Digital sculpting, also known as sculpt modeling or 3D sculpting, is the use of software that offers tools to push, pull, smooth, grab, pinch or otherwise manipulate a digital object as if it were made of a real-life substance such as clay. == Sculpting technology == The geometry used in digital sculpting programs to represent the model can vary; each offers different benefits and limitations. The majority of digital sculpting tools on the market use mesh-based geometry, in which an object is represented by an interconnected surface mesh of polygons that can be pushed and pulled around. This is somewhat similar to the physical process of beating copper plates to sculpt a scene in relief. Other digital sculpting tools use voxel-based geometry, in which the volume of the object is the basic element. Material can be added and removed, much like sculpting in clay. Still other tools make use of more than one basic geometry representation. A benefit of mesh-based programs is that they support sculpting at multiple resolutions on a single model. Areas of the model that are finely detailed can have very small polygons while other areas can have larger polygons. In many mesh-based programs, the mesh can be edited at different levels of detail, and the changes at one level will propagate to higher and lower levels of model detail. A limitation of mesh-based sculpting is the fixed topology of the mesh; the specific arrangement of the polygons can limit the ways in which detail can be added or manipulated. A benefit of voxel-based sculpting is that voxels allow complete freedom over form. The topology of a model can be altered continually during the sculpting process as material is added and subtracted, which frees the sculptor from considering the layout of polygons on the model's surface. After sculpting, it may be necessary to retopologize the model to obtain a clean mesh for use in animation or real-time rendering. Voxels, however, are more limited in handling multiple levels of detail. Unlike mesh-based modeling, broad changes made to voxels at a low level of detail may completely destroy finer details. == Uses == Sculpting can often introduce details to meshes that would otherwise have been difficult or impossible to create using traditional 3D modeling techniques. This makes it preferable for achieving photorealistic and hyperrealistic results, though, many stylized results are achieved as well. Sculpting is primarily used in high poly organic modeling (the creation of 3D models which consist mainly of curves or irregular surfaces, as opposed to hard surface modeling). It is also used by auto manufacturers in their design of new cars. It can create the source meshes for low poly game models used in video games. In conjunction with other 3D modeling and texturing techniques and Displacement and Normal mapping, it can greatly enhance the appearance of game meshes often to the point of photorealism. Some sculpting programs like 3D-Coat, Zbrush, and Mudbox offer ways to integrate their workflows with traditional 3D modeling and rendering programs. Conversely, 3D modeling applications like 3ds Max, Maya and MODO are now incorporating sculpting capability as well, though these are usually less advanced than tools found in sculpting-specific applications. High poly sculpts are also extensively used in CG artwork for movies, industrial design, art, photorealistic illustrations, and for prototyping in 3D printing. == 3D print == Sculptors and digital artists use digital sculpting to create a model (or Digital Twin) to be materialized through CNC technologies including 3D printing. The final sculptures are often called Digital Sculpture or 3D printed art. While digital technologies have emerged in many art disciplines (painting, photography), this is less the case for digital sculpture due to the higher complexity and technology limitations to produce the final sculpture. == Sculpting Process == The best way to learn sculpture is by understanding primary, secondary and tertiary forms. First, break down the object you want to make down its basic shapes, such as a sphere or cube. Focus on making the large, overall shape of the object. After that, work on the bigger shapes on top of or inside the object. These can be protrusions or cut outs. Then, do a final detail pass, such as pores or lines to break up the shape. == Sculpting programs == There are a number of digital sculpting tools available. Some popular tools for creating are: Traditional 3D modeling suites are also beginning to include sculpting capability. 3D modeling programs which currently feature some form of sculpting include the following:
Manifold hypothesis
