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doc/tutorials/ml/introduction_to_svm/introduction_to_svm.rst
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.. _introductiontosvms:
Introduction to Support Vector Machines
***************************************
Goal
====
In this tutorial you will learn how to:
.. container:: enumeratevisibleitemswithsquare
+ Use the OpenCV functions :svms:`CvSVM::train <cvsvm-train>` to build a classifier based on SVMs and :svms:`CvSVM::predict <cvsvm-predict>` to test its performance.
What is a SVM?
==============
A Support Vector Machine (SVM) is a discriminative classifier formally defined by a separating hyperplane. In other words, given labeled training data (*supervised learning*), the algorithm outputs an optimal hyperplane which categorizes new examples.
In which sense is the hyperplane obtained optimal? Let's consider the following
simple problem:
For a linearly separable set of 2D-points which belong to one of two classes, find a separating straight line.
.. image:: images/separating-lines.png
:alt: A seperation example
:align: center
.. note:: In this example we deal with lines and points in the Cartesian plane instead of hyperplanes and vectors in a high dimensional space. This is a simplification of the problem.It is important to understand that this is done only because our intuition is better built from examples that are easy to imagine. However, the same concepts apply to tasks where the examples to classify lie in a space whose dimension is higher than two.
In the above picture you can see that there exists multiple lines that offer a solution to the problem. Is any of them better than the others? We can intuitively define a criterion to estimate the worth of the lines:
A line is bad if it passes too close to the points because it will be noise sensitive and it will not generalize correctly. Therefore, our goal should be to find the line passing as far as possible from all points.
Then, the operation of the SVM algorithm is based on finding the hyperplane that gives the largest minimum distance to the training examples. Twice, this distance receives the important name of **margin** within SVM's theory. Therefore, the optimal separating hyperplane *maximizes* the margin of the training data.
.. image:: images/optimal-hyperplane.png
:alt: The Optimal hyperplane
:align: center
How is the optimal hyperplane computed?
=======================================
Let's introduce the notation used to define formally a hyperplane:
.. math::
f(x) = \beta_{0} + \beta^{T} x,
where :math:`\beta` is known as the *weight vector* and :math:`\beta_{0}` as the *bias*.
.. seealso:: A more in depth description of this and hyperplanes you can find in the section 4.5 (*Seperating Hyperplanes*) of the book: *Elements of Statistical Learning* by T. Hastie, R. Tibshirani and J. H. Friedman.
The optimal hyperplane can be represented in an infinite number of different ways by scaling of :math:`\beta` and :math:`\beta_{0}`. As a matter of convention, among all the possible representations of the hyperplane, the one chosen is
.. math::
|\beta_{0} + \beta^{T} x| = 1
where :math:`x` symbolizes the training examples closest to the hyperplane. In general, the training examples that are closest to the hyperplane are called **support vectors**. This representation is known as the **canonical hyperplane**.
Now, we use the result of geometry that gives the distance between a point :math:`x` and a hyperplane :math:`(\beta, \beta_{0})`:
.. math::
\mathrm{distance} = \frac{|\beta_{0} + \beta^{T} x|}{||\beta||}.
In particular, for the canonical hyperplane, the numerator is equal to one and the distance to the support vectors is
.. math::
\mathrm{distance}_{\text{ support vectors}} = \frac{|\beta_{0} + \beta^{T} x|}{||\beta||} = \frac{1}{||\beta||}.
Recall that the margin introduced in the previous section, here denoted as :math:`M`, is twice the distance to the closest examples:
.. math::
M = \frac{2}{||\beta||}
Finally, the problem of maximizing :math:`M` is equivalent to the problem of minimizing a function :math:`L(\beta)` subject to some constraints. The constraints model the requirement for the hyperplane to classify correctly all the training examples :math:`x_{i}`. Formally,
.. math::
\min_{\beta, \beta_{0}} L(\beta) = \frac{1}{2}||\beta||^{2} \text{ subject to } y_{i}(\beta^{T} x_{i} + \beta_{0}) \geq 1 \text{ } \forall i,
where :math:`y_{i}` represents each of the labels of the training examples.
This is a problem of Lagrangian optimization that can be solved using Lagrange multipliers to obtain the weight vector :math:`\beta` and the bias :math:`\beta_{0}` of the optimal hyperplane.
Source Code
===========
.. literalinclude:: ../../../../samples/cpp/tutorial_code/ml/introduction_to_svm/introduction_to_svm.cpp
:language: cpp
:linenos:
:tab-width: 4
Explanation
===========
1. **Set up the training data**
The training data of this exercise is formed by a set of labeled 2D-points that belong to one of two different classes; one of the classes consists of one point and the other of three points.
