A reasoning model, also known as a reasoning language model (RLM) or large reasoning model (LRM), is a type of large language model (LLM) that has been specifically trained to solve complex tasks requiring multiple steps of logical reasoning. These models demonstrate superior performance on logic, mathematics, and programming tasks compared to standard LLMs. They possess the ability to revisit and revise earlier reasoning steps and utilize additional computation during inference as a method to scale performance, complementing traditional scaling approaches based on training data size, model parameters, and training compute. == Overview == Unlike traditional language models that generate responses immediately, reasoning models allocate additional compute, or thinking, time before producing an answer to solve multi-step problems. OpenAI introduced this terminology in September 2024 when it released the o1 series, describing the models as designed to "spend more time thinking" before responding. The company framed o1 as a reset in model naming that targets complex tasks in science, coding, and mathematics, and it contrasted o1's performance with GPT-4o on benchmarks such as AIME and Codeforces. Independent reporting the same week summarized the launch and highlighted OpenAI's claim that o1 automates chain-of-thought style reasoning to achieve large gains on difficult exams. In operation, reasoning models generate internal chains of intermediate steps, then select and refine a final answer. OpenAI reported that o1's accuracy improves as the model is given more reinforcement learning during training and more test-time compute at inference. The company initially chose to hide raw chains and instead return a model-written summary, stating that it "decided not to show" the underlying thoughts so researchers could monitor them without exposing unaligned content to end users. Commercial deployments document separate "reasoning tokens" that meter hidden thinking and a control for "reasoning effort" that tunes how much compute the model uses. These features make the models slower than ordinary chat systems while enabling stronger performance on difficult problems. == History == The research trajectory toward reasoning models combined advances in supervision, prompting, and search-style inference. Early alignment work on reinforcement learning from human feedback showed that models can be fine-tuned to follow instructions with "human feedback" and preference-based rewards. In 2022, Google Research scientists Jason Wei and Denny Zhou showed that chain-of-thought prompting "significantly improves the ability" of large models on complex reasoning tasks. Input → Step 1 → Step 2 → ⋯ → Step n ⏟ Reasoning chain → Answer {\displaystyle {\text{Input}}\rightarrow \underbrace {{\text{Step}}_{1}\rightarrow {\text{Step}}_{2}\rightarrow \cdots \rightarrow {\text{Step}}_{n}} _{\text{Reasoning chain}}\rightarrow {\text{Answer}}} A companion result demonstrated that the simple instruction "Let's think step by step" can elicit zero-shot reasoning. Follow-up work introduced self-consistency decoding, which "boosts the performance" of chain-of-thought by sampling diverse solution paths and choosing the consensus, and tool-augmented methods such as ReAct, a portmanteau of Reason and Act, that prompt models to "generate both reasoning traces" and actions. Research then generalized chain-of-thought into search over multiple candidate plans. The Tree-of-Thoughts framework from Princeton computer scientist Shunyu Yao proposes that models "perform deliberate decision making" by exploring and backtracking over a tree of intermediate thoughts. OpenAI's reported breakthrough focused on supervising reasoning processes rather than only outcomes, with Lightman et al.'s "Let's Verify Step by Step" reporting that rewarding each correct step "significantly outperforms outcome supervision" on challenging math problems and improves interpretability by aligning the chain-of-thought with human judgment. OpenAI's o1 announcement ties these strands together with a large-scale reinforcement learning algorithm that trains the model to refine its own chain of thought, and it reports that accuracy rises with more training compute and more time spent thinking at inference. Together, these developments define the core of reasoning models. They use supervision signals that evaluate the quality of intermediate steps, they exploit inference-time exploration such as consensus or tree search, and they expose controls for how much internal thinking compute to allocate. OpenAI's o1 family made this approach available at scale in September 2024 and popularized the label "reasoning model" for LLMs that deliberately think before they answer. The development of reasoning models illustrates Richard S. Sutton's "bitter lesson" that scaling compute typically