proposed. Parallel WaveNet is an inverse autoregressive flow-based model which is trained by knowledge distillation with a pre-trained teacher WaveNet model. Since such inverse autoregressive flow-based models are non-auto-regressive when performing inference, the inference speed is faster than real-time. Meanwhile, Nvidia proposed a flow-based WaveGlow model, which can also generate speech faster than real-time. However, despite the high inference speed, parallel WaveNet has the limitation of needing a pre-trained WaveNet model, so that WaveGlow takes many weeks to converge with limited computing devices. This issue has been solved by Parallel WaveGAN, which learns to produce speech through multi-resolution spectral loss and GAN learning strategies.
Landmark point
In morphometrics, landmark point or shortly landmark is a point in a shape object in which correspondences between and within the populations of the object are preserved. In other disciplines, landmarks may be known as vertices, anchor points, control points, sites, profile points, 'sampling' points, nodes, markers, fiducial markers, etc. Landmarks can be defined either manually by experts or automatically by a computer program. There are three basic types of landmarks: anatomical landmarks, mathematical landmarks or pseudo-landmarks. An anatomical landmark is a biologically-meaningful point in an organism. Usually experts define anatomical points to ensure their correspondences within the same species. Examples of anatomical landmark in shape of a skull are the eye corner, tip of the nose, jaw, etc. Anatomical landmarks determine homologous parts of an organism, which share a common ancestry. Mathematical landmarks are points in a shape that are located according to some mathematical or geometrical property, for instance, a high curvature point or an extreme point. A computer program usually determines mathematical landmarks used for an automatic pattern recognition. Pseudo-landmarks are constructed points located between anatomical or mathematical landmarks. A typical example is an equally spaced set of points between two anatomical landmarks to get more sample points from a shape. Pseudo-landmarks are useful during shape matching, when the matching process requires a large number of points.
Batch normalization
In artificial neural networks, batch normalization (also known as batch norm) is a normalization technique used to make training faster and more stable by adjusting the inputs to each layer—re-centering them around zero and re-scaling them to a standard size. It was introduced by Sergey Ioffe and Christian Szegedy in 2015. Experts still debate why batch normalization works so well. It was initially thought to tackle internal covariate shift, a problem where parameter initialization and changes in the distribution of the inputs of each layer affect the learning rate of the network. However, newer research suggests it doesn’t fix this shift but instead smooths the objective function—a mathematical guide the network follows to improve—enhancing performance. In very deep networks, batch normalization can initially cause a severe gradient explosion—where updates to the network grow uncontrollably large—but this is managed with shortcuts called skip connections in residual networks. Another theory is that batch normalization adjusts data by handling its size and path separately, speeding up training. == Internal covariate shift == Each layer in a neural network has inputs that follow a specific distribution, which shifts during training due to two main factors: the random starting values of the network’s settings (parameter initialization) and the natural variation in the input data. This shifting pattern affecting the inputs to the