The manifold hypothesis posits that many high-dimensional data sets that occur in the real world actually lie along low-dimensional latent manifolds inside that high-dimensional space. As a consequence of the manifold hypothesis, many data sets that appear to initially require many variables to describe, can actually be described by a comparatively small number of variables, linked to the local coordinate system of the underlying manifold. It is suggested that this principle underpins the effectiveness of machine learning algorithms in describing high-dimensional data sets by considering a few common features. The manifold hypothesis is related to the effectiveness of nonlinear dimensionality reduction techniques in machine learning. Many techniques of dimensional reduction make the assumption that data lies along a low-dimensional submanifold, such as manifold sculpting, manifold alignment, and manifold regularization. The major implications of this hypothesis is that Machine learning models only have to fit relatively simple, low-dimensional, highly structured subspaces within their potential input space (latent manifolds). Within one of these manifolds, it's always possible to interpolate between two inputs, that is to say, morph one into another via a continuous path along which all points fall on the manifold. The ability to interpolate between samples is the key to generalization in deep learning. == The information geometry of statistical manifolds == An empirically-motivated approach to the manifold hypothesis focuses on its correspondence with an effective theory for manifold learning under the assumption that robust machine learning requires encoding the dataset of interest using methods for data compression. This perspective gradually emerged using the tools of information geometry thanks to the coordinated effort of scientists working on the efficient coding hypothesis, predictive coding and variational Bayesian methods. The argument for reasoning about the information geometry on the latent space of distributions rests upon the existence and uniqueness of the Fisher information metric. In this general setting, we are trying to find a stochastic embedding of a statistical manifold. From the perspective of dynamical systems, in the big data regime this manifold generally exhibits certain properties such as homeostasis: We can sample large amounts of data from the underlying generative process. Machine Learning experiments are reproducible, so the statistics of the generating process exhibit stationarity. In a sense made precise by theoretical neuroscientists working on the free energy principle, the statistical manifold in question possesses a Markov blanket.
SqueezeNet
SqueezeNet is a deep neural network for image classification released in 2016. SqueezeNet was developed by researchers at DeepScale, University of California, Berkeley, and Stanford University. In designing SqueezeNet, the authors' goal was to create a smaller neural network with fewer parameters while achieving competitive accuracy. Their best-performing model achieved the same accuracy as AlexNet on ImageNet classification, but has a size 510x less than it. == Version history == SqueezeNet was originally released on February 22, 2016. This original version of SqueezeNet was implemented on top of the Caffe deep learning software framework. Shortly thereafter, the open-source research community ported SqueezeNet to a number of other deep learning frameworks. On February 26, 2016, Eddie Bell released a port of SqueezeNet for the Chainer deep learning framework. On March 2, 2016, Guo Haria released a port of SqueezeNet for the Apache MXNet framework. On June 3, 2016, Tammy Yang released a port of SqueezeNet for the Keras framework. In 2017, companies including Baidu, Xilinx, Imagination Technologies, and Synopsys demonstrated SqueezeNet running on low-power processing platforms such as smartphones, FPGAs, and custom processors. As of 2018, SqueezeNet ships "natively" as part of the source code of a number of deep learning frameworks such as PyTorch, Apache MXNet, and Apple CoreML. In addition, third party developers have created implementations of SqueezeNet that are compatible with frameworks such as TensorFlow. Below is a summary of frameworks that support SqueezeNet. == Relationship to other networks == === AlexNet === SqueezeNet was originally described in SqueezeNet: AlexNet-level accuracy with 50x fewer parameters and <0.5MB model size. AlexNet is a deep neural network that has 240 MB of parameters, and SqueezeNet has just 5 MB of parameters. This small model size can more easily fit into computer memory and can more easily be transmitted over a computer network. However, it's important to note that SqueezeNet is not a "squeezed version of AlexNet." Rather, SqueezeNet is an entirely different DNN architecture than AlexNet. What SqueezeNet and AlexNet have in common is that both of them achieve approximately the same level of accuracy when evaluated on the ImageNet image classification validation dataset. === Model compression === Model compression (e.g. quantization and pruning of model parameters) can be applied to a deep neural network after it has been trained. In the SqueezeNet paper, the authors demonstrated that a model compression technique called Deep Compression can be applied to SqueezeNet to further reduce the size of the parameter file from 5 MB to 500 KB. Deep Compression has also been applied to other DNNs, such as AlexNet and VGG. == Variants == Some of the members of the original SqueezeNet team have continued to develop resource-efficient deep neural networks for a variety of applications. A few of these works are noted in the following table. As with the original SqueezeNet model, the open-source research community has ported and adapted these newer "squeeze"-family models for compatibility with multiple deep learning frameworks. In addition, the open-source research community has extended SqueezeNet to other applications, including semantic segmentation of images and style transfer.