outperforms methods based on human-designed insights. This principle was demonstrated by researchers at the Generative AI Research Lab (GAIR), who initially attempted to replicate o1's capabilities using sophisticated methods including tree search and reinforcement learning in late 2024. Their findings, published in the "o1 Replication Journey" series, revealed that knowledge distillation, a comparatively straightforward technique that trains a smaller model to mimic o1's outputs, produced unexpectedly strong performance. This outcome illustrated how direct scaling approaches can, at times, outperform more complex engineering solutions. === Drawbacks === Reasoning models require significantly more computational resources during inference compared to non-reasoning models. Research on the American Invitational Mathematics Examination (AIME) benchmark found that reasoning models were 10 to 74 times more expensive to operate than their non-reasoning counterparts. The extended inference time is attributed to the detailed, step-by-step reasoning outputs that these models generate, which are typically much longer than responses from standard large language models that provide direct answers without showing their reasoning process. One researcher in early 2025 argued that these models may face potential additional denial-of-service concerns with "overthinking attacks." === Releases === ==== 2024 ==== In September 2024, OpenAI released o1-preview, a large language model with enhanced reasoning capabilities. The full version, o1, was released in December 2024. OpenAI initially shared preliminary results on its successor model, o3, in December 2024, with the full o3 model becoming available in 2025. Alibaba released reasoning versions of its Qwen large language models in November 2024. In December 2024, the company introduced QvQ-72B-Preview, an experimental visual reasoning model. In December 2024, Google introduced Deep Research in Gemini, a feature designed to conduct multi-step research tasks. On December 16, 2024, researchers demonstrated that by scaling test-time compute, a relatively small Llama 3B model could outperform a much larger Llama 70B model on challenging reasoning tasks. This experiment suggested that improved inference strategies can unlock reasoning capabilities even in smaller models. ==== 2025 ==== In January 2025, DeepSeek released R1, a reasoning model that achieved performance comparable to OpenAI's o1 at significantly lower computational cost. The release demonstrated the effectiveness of Group Relative Policy Optimization (GRPO), a reinforcement learning technique used to train the model. On January 25, 2025, DeepSeek enhanced R1 with web search capabilities, allowing the model to retrieve information from the internet while performing reasoning tasks. Research during this period further validated the effectiveness of knowledge distillation for creating reasoning models. The s1-32B model achieved strong performance through budget forcing and scaling methods, reinforcing findings that simpler training approaches can be highly effective for reasoning capabilities. On February 2, 2025, OpenAI released Deep Research, a feature powered by their o3 model that enables users to conduct comprehensive research tasks. The system generates detailed reports by automatically gathering and synthesizing information from multiple web sources. OpenAI called GPT-4.5 its "last non-chain-of-thought model", and implemented with GPT-5 a router model that selects a model based on the difficulty of the task. ==== 2026 ==== In January 2026, Moonshot AI released Kimi K2.5, an open-source 1 trillion parameter MoE model with 32 billion active parameters. It uses an “Agent Swarm” system that dynamically decomposes tasks into sub-agents for reasoning and execution, enabling more scalable multi-step problem solving than a single sequential reasoning chain. == Training == Reasoning models follow the familiar large-scale pretraining used for frontier language models, then diverge in the post-training and optimization. OpenAI reports that o1 is trained with a large-
ASR-complete
ASR-complete is, by analogy to "NP-completeness" in complexity theory, a term to indicate that the difficulty of a computational problem is equivalent to solving the central automatic speech recognition problem, i.e. recognize and understanding spoken language. Unlike "NP-completeness", this term is typically used informally. Such problems are hypothesised to include: Spoken natural language understanding Understanding speech from far-field microphones, i.e. handling the reverbation and background noise These problems are easy for humans to do (in fact, they are described directly in terms of imitating humans). Some systems can solve very simple restricted versions of these problems, but none can solve them in their full generality.
