
\begin{latin}
\begin{description}
\item[Blind superresolution]
Blind superresolution or superresolution
without knowledge of the PSF is a challenging topic under development
\item[Inverse problem]
An inverse problem is the task that often occurs in many branches of science 
and mathematics where the values of some model parameter(s) must be obtained 
from the observed data.\\
The inverse problem can be formulated as follows:\\
Data $\Rightarrow$ Model parameters \\
The transformation from data to model parameters is a result of the interaction
 of a physical system, e.g. the Earth, the atmosphere, gravity etc. Inverse 
problems arise for example in geophysics, medical imaging (such as computed 
axial tomography), remote sensing, ocean acoustic tomography, nondestructive 
testing, and astronomy.

Inverse problems are typically ill posed, as opposed to the well-posed problems

\item[ill-posed]
The mathematical term well-posed problem stems from a definition given by 
Hadamard. He believed that mathematical models of physical phenomena should 
have the properties that
\begin{enumerate}
\item
A solution exists 
\item
The solution is unique 
\item
The solution depends continuously on the data, in some reasonable topology. 
\end{enumerate}
Examples of archetypal well-posed problems include the Dirichlet problem for 
Laplace's equation, and the heat equation with specified initial conditions. 
These might be regarded as 'natural' problems in that there are physical 
processes that solve these problems. By contrast the backwards heat equation, 
deducing a previous distribution of temperature from final data is not 
well-posed in that the solution is highly sensitive to changes in the final 
data. Problems that are not well-posed in the sense of Hadamard are termed 
{\it ill-posed}. Inverse problems are often ill-posed.\\
Such continuum problems must often be discretized in order to obtain a 
numerical solution. While in terms of functional analysis such problems are 
typically continuous, they may suffer from numerical instability when solved 
with finite precision, or with errors in the data. Even if a problem is 
well-posed, it may still be {\it ill-conditioned}, meaning that a small error in the 
initial data can result in much larger errors in the answers. An 
ill-conditioned problem is indicated by a big condition number.

If a problem is well-posed, then it stands a good chance of solution on a 
computer using a stable algorithm. If it is not well-posed, it needs to be 
re-formulated for numerical treatment. Typically this involves including 
additional assumptions, such as smoothness of solution. This process is known 
as regularization

\item[Condition number]
In numerical analysis, the condition number associated with a problem is a 
measure of that problem's amenability to digital computation, that is, how 
numerically well-posed the problem is. A problem with a low condition number is
 said to be well-conditioned, while a problem with a high condition number is 
said to be ill-conditioned.

For example, the condition number associated with the linear equation $Ax = b$ 
gives a bound on how inaccurate the solution $x$ will be after approximate 
solution. Note that this is before the effects of round-off error are taken 
into account; conditioning is a property of the matrix, not the algorithm or 
floating point accuracy of the computer used to solve the corresponding system.
{\sl In particular, one should think of the condition number as being (very 
roughly) the rate at which the solution, $x$, will change with respect to a 
change in $b$}. Thus, if the condition number is large, even a small error in $b$ 
may cause a large error in $x$. On the other hand, if the condition number is 
small then the error in $x$ will not be much bigger than the error in $b$.

The condition number is defined more precisely to be the maximum ratio of the 
relative error in $x$ divided by the relative error in $b$.

Let $e$ be the error in $b$. Then the error in the solution $A^{-1}b$ is $A^{-1}e$. The 
ratio of the relative error in the solution to the relative error in b is
\[ \frac{\Vert A^{-1} e\Vert / \Vert A^{-1} b\Vert}{\Vert e\Vert / \Vert b\Vert} \]

A condition in computer vision when condition number is important is in Normalization
(Hartly book, page 108):The effect of normalization is related to the condition number of the set of 
DLT equations, or more precisely the ratio d of the first to the second-last singular 
value of the equation matrix A. In the presence
of noise the solution will diverge from the correct result. The effect of a 
large condition number is to amplify this divergence.

\item[Cross-validation]
Cross-validation, sometimes called rotation estimation, is the 
statistical practice of partitioning a sample of data into subsets such that 
the analysis is initially performed on a single subset, while the other 
subset(s) are retained for subsequent use in confirming and validating the 
initial analysis.

The initial subset of data is called the training set; the other subset(s) are 
called validation or testing sets.