Expectation–maximization algorithm
In statistics, an expectation–maximization (EM) algorithm is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the E step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step. It can be used, for example, to estimate a mixture of gaussians, or to solve the multiple linear regression problem. == History == The EM algorithm was explained and given its name in a classic 1977 paper by Arthur Dempster, Nan Laird, and Donald Rubin. They pointed out that the method had been "proposed many times in special circumstances" by earlier authors. One of the earliest is the gene-counting method for estimating allele frequencies by Cedric Smith. Another was proposed by H.O. Hartley in 1958, and Hartley and Hocking in 1977, from which many of the ideas in the Dempster–Laird–Rubin paper originated. Another one by S.K Ng, Thriyambakam Krishnan and G.J McLachlan in 1977. Hartley's ideas can be broadened to any grouped discrete distribution. A very detailed treatment of the EM method for exponential families was published by Rolf Sundberg in his thesis and several papers, following his collaboration with Per Martin-Löf and Anders Martin-Löf. The Dempster–Laird–Rubin paper in 1977 generalized the method and sketched a convergence analysis for a wider class of problems. The Dempster–Laird–Rubin paper established the EM method as an important tool of statistical analysis. See also Meng and van Dyk (1997). The convergence analysis of the Dempster–Laird–Rubin algorithm was flawed and a correct convergence analysis was published by C. F. Jeff Wu in 1983. Wu's proof established the EM method's convergence also outside of the exponential family, as claimed by Dempster–Laird–Rubin. == Introduction == The EM algorithm is used to find (local) maximum likelihood parameters of a statistical model in cases where the equations cannot be solved directly. Typically these models involve latent variables in addition to unknown parameters and known data observations. That is, either missing values exist among the data, or the model can be formulated more simply by assuming the existence of further unobserved data points. For example, a mixture model can be described more simply by assuming that each observed data point has a corresponding unobserved data point, or latent variable, specifying the mixture component to which each data point belongs. Finding a maximum likelihood solution typically requires taking the derivatives of the likelihood function with respect to all the unknown values, the parameters and the latent variables, and simultaneously solving the resulting equations. In statistical models with latent variables, this is usually impossible. Instead, the result is typically a set of interlocking equations in which the solution to the parameters requires the values of the latent variables and vice versa, but substituting one set of equations into the other produces an unsolvable equation. The EM algorithm proceeds from the observation that there is a way to solve these two sets of equations numerically. One can simply pick arbitrary values for one of the two sets of unknowns, use them to estimate the second set, then use these new values to find a better estimate of the first set, and then keep alternating between the two until the resulting values both converge to fixed points. It's not obvious that this will work, but it can be proven in this context. Additionally, it can be proven that the derivative of the likelihood is (arbitrarily close to) zero at that point, which in turn means that the point is either a local maximum or a saddle point. In general, multiple maxima may occur, with no guarantee that the global maximum will be found. Some likelihoods also have singularities in them, i.e., nonsensical maxima. For example, one of the solutions that may be found by EM in a mixture model involves setting one of the components to have zero variance and the mean parameter for the same component to be equal to one of the data points. == Description == === The symbols === Given the statistical model which generates a set X {\displaystyle \mathbf {X} } of observed data, a set of unobserved latent data or missing values Z {\displaystyle \mathbf {Z} } , and a vector of unknown parameters θ {\displaystyle {\boldsymbol {\theta }}} , along with a likelihood function L ( θ ; X , Z ) = p ( X , Z ∣ θ ) {\displaystyle L({\boldsymbol {\theta }};\mathbf {X} ,\mathbf {Z} )=p(\mathbf {X} ,\mathbf {Z} \mid {\boldsymbol {\theta }})} , the maximum likelihood estimate (MLE) of the unknown parameters is determined by maximizing the marginal likelihood of the observed data L ( θ ; X ) = p ( X ∣ θ ) = ∫ p ( X , Z ∣ θ ) d Z = ∫ p ( X ∣ Z , θ ) p ( Z ∣ θ ) d Z {\displaystyle {\begin{aligned}L({\boldsymbol {\theta }};\mathbf {X} )=p(\mathbf {X} \mid {\boldsymbol {\theta }})&=\int p(\mathbf {X} ,\mathbf {Z} \mid {\boldsymbol {\theta }})\,d\mathbf {Z} \\&=\int p(\mathbf {X} \mid \mathbf {Z} ,{\boldsymbol {\theta }})p(\mathbf {Z} \mid {\boldsymbol {\theta }})\,d\mathbf {Z} \end{aligned}}} However, this quantity is often intractable since Z {\displaystyle \mathbf {Z} } is unobserved and the distribution of Z {\displaystyle \mathbf {Z} } is unknown before attaining θ {\displaystyle {\boldsymbol {\theta }}} . === The EM algorithm === The EM algorithm seeks to find the maximum likelihood estimate of the marginal likelihood by iteratively applying these two steps: More succinctly, we can write it as one equation: θ ( t + 1 ) = arg max θ E Z ∼ p ( ⋅ | X , θ ( t ) ) [ log p ( X , Z | θ ) ] {\displaystyle {\boldsymbol {\theta }}^{(t+1)}=\mathop {\arg \max } _{\boldsymbol {\theta }}\operatorname {E} _{\mathbf {Z} \sim p(\cdot |\mathbf {X} ,{\boldsymbol {\theta }}^{(t)})}\left[\log p(\mathbf {X} ,\mathbf {Z} |{\boldsymbol {\theta }})\right]\,} === Interpretation of the variables === The typical models to which EM is applied use Z {\displaystyle \mathbf {Z} } as a latent variable indicating membership in one of a set of groups: The observed data points X {\displaystyle \mathbf {X} } may be discrete (taking values in a finite or countably infinite set) or continuous (taking values in an uncountably infinite set). Associated with each data point may be a vector of observations. The missing values (aka latent variables) Z {\displaystyle \mathbf {Z} } are discrete, drawn from a fixed number of values, and with one latent variable per observed unit. The parameters are continuous, and are of two kinds: Parameters that are associated with all data points, and those associated with a specific value of a latent variable (i.e., associated with all data points whose corresponding latent variable has that value). However, it is possible to apply EM to other sorts of models. The motivation is as follows. If the value of the parameters θ {\displaystyle {\boldsymbol {\theta }}} is known, usually the value of the latent variables Z {\displaystyle \mathbf {Z} } can be found by maximizing the log-likelihood over all possible values of Z {\displaystyle \mathbf {Z} } , either simply by iterating over Z {\displaystyle \mathbf {Z} } or through an algorithm such as the Viterbi algorithm for hidden Markov models. Conversely, if we know the value of the latent variables Z {\displaystyle \mathbf {Z} } , we can find an estimate of the parameters θ {\displaystyle {\boldsymbol {\theta }}} fairly easily, typically by simply grouping the observed data points according to the value of the associated latent variable and averaging the values, or some function of the values, of the points in each group. This suggests an iterative algorithm, in the case where both θ {\displaystyle {\boldsymbol {\theta }}} and Z {\displaystyle \mathbf {Z} } are unknown: First, initialize the parameters θ {\displaystyle {\boldsymbol {\theta }}} to some random values. Compute the probability of each possible value of Z {\displaystyle \mathbf {Z} } , given θ {\displaystyle {\boldsymbol {\theta }}} . Then, use the just-computed values of Z {\displaystyle \mathbf {Z} } to compute a better estimate for the parameters θ {\displaystyle {\boldsymbol {\theta }}} . Iterate steps 2 and 3 until convergence. The algorithm as just described monotonically approaches a local minimum of the cost function. == Properties == Although an EM iteration does increase the observed data (i.e., marginal) likelihood function, no guarantee exists that the sequence converges to a maximum likelihood estimator. For multimodal distributions, this means that an EM algorithm may co
Tensor product network
A tensor product network, in artificial neural networks, is a network that exploits the properties of tensors to model associative concepts such as variable assignment. Orthonormal vectors are chosen to model the ideas (such as variable names and target assignments), and the tensor product of these vectors construct a network whose mathematical properties allow the user to easily extract the association from it.