PhyCV
PhyCV is the first computer vision library which utilizes algorithms directly derived from the equations of physics governing physical phenomena. The algorithms appearing in the first release emulate the propagation of light through a physical medium with natural and engineered diffractive properties followed by coherent detection. Unlike traditional algorithms that are a sequence of hand-crafted empirical rules, physics-inspired algorithms leverage physical laws of nature as blueprints. In addition, these algorithms can, in principle, be implemented in real physical devices for fast and efficient computation in the form of analog computing. Currently PhyCV has three algorithms, Phase-Stretch Transform (PST) and Phase-Stretch Adaptive Gradient-Field Extractor (PAGE), and Vision Enhancement via Virtual diffraction and coherent Detection (VEViD). All algorithms have CPU and GPU versions. PhyCV is now available on GitHub and can be installed from pip. == History == Algorithms in PhyCV are inspired by the physics of the photonic time stretch (a hardware technique for ultrafast and single-shot data acquisition). PST is an edge detection algorithm that was open-sourced in 2016 and has 800+ stars and 200+ forks on GitHub. PAGE is a directional edge detection algorithm that was open-sourced in February, 2022. PhyCV was originally developed and open-sourced by Jalali-Lab @ UCLA in May 2022. In the initial release of PhyCV, the original open-sourced code of PST and PAGE is significantly refactored and improved to be modular, more efficient, GPU-accelerated and object-oriented. VEViD is a low-light and color enhancement algorithm that was added to PhyCV in November 2022. == Background == === Phase-Stretch Transform (PST) === Phase-Stretch Transform (PST) is a computationally efficient edge and texture detection algorithm with exceptional performance in visually impaired images. The algorithm transforms the image by emulating propagation of light through a device with engineered diffractive property followed by coherent detection. It has been applied in improving the resolution of MRI image, extracting blood vessels in retina images, dolphin identification, and waste water treatment, single molecule biological imaging, and classification of UAV using micro Doppler imaging. === Phase-Stretch Adaptive Gradient-Field Extractor (PAGE) === Phase-Stretch Adaptive Gradient-Field Extractor (PAGE) is a physics-inspired algorithm for detecting edges and their orientations in digital images at various scales. The algorithm is based on the diffraction equations of optics. Metaphorically speaking, PAGE emulates the physics of birefringent (orientation-dependent) diffractive propagation through a physical device with a specific diffractive structure. The propagation converts a real-valued image into a complex function. Related information is contained in the real and imaginary components of the output. The output represents the phase of the complex function. === Vision Enhancement via Virtual diffraction and coherent Detection (VEViD) === Vision Enhancement via Virtual diffraction and coherent Detection (VEViD) an efficient and interpretable low-light and color enhancement algorithm that reimagines a digital image as a spatially varying metaphoric light field and then subjects the field to the physical processes akin to diffraction and coherent detection. The term “Virtual” captures the deviation from the physical world. The light field is pixelated and the propagation imparts a phase with an arbitrary dependence on frequency which can be different from the quadratic behavior of physical diffraction. VEViD can be further accelerated through mathematical approximations that reduce the computation time without appreciable sacrifice in image quality. A closed-form approximation for VEViD which we call VEViD-lite can achieve up to 200 FPS for 4K video enhancement. == PhyCV on the Edge == Featuring low-dimensionality and high-efficiency, PhyCV is ideal for edge computing applications. In this section, we demonstrate running PhyCV on NVIDIA Jetson Nano in real-time. === NVIDIA Jetson Nano Developer Kit === NVIDIA Jetson Nano Developer Kit is a small- sized and power-efficient platform for edge computing applications. It is equipped with an NVIDIA Maxwell architecture GPU with 128 CUDA cores, a quad-core ARM Cortex-A57 CPU, 4GB 64-bit LPDDR4 RAM, and supports video encoding and decoding up to 4K resolution. Jetson Nano also offers a variety of interfaces for connectivity and expansion, making it ideal for a wide range of AI and IoT applications. In our setup, we connect a USB camera to the Jetson Nano to acquire videos and demonstrate using PhyCV to process the videos in real-time. === Real-time PhyCV on Jetson Nano === We use the Jetson Nano (4GB) with NVIDIA JetPack SDK version 4.6.1, which comes with pre- installed Python 3.6, CUDA 10.2, and OpenCV 4.1.1. We further install PyTorch 1.10 to enable the GPU accelerated PhyCV. We demonstrate the results and metrics of running PhyCV on Jetson Nano in real-time for edge detection and low-light enhancement tasks. For 480p videos, both operations achieve beyond 38 FPS, which is sufficient for most cameras that capture videos at 30 FPS. For 720p videos, PhyCV low-light enhancement can operate at 24 FPS and PhyCV edge detection can operate at 17 FPS. == Highlights == === Modular Code Architecture === The code in PhyCV has a modular design which faithfully follows the physical process from which the algorithm was originated. Both PST and PAGE modules in the PhyCV library emulate the propagation of the input signal (original digital image) through a device with engineered diffractive property followed by coherent (phase) detection. The dispersive propagation applies a phase kernel to the frequency domain of the original image. This process has three steps in general, loading the image, initializing the kernel and applying the kernel. In the implementation of PhyCV, each algorithm is represented as a class in Python and each class has methods that simulate the steps described above. The modular code architecture follows the physics behind the algorithm. Please refer to the source code on GitHub for more details. === GPU Acceleration === PhyCV supports GPU acceleration. The GPU versions of PST and PAGE are built on PyTorch accelerated by the CUDA toolkit. The acceleration is beneficial for applying the algorithms in real-time image video processing and other deep learning tasks. The running time per frame of PhyCV algorithms on CPU (Intel i9-9900K) and GPU (NVIDIA TITAN RTX) for videos at different resolutions are shown below. Note that the PhyCV low-light enhancement operates in the HSV color space, so the running time also includes RGB to HSV conversion. However, for all running times using GPUs, we ignore the time of moving data from CPUs to GPUs and count the algorithm operation time only. == Installation and Examples == Please refer to the GitHub README file for a detailed technical documentation. == Current Limitations == === I/O (Input/Output) Bottleneck for Real-time Video Processing === When dealing with real-time video streams from cameras, the frames are captured and buffered in CPU and have to be moved to GPU to run the GPU-accelerated PhyCV algorithms. This process is time-consuming and it is a common bottleneck for real-time video-processing algorithms. === Lack of Parameter Adaptivity for Different Images === Currently, the parameters of PhyCV algorithms have to be manually tuned for different images. Although a set of pre-selected parameters work relatively well for a wide range of images, the lack of parameter adaptivity for different images remains a limitation for now.
Text normalization
Text normalization is the process of transforming text into a single canonical form that it might not have had before. Normalizing text before storing or processing it allows for separation of concerns, since input is guaranteed to be consistent before operations are performed on it. Text normalization requires being aware of what type of text is to be normalized and how it is to be processed afterwards; there is no all-purpose normalization procedure. == Applications == Text normalization is frequently used when converting text to speech. Numbers, dates, acronyms, and abbreviations are non-standard "words" that need to be pronounced differently depending on context. For example: "$200" would be pronounced as "two hundred dollars" in English, but as "lua selau tālā" in Samoan. "vi" could be pronounced as "vie," "vee," or "the sixth" depending on the surrounding words. Text can also be normalized for storing and searching in a database. For instance, if a search for "resume" is to match the word "résumé," then the text would be normalized by removing diacritical marks; and if "john" is to match "John", the text would be converted to a single case. To prepare text for searching, it might also be stemmed (e.g. converting "flew" and "flying" both into "fly"), canonicalized (e.g. consistently using American or British English spelling), or have stop words removed. == Techniques == For simple, context-independent normalization, such as removing non-alphanumeric characters or diacritical marks, regular expressions would suffice. For example, the sed script sed ‑e "s/\s+/ /g" inputfile would normalize runs of whitespace characters into a single space. More complex normalization requires correspondingly complicated algorithms, including domain knowledge of the language and vocabulary being normalized. Among other approaches, text normalization has been modeled as a problem of tokenizing and tagging streams of text and as a special case of machine translation. == Textual scholarship == In the field of textual scholarship and the editing of historic texts, the term "normalization" implies a degree of modernization and standardization – for example in the extension of scribal abbreviations and the transliteration of the archaic glyphs typically found in manuscript and early printed sources. A normalized edition is therefore distinguished from a diplomatic edition (or semi-diplomatic edition), in which some attempt is made to preserve these features. The aim is to strike an appropriate balance between, on the one hand, rigorous fidelity to the source text (including, for example, the preservation of enigmatic and ambiguous elements); and, on the other, producing a new text that will be comprehensible and accessible to the modern reader. The extent of normalization is therefore at the discretion of the editor, and will vary. Some editors, for example, choose to modernize archaic spellings and punctuation, but others do not. An edition of a text might be normalized based on internal criteria, where orthography is standardized according to the language of the original, or external criteria, where the norms of a different time period are applied. For an example of the latter, a published edition of a medieval Icelandic manuscript might be normalized to the conventions of modern Icelandic, or it might be normalized to Classical Old Icelandic. Standards of normalization vary based on language of the edition as well as the specific conventions of the publisher.