.. code-block:: cpp
float labels[4] = {1.0, -1.0, -1.0, -1.0};
float trainingData[4][2] = {{501, 10}, {255, 10}, {501, 255}, {10, 501}};
The function :svms:`CvSVM::train <cvsvm-train>` that will be used afterwards requires the training data to be stored as :basicstructures:`Mat <mat>` objects of floats. Therefore, we create these objects from the arrays defined above:
.. code-block:: cpp
Mat trainingDataMat(4, 2, CV_32FC1, trainingData);
Mat labelsMat (4, 1, CV_32FC1, labels);
2. **Set up SVM's parameters**
In this tutorial we have introduced the theory of SVMs in the most simple case, when the training examples are spread into two classes that are linearly separable. However, SVMs can be used in a wide variety of problems (e.g. problems with non-linearly separable data, a SVM using a kernel function to raise the dimensionality of the examples, etc). As a consequence of this, we have to define some parameters before training the SVM. These parameters are stored in an object of the class :svms:`CvSVMParams <cvsvmparams>` .
.. code-block:: cpp
ml::SVM::Params params;
params.svmType = ml::SVM::C_SVC;
params.kernelType = ml::SVM::LINEAR;
params.termCrit = TermCriteria(TermCriteria::MAX_ITER, 100, 1e-6);
* *Type of SVM*. We choose here the type **ml::SVM::C_SVC** that can be used for n-class classification (n :math:`\geq` 2). This parameter is defined in the attribute *ml::SVM::Params.svmType*.
.. note:: The important feature of the type of SVM **CvSVM::C_SVC** deals with imperfect separation of classes (i.e. when the training data is non-linearly separable). This feature is not important here since the data is linearly separable and we chose this SVM type only for being the most commonly used.
* *Type of SVM kernel*. We have not talked about kernel functions since they are not interesting for the training data we are dealing with. Nevertheless, let's explain briefly now the main idea behind a kernel function. It is a mapping done to the training data to improve its resemblance to a linearly separable set of data. This mapping consists of increasing the dimensionality of the data and is done efficiently using a kernel function. We choose here the type **ml::SVM::LINEAR** which means that no mapping is done. This parameter is defined in the attribute *ml::SVMParams.kernel_type*.
* *Termination criteria of the algorithm*. The SVM training procedure is implemented solving a constrained quadratic optimization problem in an **iterative** fashion. Here we specify a maximum number of iterations and a tolerance error so we allow the algorithm to finish in less number of steps even if the optimal hyperplane has not been computed yet. This parameter is defined in a structure :oldbasicstructures:`cvTermCriteria <cvtermcriteria>`.
3. **Train the SVM**
We call the method `CvSVM::train <http://docs.opencv.org/modules/ml/doc/support_vector_machines.html#cvsvm-train>`_ to build the SVM model.
.. code-block:: cpp
CvSVM SVM;
SVM.train(trainingDataMat, labelsMat, Mat(), Mat(), params);
4. **Regions classified by the SVM**
The method :svms:`CvSVM::predict <cvsvm-predict>` is used to classify an input sample using a trained SVM. In this example we have used this method in order to color the space depending on the prediction done by the SVM. In other words, an image is traversed interpreting its pixels as points of the Cartesian plane. Each of the points is colored depending on the class predicted by the SVM; in green if it is the class with label 1 and in blue if it is the class with label -1.
.. code-block:: cpp
Vec3b green(0,255,0), blue (255,0,0);
for (int i = 0; i < image.rows; ++i)
for (int j = 0; j < image.cols; ++j)
{
Mat sampleMat = (Mat_<float>(1,2) << i,j);
float response = SVM.predict(sampleMat);
if (response == 1)
image.at<Vec3b>(j, i) = green;
else
if (response == -1)
image.at<Vec3b>(j, i) = blue;
}
5. **Support vectors**
We use here a couple of methods to obtain information about the support vectors. The method :svms:`CvSVM::get_support_vector_count <cvsvm-get-support-vector>` outputs the total number of support vectors used in the problem and with the method :svms:`CvSVM::get_support_vector <cvsvm-get-support-vector>` we obtain each of the support vectors using an index. We have used this methods here to find the training examples that are support vectors and highlight them.
.. code-block:: cpp
int c = SVM.get_support_vector_count();
for (int i = 0; i < c; ++i)
{
const float* v = SVM.get_support_vector(i); // get and then highlight with grayscale
circle( image, Point( (int) v[0], (int) v[1]), 6, Scalar(128, 128, 128), thickness, lineType);
}
Results
=======
.. container:: enumeratevisibleitemswithsquare
* The code opens an image and shows the training examples of both classes. The points of one class are represented with white circles and black ones are used for the other class.