Eigenface
An eigenface ( EYE-gən-) is the name given to a set of eigenvectors when used in the computer vision problem of human face recognition. The approach of using eigenfaces for recognition was developed by Sirovich and Kirby and used by Matthew Turk and Alex Pentland in face classification. The eigenvectors are derived from the covariance matrix of the probability distribution over the high-dimensional vector space of face images. The eigenfaces themselves form a basis set of all images used to construct the covariance matrix. This produces dimension reduction by allowing the smaller set of basis images to represent the original training images. Classification can be achieved by comparing how faces are represented by the basis set. == History == The eigenface approach began with a search for a low-dimensional representation of face images. Sirovich and Kirby showed that principal component analysis could be used on a collection of face images to form a set of basis features. These basis images, known as eigenpictures, could be linearly combined to reconstruct images in the original training set. If the training set consists of M images, principal component analysis could form a basis set of N images, where N < M. The reconstruction error is reduced by increasing the number of eigenpictures; however, the number needed is always chosen less than M. For example, if you need to generate a number of N eigenfaces for a training set of M face images, you can say that each face image can be made up of "proportions" of all the K "features" or eigenfaces: Face image1 = (23% of E1) + (2% of E2) + (51% of E3) + ... + (1% En). In 1991 M. Turk and A. Pentland expanded these results and presented the eigenface method of face recognition. In addition to designing a system for automated face recognition using eigenfaces, they showed a way of calculating the eigenvectors of a covariance matrix such that computers of the time could perform eigen-decomposition on a large number of face images. Face images usually occupy a high-dimensional space and conventional principal component analysis was intractable on such data sets. Turk and Pentland's paper demonstrated ways to extract the eigenvectors based on matrices sized by the number of images rather than the number of pixels. Once established, the eigenface method was expanded to include methods of preprocessing to improve accuracy. Multiple manifold approaches were also used to build sets of eigenfaces for different subjects and different features, such as the eyes. == Generation == A set of eigenfaces can be generated by performing a mathematical process called principal component analysis (PCA) on a large set of images depicting different human faces. Informally, eigenfaces can be considered a set of "standardized face ingredients", derived from statistical analysis of many pictures of faces. Any human face can be considered to be a combination of these standard faces. For example, one's face might be composed of the average face plus 10% from eigenface 1, 55% from eigenface 2, and even −3% from eigenface 3. Remarkably, it does not take many eigenfaces combined together to achieve a fair approximation of most faces. Also, because a person's face is not recorded by a digital photograph, but instead as just a list of values (one value for each eigenface in the database used), much less space is taken for each person's face. The eigenfaces that are created will appear as light and dark areas that are arranged in a specific pattern. This pattern is how different features of a face are singled out to be evaluated and scored. There will be a pattern to evaluate symmetry, whether there is any style of facial hair, where the hairline is, or an evaluation of the size of the nose or mouth. Other eigenfaces have patterns that are less simple to identify, and the image of the eigenface may look very little like a face. The technique used in creating eigenfaces and using them for recognition is also used outside of face recognition: handwriting recognition, lip reading, voice recognition, sign language/hand gestures interpretation and medical imaging analysis. Therefore, some do not use the term eigenface, but prefer to use 'eigenimage'. === Practical implementation === To create a set of eigenfaces, one must: Prepare a training set of face images. The pictures constituting the training set should have been taken under the same lighting conditions, and