Common types of cross-validation are:
\begin{enumerate}
\item Holdout validation 
\item K-fold cross-validation 
\item Leave-one-out cross-validation 
\end{enumerate}

\item[Overdetermined system]
In mathematics, a system of linear equations is considered overdetermined if there are more equations than unknowns. The terminology can be described in terms of the concept of counting constants. Each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom.
Therefore the critical case occurs when the number of equations and the number of independent variables are equivalent. For every degree of freedom, there exists a corresponding restraint. The overdetermined case occurs when the system has been overconstrained. i.e. The number of equations outnumbers the  number of the unknowns.
% The mixels are overdetermined by the pixels when there is no value for the mixels which can exactly predict all the pixel values. The overdetermined situation means that the mixel grid is at a coarser spatial resolution than is otherwise achievable. If the ML approach is tried at too fine a resolution, the mixel values are underconstrained—i.e., there are many mixel grids that would predict the pixel values exactly, and there is no principled way of choosing among them.\cite{Cheezman96superresolved}
The method of least squares can be used to find an approximate solutions to  overdetermined systems.For the system $Ax=b$, the least squares formula can be  written $\hat{x}=(A^TA+\Gamma^T\Gamma)^{-1}A^Tb$ with this formula an approximate solution is found for \[ min_x ||Ax-b|| \]
A system of linear equations is considered {\it underdetermined} if  there are more unknowns than equations.

\item[Regularization]
In mathematics, inverse problems are often ill-posed. To solve these problems 
numerically one must introduce some additional information about the solution, 
such as an assumption on the smoothness or a bound on the norm. This process is
 known as regularization.

\item[Tikhonov regularization]
Tikhonov regularization is the most commonly used method of regularization of 
ill-posed problems. In statistics, the method is also known as ridge 
regression. It is related to the Levenberg-Marquardt algorithm for non-linear 
least squares problems.

The standard approach to solve an overdetermined system of linear equations 
given as $Ax=b$ is known as linear least squares and seeks to minimize the residual
$||Ax-b||^2$
 
where $||.||$ is the Euclidean norm. However, the matrix $A$ may be ill-conditioned or 
singular yielding a large number of solutions. In order to give preference to a
 particular solution with desirable properties, the regularization term is 
included in this minimization:
\[||Ax-b||^2+||\Gamma x||^2 \]
 
for some suitably chosen Tikhonov matrix $\Gamma$. In many cases, this matrix is 
chosen as the identity matrix$\Gamma = I$, giving preference to solutions with 
smaller norms. In other cases, highpass operators (e.g. a difference operator 
operator or a weighted fourier operator) may be used to enforce smoothness if 
the underlying vector is believed to be mostly continuous. This regularization 
improves the conditioning of the problem, thus enabling a numerical solution. 
An explicit solution, denoted by $\hat{x}$, is given by:
\[\hat{x}=(A^TA+\Gamma^T\Gamma)^{-1}A^Tb\]
 
The effect of regularization may be varied via the scale of matrix $\Gamma$ (e.g. $\Gamma=I$)
. For $\Gamma=0$ this reduces to the unregularized least squares solution 
provided that $(A^TA)^{-1}$ exists.
\item[Row space]
In linear algebra, the row space of a matrix is the set of all possible linear 
combinations of its row vectors. The row space of an $m \times n$ matrix is a subspace
 of n-dimensional Euclidean space. The dimension of the row space is called the
 rank of the matrix.
\item[Null space]
In linear algebra, the kernel or null space (also nullspace) of a matrix A is 
the set of all vectors x for which Ax = 0. The null space of a matrix with n 
columns is a linear subspace of n-dimensional Euclidean space.
\item[Isotropy]
Isotropy is the quality of a property which does not depend on the direction.
\item[Anisotropy]
Anisotropy is the property of being directionally dependent, as opposed to isotropy, 
which means homogeneity in all directions.
%\item[Artificial Intelligence]
%A name for a paradigm in which people attempt to elicit intelligence from machines.
%\item[Computational Intelligence]
%Practical adaptation concepts, paradigms, algorithms, and
%implementations that enable or facilitate appropriate actions
%(intelligent behavior) in complex and changing environments. CI is different than AI. It 
% deals well with complex, dynamic, poorly defined problems that AI has problems with.
\end{description}
\end{latin}
اصطلاحات زیر از  {\cite{Dubrofsky07mthesis}} اخذ شده‌اند:
\begin{latin}
An isometry is a transformation that preserves Euclidian distance. This
means that the distance between two points in one image will be the same
as the distance between their corresponding points in the mapped image.
The same goes for the angles between lines and areas. Isometries are made
up of only 2D rotations and 2D translations and therefore have only 3 degrees
of freedom. An isometry can be written as:
\begin{equation}
x'=\left(\begin{array}{cc}
R & t \\ 
0^T & 1
\end{array} \right)x
\end{equation}
where R is a 2x2 rotation matrix, t is a translation 2-vector and 0T is a
row of 2 zeros.
\textbf{ Similarity transformation}
A similarity transform is similar to an isometry except it also contains
isotropic scaling. Isotropic means that the scaling is invariant with respect
to direction. The scale adds an additional degree of freedom so a similarity
transform contains 4 degrees of freedom overall. Like with isometries, angles
are not affected by this transformation. The distance between points are
no longer invariant, but the ratio of distances is preserved under similarity
transformations since any scale change cancels out. A similarity transform
can be written as:
\begin{equation}
x'=\left(\begin{array}{cc}
sR & t \\ 
0^T & 1
\end{array} \right)x
\end{equation}
where s is a scalar and represents the isotropic scaling.
%\begin{description}
%\item[مستقل خطی abc]
%یک مجموعه از بردارهای ${a_1,\dots,a_n}$ در $R$ 
%\item[دومی] توضیح دومی 
%\end{description}