* The SVM is trained and used to classify all the pixels of the image. This results in a division of the image in a blue region and a green region. The boundary between both regions is the optimal separating hyperplane.
* Finally the support vectors are shown using gray rings around the training examples.
.. image:: images/svm_intro_result.png
:alt: The seperated planes
:align: center
doc/tutorials/ml/non_linear_svms/non_linear_svms.rst
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.. _nonLinearSvmS:
Support Vector Machines for Non-Linearly Separable Data
*******************************************************
Goal
====
In this tutorial you will learn how to:
.. container:: enumeratevisibleitemswithsquare
+ Define the optimization problem for SVMs when it is not possible to separate linearly the training data.
+ How to configure the parameters in :svms:`CvSVMParams <cvsvmparams>` to adapt your SVM for this class of problems.
Motivation
==========
Why is it interesting to extend the SVM optimation problem in order to handle non-linearly separable training data? Most of the applications in which SVMs are used in computer vision require a more powerful tool than a simple linear classifier. This stems from the fact that in these tasks **the training data can be rarely separated using an hyperplane**.
Consider one of these tasks, for example, face detection. The training data in this case is composed by a set of images that are faces and another set of images that are non-faces (*every other thing in the world except from faces*). This training data is too complex so as to find a representation of each sample (*feature vector*) that could make the whole set of faces linearly separable from the whole set of non-faces.
Extension of the Optimization Problem
=====================================
Remember that using SVMs we obtain a separating hyperplane. Therefore, since the training data is now non-linearly separable, we must admit that the hyperplane found will misclassify some of the samples. This *misclassification* is a new variable in the optimization that must be taken into account. The new model has to include both the old requirement of finding the hyperplane that gives the biggest margin and the new one of generalizing the training data correctly by not allowing too many classification errors.
We start here from the formulation of the optimization problem of finding the hyperplane which maximizes the **margin** (this is explained in the :ref:`previous tutorial <introductiontosvms>`):
.. math::
\min_{\beta, \beta_{0}} L(\beta) = \frac{1}{2}||\beta||^{2} \text{ subject to } y_{i}(\beta^{T} x_{i} + \beta_{0}) \geq 1 \text{ } \forall i
There are multiple ways in which this model can be modified so it takes into account the misclassification errors. For example, one could think of minimizing the same quantity plus a constant times the number of misclassification errors in the training data, i.e.:
.. math::
\min ||\beta||^{2} + C \text{(\# misclassication errors)}
However, this one is not a very good solution since, among some other reasons, we do not distinguish between samples that are misclassified with a small distance to their appropriate decision region or samples that are not. Therefore, a better solution will take into account the *distance of the misclassified samples to their correct decision regions*, i.e.:
.. math::
\min ||\beta||^{2} + C \text{(distance of misclassified samples to their correct regions)}
For each sample of the training data a new parameter :math:`\xi_{i}` is defined. Each one of these parameters contains the distance from its corresponding training sample to their correct decision region. The following picture shows non-linearly separable training data from two classes, a separating hyperplane and the distances to their correct regions of the samples that are misclassified.
.. image:: images/sample-errors-dist.png
:alt: Samples misclassified and their distances to their correct regions
:align: center
.. note:: Only the distances of the samples that are misclassified are shown in the picture. The distances of the rest of the samples are zero since they lay already in their correct decision region.
The red and blue lines that appear on the picture are the margins to each one of the decision regions. It is very **important** to realize that each of the :math:`\xi_{i}` goes from a misclassified training sample to the margin of its appropriate region.
Finally, the new formulation for the optimization problem is:
.. math::
\min_{\beta, \beta_{0}} L(\beta) = ||\beta||^{2} + C \sum_{i} {\xi_{i}} \text{ subject to } y_{i}(\beta^{T} x_{i} + \beta_{0}) \geq 1 - \xi_{i} \text{ and } \xi_{i} \geq 0 \text{ } \forall i
How should the parameter C be chosen? It is obvious that the answer to this question depends on how the training data is distributed. Although there is no general answer, it is useful to take into account these rules:
.. container:: enumeratevisibleitemswithsquare
* Large values of C give solutions with *less misclassification errors* but a *smaller margin*. Consider that in this case it is expensive to make misclassification errors. Since the aim of the optimization is to minimize the argument, few misclassifications errors are allowed.
* Small values of C give solutions with *bigger margin* and *more classification errors*. In this case the minimization does not consider that much the term of the sum so it focuses more on finding a hyperplane with big margin.