must be normalized to have the eyes and mouths aligned across all images. They must also be all resampled to a common pixel resolution (r × c). Each image is treated as one vector, simply by concatenating the rows of pixels in the original image, resulting in a single column with r × c elements. For this implementation, it is assumed that all images of the training set are stored in a single matrix T, where each column of the matrix is an image. Subtract the mean. The average image a has to be calculated and then subtracted from each original image in T. Calculate the eigenvectors and eigenvalues of the covariance matrix S. Each eigenvector has the same dimensionality (number of components) as the original images, and thus can itself be seen as an image. The eigenvectors of this covariance matrix are therefore called eigenfaces. They are the directions in which the images differ from the mean image. Usually this will be a computationally expensive step (if at all possible), but the practical applicability of eigenfaces stems from the possibility to compute the eigenvectors of S efficiently, without ever computing S explicitly, as detailed below. Choose the principal components. Sort the eigenvalues in descending order and arrange eigenvectors accordingly. The number of principal components k is determined arbitrarily by setting a threshold ε on the total variance. Total variance v = ( λ 1 + λ 2 + . . . + λ n ) {\displaystyle v=(\lambda _{1}+\lambda _{2}+...+\lambda _{n})} , n = number of components, and λ {\displaystyle \lambda } represents component eigenvalue. k is the smallest number that satisfies ( λ 1 + λ 2 + . . . + λ k ) v > ϵ {\displaystyle {\frac {(\lambda _{1}+\lambda _{2}+...+\lambda _{k})}{v}}>\epsilon } These eigenfaces can now be used to represent both existing and new faces: we can project a new (mean-subtracted) image on the eigenfaces and thereby record how that new face differs from the mean face. The eigenvalues associated with each eigenface represent how much the images in the training set vary from the mean image in that direction. Information is lost by projecting the image on a subset of the eigenvectors, but losses are minimized by keeping those eigenfaces with the largest eigenvalues. For instance, working with a 100 × 100 image will produce 10,000 eigenvectors. In practical applications, most faces can typically be identified using a projection on between 100 and 150 eigenfaces, so that most of the 10,000 eigenvectors can be discarded. === Matlab example code === Here is an example of calculating eigenfaces with Extended Yale Face Database B. To evade computational and storage bottleneck, the face images are sampled down by a factor 4×4=16. Note that although the covariance matrix S generates many eigenfaces, only a fraction of those are needed to represent the majority of the faces. For example, to represent 95% of the total variation of all face images, only the first 43 eigenfaces are needed. To calculate this result, implement the following code: === Computing the eigenvectors === Performing PCA directly on the covariance matrix of the images is often computationally infeasible. If small images are used, say 100 × 100 pixels, each image is a point in a 10,000-dimensional space and the covariance matrix S is a matrix of 10,000 × 10,000 = 108 elements. However the rank of the covariance matrix is limited by the number of training examples: if there are N training examples, there will be at most N − 1 eigenvectors with non-zero eigenvalues. If the number of training examples is smaller than the dimensionality of the images, the principal components can be computed more easily as follows. Let T be the matrix of preprocessed training examples, where each column contains one mean-subtracted image. The covariance matrix can then be computed as S = TTT and the eigenvector decomposition of S is given by S v i = T T T v i = λ i v i {\displaystyle \mathbf {Sv} _{i}=\mathbf {T} \mathbf {T} ^{T}\mathbf {v} _{i}=\lambda _{i}\mathbf {v} _{i}} However TTT is a large matrix, and if instead we take the eigenvalue decomposition of T T T u i = λ i u i {\displaystyle \mathbf {T} ^{T}\mathbf {T} \mathbf {u} _{i}=\lambda _{i}\mathbf {u} _{i}} then we notice that by pre-multiplying both sides of the equation with T, we obtain T T T T u i = λ i T u i {\displaystyle \mathbf {T} \mathbf {T} ^{T}\mathbf {T} \mathbf {u} _{i}=\lambda _{i}\mathbf {T} \mathbf {u} _{i}} Meaning that, if ui is an eigenvector of TTT, then vi = Tui is an eigenvector of S. If we have