Video quality, From Wikipedia:
Objective video quality

Objective video evaluation techniques are mathematical models that approximate results of subjective quality assessment, but are based on criteria and metrics that can be measured objectively and automatically evaluated by a computer program. Objective methods are classified based on the availability of the original video signal, which is considered to be of high quality (generally not compressed). Therefore, they can be classified as Full Reference Methods (FR), Reduced Reference Methods (RR) and No-Reference Methods (NR). FR metrics compute the quality difference by comparing every pixel in each image of the distorted video to its corresponding pixel in the original video. RR metrics extract some features of both videos and compare them to give a quality score. They are used when all the original video is not available, e.g. in a transmission with a limited bandwidth. NR metrics try to assess the quality of a distorted video without any reference to the original video. These metrics are usually used when the video coding method is known.

The most traditional ways of evaluating quality of digital video processing system (e.g. video codec like DivX, XviD) are calculation of the signal-to-noise ratio (SNR) and peak signal-to-noise ratio (PSNR) between the original video signal and signal passed through this system. PSNR is the most widely used objective video quality metric. However, PSNR values do not perfectly correlate with a perceived visual quality due to non-linear behavior of human visual system. Recently a number of more complicated and precise metrics were developed, for example UQI, VQM, PEVQ, SSIM and CZD. Based on a benchmark by the Video Quality Experts Group (VQEG) in the course of the Multimedia Test Phase 2007-2008 some metrics were standardized as ITU-T Rec. J.246 (RR) and J.247 (FR) in 2008.

The performance of an objective video quality metric is evaluated by computing the correlation between the objective scores and the subjective test results. The latter is called mean opinion score (MOS). The most frequently used correlation coefficients are : linear correlation coefficient, Spearman's rank correlation coefficient, kurtosis, kappa coefficient and outliers ratio.

When estimating quality of a video codec, all the mentioned objective methods may require repeating post-encoding tests in order to determine the encoding parameters that satisfy a required level of visual quality, making them time consuming, complex and impractical for implementation in real commercial applications. For this reason, much research has been focused on developing novel objective evaluation methods which enable prediction of the perceived quality level of the encoded video before the actual encoding is performed [1].
[edit] Subjective video quality

Subjective video quality

The main goal of many objective video quality metrics is to automatically estimate average user (viewer) opinion on a quality of video processed by the system. Sometimes however, measurement of subjective video quality can also be challenging because it may require a trained expert to judge it. Many “subjective video quality measurements” are described in ITU-T recommendation BT.500. Their main idea is the same as in Mean Opinion Score for audio: video sequences are shown to the group of viewers and then their opinion is recorded and averaged to evaluate the quality of each video sequence. However, details of testing may vary greatly.
\end{latin}
تبدیل‌های آفین و پروجکتیو و پرسپکتیو هم در همان مرجع توضیح داده شده‌اند. (پرسپکتیو در واقع همان ماتریسهایی است که در بینایی ماشین برای تبدیل مختصات فضای سه بعدی به دوبعدی داشتیم - ترکیب مشخصه‌های داخلی و خارجی دوربین)