Source Code
===========
You may also find the source code and these video file in the :file:`samples/cpp/tutorial_code/gpu/non_linear_svms/non_linear_svms` folder of the OpenCV source library or :download:`download it from here <../../../../samples/cpp/tutorial_code/ml/non_linear_svms/non_linear_svms.cpp>`.
.. literalinclude:: ../../../../samples/cpp/tutorial_code/ml/non_linear_svms/non_linear_svms.cpp
:language: cpp
:linenos:
:tab-width: 4
:lines: 1-12, 23-24, 27-
Explanation
===========
1. **Set up the training data**
The training data of this exercise is formed by a set of labeled 2D-points that belong to one of two different classes. To make the exercise more appealing, the training data is generated randomly using a uniform probability density functions (PDFs).
We have divided the generation of the training data into two main parts.
In the first part we generate data for both classes that is linearly separable.
.. code-block:: cpp
// Generate random points for the class 1
Mat trainClass = trainData.rowRange(0, nLinearSamples);
// The x coordinate of the points is in [0, 0.4)
Mat c = trainClass.colRange(0, 1);
rng.fill(c, RNG::UNIFORM, Scalar(1), Scalar(0.4 * WIDTH));
// The y coordinate of the points is in [0, 1)
c = trainClass.colRange(1,2);
rng.fill(c, RNG::UNIFORM, Scalar(1), Scalar(HEIGHT));
// Generate random points for the class 2
trainClass = trainData.rowRange(2*NTRAINING_SAMPLES-nLinearSamples, 2*NTRAINING_SAMPLES);
// The x coordinate of the points is in [0.6, 1]
c = trainClass.colRange(0 , 1);
rng.fill(c, RNG::UNIFORM, Scalar(0.6*WIDTH), Scalar(WIDTH));
// The y coordinate of the points is in [0, 1)
c = trainClass.colRange(1,2);
rng.fill(c, RNG::UNIFORM, Scalar(1), Scalar(HEIGHT));
In the second part we create data for both classes that is non-linearly separable, data that overlaps.
.. code-block:: cpp
// Generate random points for the classes 1 and 2
trainClass = trainData.rowRange( nLinearSamples, 2*NTRAINING_SAMPLES-nLinearSamples);
// The x coordinate of the points is in [0.4, 0.6)
c = trainClass.colRange(0,1);
rng.fill(c, RNG::UNIFORM, Scalar(0.4*WIDTH), Scalar(0.6*WIDTH));
// The y coordinate of the points is in [0, 1)
c = trainClass.colRange(1,2);
rng.fill(c, RNG::UNIFORM, Scalar(1), Scalar(HEIGHT));
2. **Set up SVM's parameters**
.. seealso::
In the previous tutorial :ref:`introductiontosvms` there is an explanation of the atributes of the class :svms:`CvSVMParams <cvsvmparams>` that we configure here before training the SVM.
.. code-block:: cpp
CvSVMParams params;
params.svm_type = SVM::C_SVC;
params.C = 0.1;
params.kernel_type = SVM::LINEAR;
params.term_crit = TermCriteria(TermCriteria::ITER, (int)1e7, 1e-6);
There are just two differences between the configuration we do here and the one that was done in the :ref:`previous tutorial <introductiontosvms>` that we use as reference.
* *CvSVM::C_SVC*. We chose here a small value of this parameter in order not to punish too much the misclassification errors in the optimization. The idea of doing this stems from the will of obtaining a solution close to the one intuitively expected. However, we recommend to get a better insight of the problem by making adjustments to this parameter.
.. note:: Here there are just very few points in the overlapping region between classes, giving a smaller value to **FRAC_LINEAR_SEP** the density of points can be incremented and the impact of the parameter **CvSVM::C_SVC** explored deeply.
* *Termination Criteria of the algorithm*. The maximum number of iterations has to be increased considerably in order to solve correctly a problem with non-linearly separable training data. In particular, we have increased in five orders of magnitude this value.
3. **Train the SVM**
We call the method :svms:`CvSVM::train <cvsvm-train>` to build the SVM model. Watch out that the training process may take a quite long time. Have patiance when your run the program.
.. code-block:: cpp
CvSVM svm;
svm.train(trainData, labels, Mat(), Mat(), params);
4. **Show the Decision Regions**
The method :svms:`CvSVM::predict <cvsvm-predict>` is used to classify an input sample using a trained SVM. In this example we have used this method in order to color the space depending on the prediction done by the SVM. In other words, an image is traversed interpreting its pixels as points of the Cartesian plane. Each of the points is colored depending on the class predicted by the SVM; in dark green if it is the class with label 1 and in dark blue if it is the class with label 2.
