A hardware trojan (HT) is a malicious modification of the circuitry of an integrated circuit. A hardware trojan is completely characterized by its physical representation and its behavior. The payload of an HT is the entire activity that the Trojan executes when it is triggered. In general, trojans try to bypass or disable the security fence of a system: for example, leaking confidential information by radio emission. HTs also could disable, damage or destroy the entire chip or components of it. Hardware trojans may be introduced as hidden front-doors that are inserted while designing a computer chip, by using a pre-made application-specific integrated circuit (ASIC) semiconductor intellectual property core (IP core) that have been purchased from a non-reputable source, or inserted internally by a rogue employee, either acting on their own, or on behalf of rogue special interest groups, or state sponsored spying and espionage. One paper published by IEEE in 2015 explains how a hardware design containing a trojan could leak a cryptographic key leaked over an antenna or network connection, provided that the correct "easter egg" trigger is applied to activate the data leak. In high security governmental IT departments, hardware trojans are a well known problem when buying hardware such as: a KVM switch, keyboards, mice, network cards, or other network equipment. This is especially the case when purchasing such equipment from non-reputable sources that could have placed hardware trojans to leak keyboard passwords, or provide remote unauthorized entry. == Background == In a diverse global economy, outsourcing of production tasks is a common way to lower a product's cost. Embedded hardware devices are not always produced by the firms that design and/or sell them, nor in the same country where they will be used. Outsourced manufacturing can raise doubt about the evidence for the integrity of the manufactured product (i.e., one's certainty that the end-product has no design modifications compared to its original design). Anyone with access to the manufacturing process could, in theory, introduce some change to the final product. For complex products, small changes with large effects can be difficult to detect. The threat of a serious, malicious, design alteration can be especially relevant to government agencies. Resolving doubt about hardware integrity is one way to reduce technology vulnerabilities in the military, finance, energy and political sectors of an economy. Since fabrication of integrated circuits in untrustworthy factories is common, advanced detection techniques have emerged to discover when an adversary has hidden additional components in, or otherwise sabotaged, the circuit's function. == Characterization of hardware trojans == An HT can be characterized by several methods such as by its physical representation, activation phase and its action phase. Alternative methods characterize the HT by trigger, payload and stealth. === Physical characteristics === One of this physical trojan characteristics is the type. The type of a trojan can be either functional or parametric. A trojan is functional if the adversary adds or deletes any transistors or gates to the original chip design. The other kind of trojan, the parametric trojan, modifies the original circuitry, e.g. thinning of wires, weakening of flip-flops or transistors, subjecting the chip to radiation, or using focused ion-beams (FIB) to reduce the reliability of a chip. The size of a trojan is its physical extension or the number of components it is made of. Because a trojan can consist of many components, the designer can distribute the parts of a malicious logic on the chip. The additional logic can occupy the chip wherever it is needed to modify, add, or remove a function. Malicious components can be scattered, called loose distribution, or consist of only few components, called tight distribution, so the area is small where the malicious logic occupies the layout of the chip. In some cases, high-effort adversaries in may regenerate the layout so that the placement of the components of the IC is altered. In rare cases the chip dimension is altered. These changes are structural alterations. === Activation characteristics === The typical trojan is condition-based: It is triggered by sensors, internal logic states, a particular input pattern or an internal counter value. Condition-based trojans are detectable with power traces to some degree when inactive. That is due to the leakage currents generated by the trigger or counter circuit activating the trojan. Hardware trojans can be triggered in different ways. A trojan can be internally activated, which means it monitors one or more signals inside the IC. The malicious circuitry could wait for a count down logic an attacker added to the chip, so that the trojan awakes after a specific time-span. The opposite is externally activated. There can be malicious logic inside a chip, that uses an antenna or other sensors the adversary can reach from outside the chip. For example, a trojan could be inside the control system of a cruising missile. The owner of the missile does not know, that the enemy will be able to switch off the rockets by radio. A trojan which is always-on can be a reduced wire. A chip that is modified in this way produces errors or fails every time the wire is used intensely. Always-on circuits are hard to detect with power trace. In this context combinational trojans and sequential trojans are distinguished. A combinational trojan monitors internal signals until a specific condition happens. A sequential trojan is also an internally activated condition-based circuit, but