Mating pool
Mating pool is a concept used in evolutionary algorithms and means a population of parents for the next population. The mating pool is formed by candidate solutions that the selection operators deem to have the highest fitness in the current population. Solutions that are included in the mating pool are referred to as parents. Individual solutions can be repeatedly included in the mating pool, with individuals of higher fitness values having a higher chance of being included multiple times. Crossover operators are then applied to the parents, resulting in recombination of genes recognized as superior. Lastly, random changes in the genes are introduced through mutation operators, increasing the genetic variation in the gene pool. Those two operators improve the chance of creating new, superior solutions. A new generation of solutions is thereby created, the children, who will constitute the next population. Depending on the selection method, the total number of parents in the mating pool can be different to the size of the initial population, resulting in a new population that’s smaller. To continue the algorithm with an equally sized population, random individuals from the old populations can be chosen and added to the new population. At this point, the fitness value of the new solutions is evaluated. If the termination conditions are fulfilled, processes come to an end. Otherwise, they are repeated. The repetition of the steps result in candidate solutions that evolve towards the most optimal solution over time. The genes will become increasingly uniform towards the most optimal gene, a process called convergence. If 95% of the population share the same version of a gene, the gene has converged. When all the individual fitness values have reached the value of the best individual, i.e. all the genes have converged, population convergence is achieved. == Mating pool creation == Several methods can be applied to create a mating pool. All of these processes involve the selective breeding of a particular number of individuals within a population. There are multiple criteria that can be employed to determine which individuals make it into the mating pool and which are left behind. The selection methods can be split into three general types: fitness proportionate selection, ordinal based selection and threshold based selection. === Fitness proportionate selection === In the case of fitness proportionate selection, random individuals are selected to enter the pool. However, the ones with a higher level of fitness are more likely to be picked and therefore have a greater chance of passing on their features to the next generation. One of the techniques used in this type of parental selection is the roulette wheel selection. This approach divides a hypothetical circular wheel into different slots, the size of which is equal to the fitness values of each potential candidate. Afterwards, the wheel is rotated and a fixed point determines which individual gets picked. The greater the fitness value of an individual, the higher the probability of being chosen as a parent by the random spin of the wheel. Alternatively, stochastic universal sampling can be implemented. This selection method is also based on the rotation of a spinning wheel. However, in this case there is more than one fixed point and as a result all of the mating pool members will be selected simultaneously. === Ordinal based selection === The ordinal based selection methods include the tournament and ranking selection. Tournament selection involves the random selection of individuals of a population and the subsequent comparison of their fitness levels. The winners of these “tournaments” are the ones with the highest values and will be put into the mating pool as parents. In ranking selection all the individuals are sorted based on their fitness values. Then, the selection of the parents is made according to the rank of the candidates. Every individual has a chance of being chosen, but higher ranked ones are favored === Threshold based selection === The last type of selection method is referred to as the threshold based method. This includes the truncation selection method, which sorts individuals based on their phenotypic values on a specific trait and later selects the proportion of them that are within a certain threshold as parents.