.. code-block:: cpp
Vec3b green(0,100,0), blue (100,0,0);
for (int i = 0; i < I.rows; ++i)
for (int j = 0; j < I.cols; ++j)
{
Mat sampleMat = (Mat_<float>(1,2) << i, j);
float response = svm.predict(sampleMat);
if (response == 1) I.at<Vec3b>(j, i) = green;
else if (response == 2) I.at<Vec3b>(j, i) = blue;
}
5. **Show the training data**
The method :drawingFunc:`circle <circle>` is used to show the samples that compose the training data. The samples of the class labeled with 1 are shown in light green and in light blue the samples of the class labeled with 2.
.. code-block:: cpp
int thick = -1;
int lineType = 8;
float px, py;
// Class 1
for (int i = 0; i < NTRAINING_SAMPLES; ++i)
{
px = trainData.at<float>(i,0);
py = trainData.at<float>(i,1);
circle(I, Point( (int) px, (int) py ), 3, Scalar(0, 255, 0), thick, lineType);
}
// Class 2
for (int i = NTRAINING_SAMPLES; i <2*NTRAINING_SAMPLES; ++i)
{
px = trainData.at<float>(i,0);
py = trainData.at<float>(i,1);
circle(I, Point( (int) px, (int) py ), 3, Scalar(255, 0, 0), thick, lineType);
}
6. **Support vectors**
We use here a couple of methods to obtain information about the support vectors. The method :svms:`CvSVM::get_support_vector_count <cvsvm-get-support-vector>` outputs the total number of support vectors used in the problem and with the method :svms:`CvSVM::get_support_vector <cvsvm-get-support-vector>` we obtain each of the support vectors using an index. We have used this methods here to find the training examples that are support vectors and highlight them.
.. code-block:: cpp
thick = 2;
lineType = 8;
int x = svm.get_support_vector_count();
for (int i = 0; i < x; ++i)
{
const float* v = svm.get_support_vector(i);
circle( I, Point( (int) v[0], (int) v[1]), 6, Scalar(128, 128, 128), thick, lineType);
}
Results
========
.. container:: enumeratevisibleitemswithsquare
* The code opens an image and shows the training examples of both classes. The points of one class are represented with light green and light blue ones are used for the other class.
* The SVM is trained and used to classify all the pixels of the image. This results in a division of the image in a blue region and a green region. The boundary between both regions is the separating hyperplane. Since the training data is non-linearly separable, it can be seen that some of the examples of both classes are misclassified; some green points lay on the blue region and some blue points lay on the green one.
* Finally the support vectors are shown using gray rings around the training examples.
.. image:: images/svm_non_linear_result.png
:alt: Training data and decision regions given by the SVM
:width: 300pt
:align: center
You may observe a runtime instance of this on the `YouTube here <https://www.youtube.com/watch?v=vFv2yPcSo-Q>`_.
.. raw:: html
<div align="center">
<iframe title="Support Vector Machines for Non-Linearly Separable Data" width="560" height="349" src="http://www.youtube.com/embed/vFv2yPcSo-Q?rel=0&loop=1" frameborder="0" allowfullscreen align="middle"></iframe>
</div>
doc/tutorials/ml/table_of_content_ml/table_of_content_ml.rst
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.. _Table-Of-Content-Ml:
*ml* module. Machine Learning
-----------------------------------------------------------
Use the powerfull machine learning classes for statistical classification, regression and clustering of data.
.. include:: ../../definitions/tocDefinitions.rst
+
.. tabularcolumns:: m{100pt} m{300pt}
.. cssclass:: toctableopencv
============ ==============================================
|IntroSVM| **Title:** :ref:`introductiontosvms`
*Compatibility:* > OpenCV 2.0
*Author:* |Author_FernandoI|
Learn what a Suport Vector Machine is.
============ ==============================================
.. |IntroSVM| image:: images/introduction_to_svm.png
:height: 90pt
:width: 90pt
+
.. tabularcolumns:: m{100pt} m{300pt}
.. cssclass:: toctableopencv
============ ==============================================
|NonLinSVM| **Title:** :ref:`nonLinearSvmS`
*Compatibility:* > OpenCV 2.0
*Author:* |Author_FernandoI|
Here you will learn how to define the optimization problem for SVMs when it is not possible to separate linearly the training data.
============ ==============================================
.. |NonLinSVM| image:: images/non_linear_svms.png
:height: 90pt
:width: 90pt
.. raw:: latex
\pagebreak
.. toctree::
:hidden:
../introduction_to_svm/introduction_to_svm
../non_linear_svms/non_linear_svms