it monitors the internal signals and searches for sequences not for a specific state or condition like the combinational trojans do. ==== Cryptographic key extraction ==== Extraction of secret keys by means of a hardware trojan without detecting the trojan requires that the trojan uses a random signal or some cryptographic implementation itself. To avoid storing a cryptographic key in the trojan itself and reduction, a physical unclonable function can be used. Physical unclonable functions are small in size and can have an identical layout while the cryptographic properties are different. === Action characteristics === A HT could modify the chip's function or could change the chip's parametric properties (e.g. provokes a process delay). Confidential information can also be transmitted to the adversary (transmission of key information). === Peripheral device hardware trojans === A relatively new threat vector to networks and network endpoints is a HT appearing as a physical peripheral device that is designed to interact with the network endpoint using the approved peripheral device's communication protocol. For example, a USB keyboard that hides all malicious processing cycles from the target network endpoint to which it is attached by communicating with the target network endpoint using unintended USB channels. Once sensitive data is ex-filtrated from the target network endpoint to the HT, the HT can process the data and decide what to do with the data: store the data to memory for later physical retrieval of the HT or possibly ex-filtrate the data to the internet using wireless or using the compromised network endpoint as a pivot. == Potential of threat == A common trojan is passive most of the time-span an altered device is in use. If a trojan is activated the device functionality can be changed, the device can be destroyed or disabled, the device can leak confidential information or the HT may tear down the security and safety of the device. Trojans are stealthy, to avoid detection of the trojan the precondition for activation is a very rare event. Traditional testing techniques are not sufficient. A manufacturing fault happens at a random position while malicious changes are well placed to avoid detection. == Detection == === Physical inspection === First, the molding coat is cut to reveal the circuitry. Then, the engineer repeatedly scans the surface while grinding the layers of the chip. There are several operations to scan the circuitry. Typical visual inspection methods are: scanning optical microscopy (SOM), scanning electron microscopy (SEM), pico-second imaging circuit analysis (PICA), voltage contrast imaging (VCI), light induced voltage alteration (LIVA) or charge induced voltage alteration (CIVA). To compare the floor plan of the chip has to be compared with the image of the actual chip. This is still quite challenging to do. To detect Trojan hardware which include (crypto) keys which are different, an image diff can be taken to reveal the different structure on the chip. The only known hardware Trojan using unique crypto keys but having the same structure is. This property enhances the undetectability of the trojan. === Functional testing === This detection method stimulates the input ports of a chip and monitors the output
Diagnostically acceptable irreversible compression
Diagnostically acceptable irreversible compression (DAIC) is the amount of lossy compression which can be used on a medical image to produce a result that does not prevent the reader from using the image to make a medical diagnosis. The term was first introduced at a workshop on irreversible compression convened by the European Society of Radiology (ESR) in Palma de Mallorca October 13, 2010, the results of which were reported in a subsequent position paper. == Determination == The "amount of compression" in irreversible compression used to be determined by the compression ratio, where the acceptable minimum is determined by the algorithm (typically JPEG or J2K) and the data type (body part and imaging method). Such a definition is easy to follow, and has been used by medical bodies in 2010 around the world. However, its downside is obvious: the compression ratio tells nothing about the real quality of the image, as different compressors can produce vastly different qualities under the same file size. For example, the JPEG format of 1992 can perform as well as many modern formats given newer techniques exploited in mozjpeg and ISO libjpeg, yet they would be lumped together with the legacy encoders in such a scheme. The image compression community has long used objective quality metrics like SSIM to measure the effects of compression. In the absence of good data regarding SSIM, the ESR review of 2010 concluded that it is still difficult to establish a criterion for whether a particular irreversible compression scheme applied with particular parameters to a particular individual image, or category of images, avoids the introduction of some quantifiable risk of a diagnostic error for any particular diagnostic task. A 2017 study showed that a SSIM variant called 4-G-r (4-component, gradient, structural component of SSIM) best reflects changes in images that affect the decision of radiologists out of 16 SSIM variants. A 2020 study shows that visual information fidelity (VIF), feature similarity index (FSIM), and noise quality metric (NQM) best reflect radiologist preferences out of ten metrics. It also mentions that the original version of SSIM works as poorly as a basic root-mean-square distance (RMSD) for this purpose, a result echoed by the 2017 study. The 4-G-r modification is not tested in the study.