BERT (language model)
Bidirectional encoder representations from transformers (BERT) is a language model introduced in October 2018 by researchers at Google. It learns to represent text as a sequence of vectors using self-supervised learning. It uses the encoder-only transformer architecture. BERT dramatically improved the state of the art for large language models. As of 2020, BERT is a ubiquitous baseline in natural language processing (NLP) experiments. BERT is trained by masked token prediction and next sentence prediction. With this training, BERT learns contextual, latent representations of tokens in their context, similar to ELMo and GPT-2. It found applications for many natural language processing tasks, such as coreference resolution and polysemy resolution. It improved on ELMo and spawned the study of "BERTology", which attempts to interpret what is learned by BERT. BERT was originally implemented in the English language at two model sizes, BERTBASE (110 million parameters) and BERTLARGE (340 million parameters). Both were trained on the Toronto BookCorpus (800M words) and English Wikipedia (2,500M words). The weights were released on GitHub. On March 11, 2020, 24 smaller models were released, the smallest being BERTTINY with just 4 million parameters. == Architecture == BERT is an "encoder-only" transformer architecture. At a high level, BERT consists of 4 modules: Tokenizer: This module converts a piece of English text into a sequence of integers ("tokens"). Embedding: This module converts the sequence of tokens into an array of real-valued vectors representing the tokens. It represents the conversion of discrete token types into a lower-dimensional Euclidean space. Encoder: a stack of Transformer blocks with self-attention, but without causal masking. Task head: This module converts the final representation vectors into one-shot encoded tokens again by producing a predicted probability distribution over the token types. It can be viewed as a simple decoder, decoding the latent representation into token types, or as an "un-embedding layer". The task head is necessary for pre-training, but it is often unnecessary for so-called "downstream tasks," such as question answering or sentiment classification. Instead, one removes the task head and replaces it with a newly initialized module suited for the task, and finetune the new module. The latent vector representation of the model is directly fed into this new module, allowing for sample-efficient transfer learning. === Embedding === This section describes the embedding used by BERTBASE. The other one, BERTLARGE, is similar, just larger. The tokenizer of BERT is WordPiece, which is a sub-word strategy like byte-pair encoding. Its vocabulary size is 30,000, and any token not appearing in its vocabulary is replaced by [UNK] ("unknown"). The first layer is the embedding layer, which contains three components: token type embeddings, position embeddings, and segment type embeddings. Token type: The token type is a standard embedding layer, translating a one-hot vector into a dense vector based on its token type. Position: The position embeddings are based on a token's position in the sequence. BERT uses absolute position embeddings, where each position in a sequence is mapped to a real-valued vector. Each dimension of the vector consists of a sinusoidal function that takes the position in the sequence as input. Segment type: Using a vocabulary of just 0 or 1, this embedding layer produces a dense vector based on whether the token belongs to the first or second text segment in that input. In other words, type-1 tokens are all tokens that appear after the [SEP] special token. All prior tokens are type-0. The three embedding vectors are added together representing the initial token representation as a function of these three pieces of information. After embedding, the vector representation is normalized using a LayerNorm operation, outputting a 768-dimensional vector for each input token. After this, the representation vectors are passed forward through 12 Transformer encoder blocks, and are decoded back to 30,000-dimensional vocabulary space using a basic affine transformation layer. === Architectural family === The encoder stack of BERT has 2 free parameters: L {\displaystyle L} , the number of layers, and H {\displaystyle H} , the hidden size. There are always H / 64 {\displaystyle H/64} self-attention heads, and the feed-forward/filter size is always 4 H {\displaystyle 4H} . By varying these two numbers, one obtains an entire family of BERT models. For BERT: the feed-forward size and filter size are synonymous. Both of them denote the number of dimensions in the middle layer of the feed-forward network. the hidden size and embedding size are synonymous. Both of them denote the number of real numbers used to represent a token. The notation for encoder stack is written as L/H. For example, BERTBASE is written as 12L/768H, BERTLARGE as 24L/1024H, and BERTTINY as 2L/128H. == Training == === Pre-training === BERT was pre-trained