network’s inner layers is called internal covariate shift. While a strict definition isn’t fully agreed upon, experiments show that it involves changes in the means and variances of these inputs during training. Batch normalization was first developed to address internal covariate shift. During training, as the parameters of preceding layers adjust, the distribution of inputs to the current layer changes accordingly, such that the current layer needs to constantly readjust to new distributions. This issue is particularly severe in deep networks, because small changes in shallower hidden layers will be amplified as they propagate within the network, resulting in significant shift in deeper hidden layers. Batch normalization was proposed to reduced these unwanted shifts to speed up training and produce more reliable models. Beyond possibly tackling internal covariate shift, batch normalization offers several additional advantages. It allows the network to use a higher learning rate—a setting that controls how quickly the network learns—without causing problems like vanishing or exploding gradients, where updates become too small or too large. It also appears to have a regularizing effect, improving the network’s ability to generalize to new data, reducing the need for dropout, a technique used to prevent overfitting (when a model learns the training data too well and fails on new data). Additionally, networks using batch normalization are less sensitive to the choice of starting settings or learning rates, making them more robust and adaptable. == Procedures == === Transformation === In a neural network, batch normalization is achieved through a normalization step that fixes the means and variances of each layer's inputs. Ideally, the normalization would be conducted over the entire training set, but to use this step jointly with stochastic optimization methods, it is impractical to use the global information. Thus, normalization is restrained to each mini-batch in the training process. Let us use B to denote a mini-batch of size m of the entire training set. The empirical mean and variance of B could thus be denoted as μ B = 1 m ∑ i = 1 m x i {\displaystyle \mu _{B}={\frac {1}{m}}\sum _{i=1}^{m}x_{i}} and σ B 2 = 1 m ∑ i = 1 m ( x i − μ B ) 2 {\displaystyle \sigma _{B}^{2}={\frac {1}{m}}\sum _{i=1}^{m}(x_{i}-\mu _{B})^{2}} . For a layer of the network with d-dimensional input, x = ( x ( 1 ) , . . . , x ( d ) ) {\displaystyle x=(x^{(1)},...,x^{(d)})} , each dimension of its input is then normalized (i.e. re-centered and re-scaled) separately, x ^ i ( k ) = x i ( k ) − μ B ( k ) ( σ B ( k ) ) 2 + ϵ {\displaystyle {\hat {x}}_{i}^{(k)}={\frac {x_{i}^{(k)}-\mu _{B}^{(k)}}{\sqrt {\left(\sigma _{B}^{(k)}\right)^{2}+\epsilon }}}} , where k ∈ [ 1 , d ] {\displaystyle k\in [1,d]} and i ∈ [ 1 , m ] {\displaystyle i\in [1,m]} ; μ B ( k ) {\displaystyle \mu _{B}^{(k)}} and σ B ( k ) {\displaystyle \sigma _{B}^{(k)}} are the per-dimension mean and standard deviation, respectively. ϵ {\displaystyle \epsilon } is added in the denominator for numerical stability and is an arbitrarily small positive constant. The resulting normalized activation x ^ ( k ) {\displaystyle {\hat {x}}^{(k)}} have zero mean and unit variance, if ϵ {\displaystyle \epsilon } is not taken into account. To restore the representation power of the network, a transformation step then follows as y i ( k ) = γ ( k ) x ^ i ( k ) + β ( k ) {\displaystyle y_{i}^{(k)}=\gamma ^{(k)}{\hat {x}}_{i}^{(k)}+\beta ^{(k)}} , where the parameters γ ( k ) {\displaystyle \gamma ^{(k)}} and β ( k ) {\displaystyle \beta ^{(k)}} are subsequently learned in the optimization process. Formally, the operation that implements batch normalization is a transform B N γ ( k ) , β ( k ) : x 1... m ( k ) → y 1... m ( k ) {\displaystyle BN_{\gamma ^{(k)},\beta ^{(k)}}:x_{1...m}^{(k)}\rightarrow y_{1...m}^{(k)}} called the Batch Normalizing transform. The output of the BN transform y ( k ) = B N γ ( k ) , β ( k ) ( x ( k ) ) {\displaystyle y^{(k)}=BN_{\gamma ^{(k)},\beta ^{(k)}}(x^{(k)})} is then passed to other network layers, while the normalized output x ^ i ( k ) {\displaystyle {\hat {x}}_{i}^{(k)}} remains internal to the current layer. === Backpropagation === The described BN transform is a differentiable operation, and the gradient of the loss l {\displaystyle l} with respect to the different parameters can be computed directly with the chain rule. Specifically, ∂ l ∂ y i ( k ) {\displaystyle {\frac {\partial l}{\partial y_{i}^{(k)}}}} depends on the choice of activation function, and the gradient against other parameters could be expressed as a function of ∂ l ∂ y i ( k ) {\displaystyle {\frac {\partial l}{\partial y_{i}^{(k)}}}} : ∂ l ∂ x ^ i ( k ) = ∂ l ∂ y i ( k ) γ ( k ) {\displaystyle {\frac {\partial l}{\partial {\hat {x}}_{i}^{(k)}}}={\frac {\partial l}{\partial y_{i}^{(k)}}}\gamma ^{(k)}} , ∂ l ∂ γ ( k ) = ∑ i = 1 m ∂ l ∂ y i ( k ) x ^ i ( k ) {\displaystyle {\frac {\partial l}{\partial \gamma ^{(k)}}}=\sum _{i=1}^{m}{\frac {\partial l}{\partial y_{i}^{(k)}}}{\hat {x}}_{i}^{(k)}} , ∂ l ∂ β ( k ) = ∑ i = 1 m ∂ l ∂ y i ( k ) {\displaystyle {\frac {\partial l}{\partial \beta ^{(k)}}}=\sum _{i=1}^{m}{\frac {\partial l}{\partial y_{i}^{(k)}}}} , ∂ l ∂ σ B ( k ) 2 = ∑ i = 1 m ∂ l ∂ y i ( k ) ( x i ( k ) − μ B ( k ) ) ( − γ ( k ) 2 ( σ B ( k ) 2 + ϵ ) − 3 / 2 ) {\displaystyle {\frac {\partial l}{\partial \sigma _{B}^{(k)^{2}}}}=\sum _{i=1}^{m}{\frac {\partial l}{\partial y_{i}^{(k)}}}(x_{i}^{(k)}-\mu _{B}^{(k)})\left(-{\frac {\gamma ^{(k)}}{2}}(\sigma _{B}^{(k)^{2}}+\epsilon )^{-3/2}\right)} , ∂ l ∂ μ B ( k ) = ∑ i = 1 m ∂ l ∂ y i ( k ) − γ ( k ) σ B ( k ) 2 + ϵ + ∂ l ∂ σ B ( k ) 2 1 m ∑ i = 1 m ( − 2 ) ⋅ ( x i ( k ) − μ B ( k ) ) {\displaystyle {\frac {\partial l}{\partial \mu _{B}^{(k)}}}=\sum _{i=1}^{m}{\frac {\partial l}{\partial y_{i}^{(k)}}}{\frac {-\gamma ^{(k)}}{\sqrt {\sigma _{B}^{(k)^{2}}+\epsilon }}}+{\frac {\partial l}{\partial \sigma _{B}^{(k)^{2}}}}{\frac {1}{m}}\sum _{i=1}^{m}(-2)\cdot (x_{i}^{(k)}-\mu _{B}^{(k)})} , and ∂ l ∂ x i ( k ) = ∂ l ∂ x ^ i ( k ) 1 σ B ( k ) 2 + ϵ + ∂ l ∂ σ B ( k ) 2 2 ( x i ( k ) − μ B ( k ) ) m + ∂ l ∂ μ B ( k ) 1 m {\displaystyle {\frac {\partial l}{\partial x_{i}^{(k)}}}={\frac {\partial l}{\partial {\hat {x}}_{i}^{(k)}}}{\frac {1}{\sqrt {\sigma _{B}^{(k)^{2}}+\epsilon }}}+{\frac {\partial l}{\partial \sigma _{B}^{(k)^{2}}}}{\frac {2(x_{i}^{(k)}-\mu _{B}^{(k)})}{m}}+{\frac {\partial l}{\partial \mu _{B}^{(k)}}}{\frac {1}{m}}} . === Inference === During the training stage, the normalization steps depend on the mini-batches to ensure efficient and reliable training. However, in the inference stage, this dependence is not useful any more. Instead, the normalization step in this stage is computed with the population statistics such that the output could depend on the input in a deterministic manner. The population mean, E [ x ( k ) ] {\displaystyle E[x^{(k)}]} , and variance, Var [ x ( k ) ] {\displaystyle \operatorname {Var} [x^{(k)}]} , are computed as: E [ x ( k ) ] = E B [ μ B ( k ) ] {\displaystyle E[x^{(k)}]=E_{B}[\mu _{B}^{(k)}]} , and Var [ x ( k ) ] = m m − 1 E B [ ( σ B ( k ) ) 2 ] {\displaystyle \operatorname {Var} [x^{(k)}]={\frac {m}{m-1}}E_{B}[\left(\sigma _{B}^{(k)}\right)^{2}]} . The population statistics thus is a complete representation of the mini-batches. The BN transform in the inference step thus becomes y ( k ) = B N γ ( k ) , β ( k ) inf ( x ( k ) ) = γ ( k ) x ( k ) − E [ x ( k ) ] Var [ x ( k ) ] + ϵ + β