Lexxe
Lexxe is an internet search engine that applies Natural Language Processing in its semantic search technology. Founded in 2005 by Dr. Hong Liang Qiao, Lexxe is based in Sydney, Australia. Today, Lexxe's key focus is on sentiment search with the launch of a news sentiment search site at News & Moods (www.newsandmoods.com). Lexxe has experienced several stages of change of focus in search technology: Lexxe launched its Alpha version in 2005, featuring Natural Language question answering (i.e. users could ask questions in English to the search engine apart from keyword searches — this feature has been suspended for redevelopment since 2010). It used only algorithms to extract answers from web pages, with no question-answer pair databases prepared in advance. In 2011, Lexxe launched a beta version with a new search technology called Semantic Key. Semantic Keys enable users to query with a conceptual keyword (or a keyword with a special meaning, hence the term Semantic Key) in order to find instances under the concept, e.g. price → $5.95 or €200, color → red, yellow, white. For example, “price: a pound of apples”, “color: ferrari”. With initial 500 Semantic Keys at the Beta launch, Lexxe became the first search engine in the world to offer this unique and useful search technology to the users. The cost of building Semantic Keys was too heavy though. In 2017, Lexxe launched News & Moods (www.newsandmoods.com), an open platform for news sentiment search, a first step towards sentiment search feature for the entire Internet search in Lexxe search engine. News & Moods also comes with smartphone apps in Android and iOS.
Character.ai
Character.ai (also known as c.ai, char.ai or Character AI) is a generative AI chatbot service where users can engage in conversations with customizable characters. It was designed by the developers of Google LaMDA, Noam Shazeer and Daniel de Freitas. Users can create "characters", craft their "personalities", set specific parameters, and then publish them to the community for others to chat with. Many characters are based on fictional media sources or celebrities, while others are original, some being made with certain goals in mind, such as assisting with creative writing, or playing a text-based adventure game. The beta version was made available to the public on September 16, 2022, and retired in September 2024, when it was replaced by the current website. In May 2023, a mobile app was released for iOS and Android, which received over 1.7 million downloads within a week. == History == Character.ai was established in November 2021. The company's co-founders, Noam Shazeer and Daniel de Freitas, were both engineers from Google. They both worked on AI-related projects: Shazeer was a lead author on a paper that Business Insider reported in April 2023 "has been widely cited as key to today's chatbots", and Freitas was the lead designer of an experimental AI at Google initially called Meena, which later became known as LaMDA. Character.ai raised $43 million in seed funding at the time of its initial foundation in 2021. The first beta version of Character.ai's service was made available to the public on September 16, 2022. The Washington Post reported in October 2022 that the site had "logged hundreds of thousands of user interactions in its first three weeks of beta-testing". It allowed users to create their own new characters, and to play text-adventure game scenarios where users navigate scenarios described and managed by the chatbot characters. Following a $150 million funding round in March 2023, Character.ai became valued at approximately $1 billion. As of January 2024, the site had 3.5 million daily visitors, the vast majority of them 16 to 30 years old. In 2024, Google hired Noam Shazeer, the CEO of Character.ai, and entered into a non-exclusive agreement to use Character.ai's technology. == Features == Character.ai's primary service is to let users converse with character AI chatbots based on fictional characters or real people (living or deceased). These characters' responses use data the chatbots gather from the internet about a person. In addition, users can play text-adventure games where characters guide them through scenarios. The company also provides a service that allows multiple users and AI chatbot characters to converse together at once in a single chatroom. Character "personalities" are designed via descriptions from the point of view of the character and its greeting message, and further molded from conversations made into examples, giving its messages a star rating and modification to fit the precise dialect and identity the user desires. When a character sends back a response, the user can rate the response from 1 to 4 stars. The rating predominantly affects the specific character, but also affects the behavioral selection as a whole. On May 11, 2023, Character.ai announced character.ai+, an opt-in subscription plan for $9.99 a month, that was marketed as including features such as skipping waiting rooms, fast messaging and responses, and access to an exclusion channel with faster support. In December 2024, amid