simultaneously on two tasks: Masked language modeling (MLM): In this task, BERT ingests a sequence of words, where one word may be randomly changed ("masked"), and BERT tries to predict the original words that had been changed. For example, in the sentence "The cat sat on the [MASK]," BERT would need to predict "mat." This helps BERT learn bidirectional context, meaning it understands the relationships between words not just from left to right or right to left but from both directions at the same time. Next sentence prediction (NSP): In this task, BERT is trained to predict whether one sentence logically follows another. For example, given two sentences, "The cat sat on the mat" and "It was a sunny day", BERT has to decide if the second sentence is a valid continuation of the first one. This helps BERT understand relationships between sentences, which is important for tasks like question answering or document classification. ==== Masked language modeling ==== In masked language modeling, 15% of tokens would be randomly selected for masked-prediction task, and the training objective was to predict the masked token given its context. In more detail, the selected token is: replaced with a [MASK] token with probability 80%, replaced with a random word token with probability 10%, not replaced with probability 10%. The reason not all selected tokens are masked is to avoid the dataset shift problem. The dataset shift problem arises when the distribution of inputs seen during training differs significantly from the distribution encountered during inference. A trained BERT model might be applied to word representation (like Word2Vec), where it would be run over sentences not containing any [MASK] tokens. It is later found that more diverse training objectives are generally better. As an illustrative example, consider the sentence "my dog is cute". It would first be divided into tokens like "my1 dog2 is3 cute4". Then a random token in the sentence would be picked. Let it be the 4th one "cute4". Next, there would be three possibilities: with probability 80%, the chosen token is masked, resulting in "my1 dog2 is3 [MASK]4"; with probability 10%, the chosen token is replaced by a uniformly sampled random token, such as "happy", resulting in "my1 dog2 is3 happy4"; with probability 10%, nothing is done, resulting in "my1 dog2 is3 cute4". After processing the input text, the model's 4th output vector is passed to its decoder layer, which outputs a probability distribution over its 30,000-dimensional vocabulary space. ==== Next sentence prediction ==== Given two sentences, the model predicts if they appear sequentially in the training corpus, outputting either [IsNext] or [NotNext]. During training, the algorithm sometimes samples two sentences from a single continuous span in the training corpus, while at other times, it samples two sentences from two discontinuous spans. The first sentence starts with a special token, [CLS] (for "classify"). The two sentences are separated by another special token, [SEP] (for "separate"). After processing the two sentences, the final vector for the [CLS] token is passed to a linear layer for binary classification into [IsNext] and [NotNext]. For example: Given "[CLS] my dog is cute [SEP] he likes playing [SEP]", the model should predict [IsNext]. Given "[CLS] my dog is cute [SEP] how do magnets work [SEP]", the model should predict [NotNext]. === Fine-tuning === BERT is meant as a general pretrained model for various applications in natural language processing. That is, after pre-training, BERT can be fine-tuned with fewer resources on smaller datasets to optimize its performance on specific tasks such as natural language inference and text classification, and sequence-to-sequence-based language generation tasks such as question answering and conversational response generation. The original BERT paper published results demonstrating that a small amount of fine
Averbis
Averbis has a focus on healthcare, pharma, automotive and intellectual property analytics. Averbis is involved in various research projects of the German Federal Ministry of Economics and Energy and the European Union such as DebugIT, EUCases, Mantra and SEMCARE. In addition to these projects, Averbis was also involved in the following projects: Greenpilot is a virtual library, which provides technical information in the fields of nutrition, environment and agriculture. Medpilot is a virtual library, which provides information about medicine and related sciences. In 2013, Averbis has been nominated for the German Founder Prize 2013. Averbis GmbH provides text analytics and text mining software to transform unstructured text into actionable information. It was founded in 2007 by IT experts after years of relevant scientific experience in the field of text mining and multilingual information retrieval. Averbis works in the field of terminology management, natural language processing, machine learning and semantic search. Its text mining software is embedded into the text mining framework UIMA.
