Offline Handwritten Signature Verification - Literature Review <s> Hidden Markov Models <s> This tutorial provides an overview of the basic theory of hidden Markov models (HMMs) as originated by L.E. Baum and T. Petrie (1966) and gives practical details on methods of implementation of the theory along with a description of selected applications of the theory to distinct problems in speech recognition. Results from a number of original sources are combined to provide a single source of acquiring the background required to pursue further this area of research. The author first reviews the theory of discrete Markov chains and shows how the concept of hidden states, where the observation is a probabilistic function of the state, can be used effectively. The theory is illustrated with two simple examples, namely coin-tossing, and the classic balls-in-urns system. Three fundamental problems of HMMs are noted and several practical techniques for solving these problems are given. The various types of HMMs that have been studied, including ergodic as well as left-right models, are described. > <s> BIB001 </s> Offline Handwritten Signature Verification - Literature Review <s> Hidden Markov Models <s> The objective of this work is to present an off-line signature verification system. It is basically divided into three parts. The first one describes a pre-processing process, a segmentation process and a feature extraction process, in which the main aim is to obtain the maximum performance quality of the process of verification of random falsifications, in the false acceptance and false rejection concept. The second presents a learning process based on HMM, where the aim is obtaining the best model, that is, one that is capable of representing each writer's signature, absorbing yet at the same time discriminating, at most the intra-personal variation and the interpersonal variation. A third and last part, presents a signature verification process that uses the models generated by the learning process without using any prior knowledge of test data, in other words, using an automatic derivation process of the decision thresholds. <s> BIB002 </s> Offline Handwritten Signature Verification - Literature Review <s> Hidden Markov Models <s> In practice, each writer provides only a limited number of signature samples to design a signature verification (SV) system. Hybrid generative-discriminative ensembles of classifiers (EoCs) are proposed in this paper to design an off-line SV system from few samples, where the classifier selection process is performed dynamically. To design the generative stage, multiple discrete left-to-right Hidden Markov Models (HMMs) are trained using a different number of states and codebook sizes, allowing the system to learn signatures at different levels of perception. To design the discriminative stage, HMM likelihoods are measured for each training signature, and assembled into feature vectors that are used to train a diversified pool of two-class classifiers through a specialized Random Subspace Method. During verification, a new dynamic selection strategy based on the K-nearest-oracles (KNORA) algorithm and on Output Profiles selects the most accurate EoCs to classify a given input signature. This SV system is suitable for incremental learning of new signature samples. Experiments performed with real-world signature data (composed of genuine samples, and random, simple and skilled forgeries) indicate that the proposed dynamic selection strategy can significantly reduce the overall error rates, with respect to other EoCs formed using well-known dynamic and static selection strategies. Moreover, the performance of the SV system proposed in this paper is significantly greater than or comparable to that of related systems found in the literature. <s> BIB003
Several authors have proposed using Hidden Markov Models for the task of signature verification BIB002 ,  , BIB003 . HMMs are generative models that attempt to learn the joint distribution P (X, Y ), where X are the features and Y are the labels, instead of just the conditional distribution P (Y |X). In particular, HMMs with a left-to-right topology have been mostly studied, as they match the dynamic characteristics of American and European handwriting (with hand movements from left to right). In the work from Justino BIB002 , Oliveira  and Batista BIB003 , the signatures are divided in a grid format. Each column of the grid is used as an observation of the HMM, and features are extracted from the different cells within each column -that is, a signature image I is converted to a sequence of feature vectors F = {f 1 , ..., f C }, where C is the number of columns in the grid (observations). To quantize (discretize) the sequence of observations, commonly the K-means algorithm is used on the feature vectors (of a subset of the dataset) to form a a codebook Q = {Q 1 , ..., Q k }. The sequence of the feature vectors for a signature image is then quantized using this codebook, forming a sequence of observations O = {O 1 , ..., O C }, where each observation is a symbol from the codebook O i ∈ Q. With the sequence of discrete observations, a HMM λ is trained, commonly using the Baum-Welch Forward-Backward algorithm BIB001 . This process is illustrated in Figure 13 . In the verification phase, a sequence of feature vectors is extracted from the signature and quantized using the codebook. The HMM is then used to calculate the likelihood of the observations given the model P (O|λ). After calculating the likelihood, a simple threshold can be used to discriminate between genuine signatures and forgeries BIB002 , or the likelihood itself can be used for more complex classification mechanisms, such as the work from Batista BIB003 that used HMMs trained on the genuine class, and HMMs trained on the forgery (random forgery) class, and used the likelihoods obtained by the different HMMs as a feature vector for another classifier to make the decision.