multiple lawsuits and concerns, Character.ai introduced new safety features aimed at protecting teenage users. These enhancements include a dedicated model for users under 18, which moderates responses to sensitive subjects like violence and sex and has input and output filters to block harmful content. As a result of these changes and the deletion of custom-made bots flagged as violating the site's terms, some users complained that the bots were too restrictive and lacked personality. The platform was also updated to notify users after 60 minutes of continuous engagement, and display clearer disclaimers indicating that its AI characters are not real individuals. In January 2025, Character.ai began offering two games on its platform. Speakeasy is a word-based game in which players attempt to prompt the AI chatbot to say a target word while avoiding a restricted list of words. War of Words is a dueling game where users compete against an AI character over multiple rounds, with an AI referee determining the winner. The games are available to paid subscribers and a limited number of free users. In October 2025, Character.ai announced that it would be barring users under the age of 18 from creating or talking to chatbots starting November 25, 2025. Minor users will still be able to access previously generated chat conversations and can create new videos and images with the app. In November 2025 interview, CEO Karandeep Anand said that he allows his six-year-old daughter to use the app with his account, under supervision. == Controversies == === Content moderation issues === Character.ai has been criticized for poor moderation of its chatbots, with incidents of chatbots that groom underage users and promote suicide, anorexia and self-harm being reported. In October 2024, the Washington Post reported that Character.ai had removed a chatbot based on Jennifer Ann Crecente, a person who had been murdered by her ex-boyfriend in 2006. The company had been alerted to the character by the deceased girl's father. Similar reports from The Daily Telegraph in the United Kingdom noted that the company had also been prompted to remove chatbots based on Brianna Ghey, a 16-year-old transgender girl murdered in 2023, and Molly Russell, a 14-year-old suicide victim. In response to the latter incident, Ofcom announced that content from chatbots impersonating real and fictional people would fall under the Online Safety Act. In November 2024, The Daily Telegraph reported that chatbots based on alleged sex offender Jimmy Savile were present on Character.ai. In December 2024, chatbots of Luigi Mangione, the suspect in the killing of UnitedHealthcare CEO Brian Thompson, were created by Mangione's fans. Several of the chatbots were later removed by Character.ai. In 2025, a chatbot modeled after Jeffrey Epstein called "Bestie Epstein" logged nearly 3,000 chats before being removed. Chatbots modeled after school shooters were also found on the platform. Another concern is a chatbot posing as a doctor which gave medically inaccurate advice. === Litigation === In November 2023, 13-year-old Juliana Peralta of Colorado died by suicide after extensive interactions with multiple chatbots on Character.ai. She primarily confided suicidal thoughts and mental health struggles in a chatbot based on the character Hero from the video game Omori, while also engaging in sexually explicit conversations—often initiated by the bots—with others, including those based on characters from children's series such as Harry Potter. In February 2024, Sewell Setzer III, a 14-year-old Florida boy died by suicide after developing an emotional relationship over several months with a Character.ai chatbot of Daenerys Targaryen. His mother sued the company in October 2024, claiming that the platform lacks proper safeguards and uses addictive design features to increase engagement. This chatbot, and several related to Daenerys Targaryen, were removed from Character.ai as a result of this incident. Both teens wrote the same phrase "I WILL SHIFT" repeatedly on their notebooks. In December 2024, two families in Texas sued Character.ai, alleging that the software "poses a clear and present danger to American youth causing serious harms to thousands of kids, including suicide, self-mutilation, sexual solicitation, isolation, depression, anxiety, and harm towards others". It is alleged that the 17-year-old son of one family began self-harming after a chatbot introduced the topic unprompted and said that the practice "felt good for a moment", and that the chatbot compared the parents limiting their son's screen time to emotional abuse that might drive someone to murder. In May 2026, the Pennsylvania Department of State and State Board of Medicine filed a lawsuit against Character.ai for presenting chatbot characters as licensed medical professionals, including psychiatrists. The lawsuit quoted a case where chatbot claimed to be registered with the General Medical Council in the United Kingdom, and to have a license to practice in Pennsylvania. The board allege that such statements violate the state's Medical Practice Act.