Point distribution model
The point distribution model is a model for representing the mean geometry of a shape and some statistical modes of geometric variation inferred from a training set of shapes. == Background == The point distribution model concept has been developed by Cootes, Taylor et al. and became a standard in computer vision for the statistical study of shape and for segmentation of medical images where shape priors really help interpretation of noisy and low-contrasted pixels/voxels. The latter point leads to active shape models (ASM) and active appearance models (AAM). Point distribution models rely on landmark points. A landmark is an annotating point posed by an anatomist onto a given locus for every shape instance across the training set population. For instance, the same landmark will designate the tip of the index finger in a training set of 2D hands outlines. Principal component analysis (PCA), for instance, is a relevant tool for studying correlations of movement between groups of landmarks among the training set population. Typically, it might detect that all the landmarks located along the same finger move exactly together across the training set examples showing different finger spacing for a flat-posed hands collection. == Details == First, a set of training images are manually landmarked with enough corresponding landmarks to sufficiently approximate the geometry of the original shapes. These landmarks are aligned using the generalized procrustes analysis, which minimizes the least squared error between the points. k {\displaystyle k} aligned landmarks in two dimensions are given as X = ( x 1 , y 1 , … , x k , y k ) {\displaystyle \mathbf {X} =(x_{1},y_{1},\ldots ,x_{k},y_{k})} . It's important to note that each landmark i ∈ { 1 , … k } {\displaystyle i\in \lbrace 1,\ldots k\rbrace } should represent the same anatomical location. For example, landmark #3, ( x 3 , y 3 ) {\displaystyle (x_{3},y_{3})} might represent the tip of the ring finger across all training images. Now the shape outlines are reduced to sequences of k {\displaystyle k} landmarks, so that a given training shape is defined as the vector X ∈ R 2 k {\displaystyle \mathbf {X} \in \mathbb {R} ^{2k}} . Assuming the scattering is gaussian in this space, PCA is used to compute normalized eigenvectors and eigenvalues of the covariance matrix across all training shapes. The matrix of the top d {\displaystyle d} eigenvectors is given as P ∈ R 2 k × d {\displaystyle \mathbf {P} \in \mathbb {R} ^{2k\times d}} , and each eigenvector describes a principal mode of variation along the set. Finally, a linear combination of the eigenvectors is used to define a new shape X ′ {\displaystyle \mathbf {X} '} , mathematically defined as: X ′ = X ¯ + P b {\displaystyle \mathbf {X} '={\overline {\mathbf {X} }}+\mathbf {P} \mathbf {b} } where X ¯ {\displaystyle {\overline {\mathbf {X} }}} is defined as the mean shape across all training images, and b {\displaystyle \mathbf {b} } is a vector of scaling values for each principal component. Therefore, by modifying the variable b {\displaystyle \mathbf {b} } an infinite number of shapes can be defined. To ensure that the new shapes are all within the variation seen in the training set, it is common to only allow each element of b {\displaystyle \mathbf {b} } to be within ± {\displaystyle \pm } 3 standard deviations, where the standard deviation of a given principal component is defined as the square root of its corresponding eigenvalue. PDM's can be extended to any arbitrary number of dimensions, but are typically used in 2D image and 3D volume applications (where each landmark point is R 2 {\displaystyle \mathbb {R} ^{2}} or R 3 {\displaystyle \mathbb {R} ^{3}} ). == Discussion == An eigenvector, interpreted in euclidean space, can be seen as a sequence of k {\displaystyle k} euclidean vectors associated to corresponding landmark and designating a compound move for the whole shape. Global nonlinear variation is usually well handled provided nonlinear variation is kept to a reasonable level. Typically, a twisting nematode worm is used as an example in the teaching of kernel PCA-based methods. Due to the PCA properties: eigenvectors are mutually orthogonal, form a basis of the training set cloud in the shape space, and cross at the 0 in this space, which represents the mean shape. Also, PCA is a traditional way of fitting a closed ellipsoid to a Gaussian cloud of points (whatever their dimension): this suggests the concept of bounded variation. The idea behind PDMs is that eigenvectors can be linearly combined to create an infinity of new shape instances that will 'look like' the one in the training set. The coefficients are bounded alike the values of the corresponding eigenvalues, so as to ensure the generated 2n/3n-dimensional dot will remain into the hyper-ellipsoidal allowed domain—allowable shape domain (ASD).