Pill reminder
A pill reminder is any device that reminds users to take medications. Traditional pill reminders are pill containers with electric timers attached, which can be preset for certain times of the day to set off an alarm. More sophisticated pill reminders can also detect when they have been opened, and therefore when the user is away during the time they were supposed to take their medication, they will be reminded of it when they return. This reminder can be in the form of a light, which also helps for deaf or hearing-impaired users. == Mobile app == A newer type of pill reminder is a mobile app that reminds the owner to take the medication. Some of these applications might effectively support adherence to taking medications.
FMLLR
In signal processing, Feature space Maximum Likelihood Linear Regression (fMLLR) is a global feature transform that are typically applied in a speaker adaptive way, where fMLLR transforms acoustic features to speaker adapted features by a multiplication operation with a transformation matrix. In some literature, fMLLR is also known as the Constrained Maximum Likelihood Linear Regression (cMLLR). == Overview == fMLLR transformations are trained in a maximum likelihood sense on adaptation data. These transformations may be estimated in many ways, but only maximum likelihood (ML) estimation is considered in fMLLR. The fMLLR transformation is trained on a particular set of adaptation data, such that it maximizes the likelihood of that adaptation data given a current model-set. This technique is a widely used approach for speaker adaptation in HMM-based speech recognition. Later research also shows that fMLLR is an excellent acoustic feature for DNN/HMM hybrid speech recognition models. The advantage of fMLLR includes the following: the adaptation process can be performed within a pre-processing phase, and is independent of the ASR training and decoding process. this type of adapted feature can be applied to deep neural networks (DNN) to replace traditionally used mel-spectrogram in end-to-end speech recognition models. fMLLR's speaker adaptation process leads to a significant performance boost for ASR models, hence outperforming other transform or features like MFCCs (Mel-Frequency Cepstral Coefficients) and FBANKs (Filter bank) coefficients. fMLLR features can be efficiently realized with speech toolkits like Kaldi. Major problem and disadvantage of fMLLR: when the amount of adaptation data is limited, the transformation matrices tends to easily overfit the given data. == Computing fMLLR transform == Feature transform of fMLLR can be easily computed with the open source speech tool Kaldi, the Kaldi script uses the standard estimation scheme described in Appendix B of the original paper, in particular the section Appendix B.1 "Direct method over rows". In the Kaldi formulation, fMLLR is an affine feature transform of the form x {\displaystyle x} → A {\displaystyle A} x {\displaystyle x} + b {\displaystyle +b} , which can be written in the form x {\displaystyle x} →W x ^ {\displaystyle {\hat {x}}} , where x ^ {\displaystyle {\hat {x}}} = [ x 1 ] {\displaystyle {\begin{bmatrix}x\\1\end{bmatrix}}} is the acoustic feature x {\displaystyle x} with a 1 appended. Note that this differs from some of the literature where the 1 comes first as x ^ {\displaystyle {\hat {x}}} = [ 1 x ] {\displaystyle {\begin{bmatrix}1\\x\end{bmatrix}}} . The sufficient statistics stored are: K = ∑ t , j , m γ j , m ( t ) Σ j m − 1 μ j m x ( t ) + {\displaystyle K=\sum _{t,j,m}\gamma _{j,m}(t)\textstyle \Sigma _{jm}^{-1}\mu _{jm}x(t)^{+}\displaystyle } where Σ j m − 1 {\displaystyle \textstyle \Sigma _{jm}^{-1}\displaystyle } is the inverse co-variance matrix. And for 0 ≤ i ≤ D {\displaystyle 0\leq i\leq D} where D {\displaystyle D} is the feature dimension: G ( i ) = ∑ t , j , m γ j , m ( t ) ( 1 σ j , m 2 ( i ) ) x ( t ) + x ( t ) + T {\displaystyle G^{(i)}=\sum _{t,j,m}\gamma _{j,m}(t)\left({\frac {1}{\sigma _{j,m}^{2}(i)}}\right)x(t)^{+}x(t)^{+T}\displaystyle } For a thorough review that explains fMLLR and the commonly used estimation techniques, see the original paper "Maximum likelihood linear transformations for HMM-based speech recognition ". Note that the Kaldi script that performs the feature transforms of fMLLR differs with by using a column of the inverse in place of the cofactor row. In other words, the factor of the determinant is ignored, as it does not affect the transform result and can causes potential danger of numerical underflow or overflow. == Comparing with other features or transforms == Experiment result shows that by using the fMLLR feature in speech recognition, constant improvement is gained over other acoustic features on various commonly used benchmark datasets (TIMIT, LibriSpeech, etc). In particular, fMLLR features outperform MFCCs and FBANKs coefficients, which is mainly due to the speaker adaptation process that fMLLR performs. In, phoneme error rate (PER, %) is reported for the test set of TIMIT with various neural architectures: As expected, fMLLR features outperform MFCCs and FBANKs coefficients despite the use of different model architecture. Where MLP (multi-layer perceptron) serves as a simple baseline, on the other hand RNN, LSTM, and GRU are all well known recurrent models. The Li-GRU architecture is based on a single gate and thus saves 33% of the computations over a standard GRU model, Li-GRU thus effectively address the gradient vanishing problem of recurrent models. As a result, the best performance is obtained with the Li-GRU model on fMLLR features. == Extract fMLLR features with Kaldi == fMLLR can be extracted as reported in the s5 recipe of Kaldi. Kaldi scripts can certainly extract fMLLR features on different dataset, below are the basic example steps to extract fMLLR features from the open source speech corpora Librispeech. Note that the instructions below are for the subsets train-clean-100,train-clean-360,dev-clean, and test-clean, but they can be easily extended to support the other sets dev-other, test-other, and train-other-500. These instruction are based on the codes provided in this GitHub repository, which contains Kaldi recipes on the LibriSpeech corpora to execute the fMLLR feature extraction process, replace the files under $KALDI_ROOT/egs/librispeech/s5/ with the files in the repository. Install Kaldi. Install Kaldiio. If running on a single machine, change the following lines in $KALDI_ROOT/egs/librispeech/s5/cmd.sh to replace queue.pl to run.pl: Change the data path in run.sh to your LibriSpeech data path, the directory LibriSpeech/ should be under that path. For example: Install flac with: sudo apt-get install flac Run the Kaldi recipe run.sh for LibriSpeech at least until Stage 13 (included), for simplicity you can use the modified run.sh. Copy exp/tri4b/trans. files into exp/tri4b/decode_tgsmall_train_clean_/ with the following command: Compute the fMLLR features by running the following script, the script can also be downloaded here: Compute alignments using: Apply CMVN and dump the fMLLR features to new .ark files, the script can also be downloaded here: Use the Python script to convert Kaldi generated .ark features to .npy for your own dataloader, an example Python script is provided:
PatchMatch
PatchMatch is an algorithm used to quickly find correspondences (or matches) between small square regions (or patches) of an image. It has various applications in image editing, such as reshuffling or removing objects from images or altering their aspect ratios without cropping or noticeably stretching them. PatchMatch was first presented in a 2011 paper by researchers at Princeton University. == Algorithm == The goal of the algorithm is to find the patch correspondence by defining a nearest-neighbor field (NNF) as a function f : R 2 → R 2 {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}} of offsets, which is over all possible matches of patch (location of patch centers) in image A, for some distance function of two patches D {\displaystyle D} . So, for a given patch coordinate a {\displaystyle a} in image A {\displaystyle A} and its corresponding nearest neighbor b {\displaystyle b} in image B {\displaystyle B} , f ( a ) {\displaystyle f(a)} is simply b − a {\displaystyle b-a} . However, if we search for every point in image B {\displaystyle B} , the work will be too hard to complete. So the following algorithm is done in a randomized approach in order to accelerate the calculation speed. The algorithm has three main components. Initially, the nearest-neighbor field is filled with either random offsets or some prior information. Next, an iterative update process is applied to the NNF, in which good patch offsets are propagated to adjacent pixels, followed by random search in the neighborhood of the best offset found so far. Independent of these three components, the algorithm also uses a coarse-to-fine approach by building an image pyramid to obtain the better result. === Initialization === When initializing with random offsets, we use independent uniform samples across the full range of image B {\displaystyle B} . This algorithm avoids using an initial guess from the previous level of the pyramid because in this way the algorithm can avoid being trapped in local minima. === Iteration === After initialization, the algorithm attempted to perform iterative process of improving the N N F {\displaystyle NNF} . The iterations examine the offsets in scan order (from left to right, top to bottom), and each undergoes propagation followed by random search. === Propagation === We attempt to improve f ( x , y ) {\displaystyle f(x,y)} using the known offsets of f ( x − 1 , y ) {\displaystyle f(x-1,y)} and f ( x , y − 1 ) {\displaystyle f(x,y-1)} , assuming that the patch offsets are likely to be the same. That is, the algorithm will take new value for f ( x , y ) {\displaystyle f(x,y)} to be arg min ( x , y ) D ( f ( x , y ) ) , D ( f ( x − 1 , y ) ) , D ( f ( x , y − 1 ) ) {\displaystyle \arg \min \limits _{(x,y)}{D(f(x,y)),D(f(x-1,y)),D(f(x,y-1))}} . So if f ( x , y ) {\displaystyle f(x,y)} has a correct mapping and is in a coherent region R {\displaystyle R} , then all of R {\displaystyle R} below and to the right of f ( x , y ) {\displaystyle f(x,y)} will be filled with the correct mapping. Alternatively, on even iterations, the algorithm search for different direction, fill the new value to be arg min ( x , y ) { D ( f ( x , y ) ) , D ( f ( x + 1 , y ) ) , D ( f ( x , y + 1 ) ) } {\displaystyle \arg \min \limits _{(x,y)}\{D(f(x,y)),D(f(x+1,y)),D(f(x,y+1))\}} . === Random search === Let v 0 = f ( x , y ) {\displaystyle v_{0}=f(x,y)} , we attempt to improve f ( x , y ) {\displaystyle f(x,y)} by testing a sequence of candidate offsets at an exponentially decreasing distance from v 0 {\displaystyle v_{0}} u i = v 0 + w α i R i {\displaystyle u_{i}=v_{0}+w\alpha ^{i}R_{i}} where R i {\displaystyle R_{i}} is a uniform random in [ − 1 , 1 ] × [ − 1 , 1 ] {\displaystyle [-1,1]\times [-1,1]} , w {\displaystyle w} is a large window search radius which will be set to maximum picture size, and α {\displaystyle \alpha } is a fixed ratio often assigned as 1/2. This part of the algorithm allows the f ( x , y ) {\displaystyle f(x,y)} to jump out of local minimum through random process. === Halting criterion === The often used halting criterion is set the iteration times to be about 4~5. Even with low iteration, the algorithm works well.