Difficulties implementing Equations

Question

I am trying to write this equations in TeXnic Center, but so far, I am not very successful. The equations I want to write are here: http://www.mathworks.com/help/vision/ref/vision.opticalflow-class.html - Horn&Schunk method.

The Horn-Schunk method, by assuming that the optical flow is smooth over the entire image, computes an estimate of the velocity field, that minimizes this equation:
\begin{flushleft}

    \[
E = \int \!\!\! \int ($I_x$u + $I_y$v + $I_t$)^2 dxdy + \alpha \int \!\!\! \int (\frac{\partial \textit{u}}{\partial x}^2 + \frac{\partial \textit{u}}{\partial y}^2 + \frac{\partial \textit{v}}{\partial x}^2 + \frac{\partial \textit{v}}{\partial y}^2)dxdy
    \]
\end{flushleft}

where \alpha is the smoothness term of the velocity field, 
    \[
    \frac{\partial \textit{u}}{\partial x}
\]
 and 
     \[
    \frac{\partial \textit{v}}{\partial x} 
 \]
 are the spatial derivatives of the optical velocity component \textit{u}. The \alpha regularization parameter controls the strength of the smoothness constraint and is usually selected heuristically. The Horn-Schunck method minimizes the previous equation to obtain the velocity field, [u v], for each pixel in the image, which is given by the following equations:

    \[
        \stackrel u{k+1}{x,y} = \stackrel u{-k}{x,y} - \frac{$I_x$[$I_x$\stackrel u{-k}{x,y} + $I_y$\stackrel v{-k}{x,y} + $I_t$]}{\alpha^2 + $I_x$^2 + $I_y$^2
    \]

    \[
        \stackrel v{k+1}{x,y} = \stackrel v{-k}{x,y} - \frac{$I_y$[$I_x$\stackrel u{-k}{x,y} + $I_y$\stackrel v{-k}{x,y} + $I_t$]}{\alpha^2 + $I_x$^2 + $I_y$^2
    \]

I am newbie in LateX, and when I compile, it doesn't puts out the expected result. Also, when I write \alpha, afterwards, the text gets formatted in a different way (see picture attached). Can someone help me?

Answer 1

I've taken a crack at cleaning up your code.

Some dos and donts:

• Don't use $...$ inside display-math mode to denote math objects; all objects in display-math mode are assumed to be math objects.

• Similarly, don't use \textit{...} to denote math items: If already in math mode, you need do nothing extra; if not, use $u$. (Some, but not all, fonts distinguish between math italics and text italics; when in math mode, use math italics.)

• The flushleft environment encasing a display-math environment achieves nothing; leave it off.

• Take care to resize various parentheses when they enclose "large" objects, using either \left and \right or, better still, optimally chosen sizing directives such as \big and \bigg.

• To write a "bar" (overline) across a variable x, write either \bar{x} or \overline{x} (if you want a heftier bar).

• Use ^ ("caret") to initiate superscript matter and _ ("underline") to initiate subscript matter.

• Don't leave extra blank lines in matter that clearly is meant to be a single logical paragraph; all-blank lines initiate a paragraph break, with the first word of the new paragraph right-indented (in the amount of \parindent).

• When you have groups of equations (such as the last two ones in your example), try using an environment such as align or align*. Please consult the user guide of the amsmath package to find out more about these environments.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
The Horn-Schunk method, by assuming that the optical flow is smooth over the entire image, computes an estimate of the velocity field, that minimizes this equation:
\[
E = \int \!\!\! \int \bigl(I_xu + I_yv + I_t\bigr)^2 dxdy + \alpha \int \!\!\! \int \biggl(\frac{\partial u}{\partial x}^2 + \frac{\partial u}{\partial y}^2 + \frac{\partial v}{\partial x}^2 + \frac{\partial v}{\partial y}^2\biggr)dxdy
\]
where $\alpha$ is the smoothness term of the velocity field and
$\frac{\partial u}{\partial x}$ and 
$\frac{\partial v}{\partial x}$
are the spatial derivatives of the optical velocity component~$u$. The $\alpha$ regularization parameter controls the strength of the smoothness constraint and is usually selected heuristically. The Horn-Schunck method minimizes the previous equation to obtain the velocity field, $[u\ v]$, for each pixel in the image, which is given by the following equations:
\begin{align*}
u^{k+1}_{x,y} &= \bar{u}^{k}_{x,y} - 
\frac{I_x[I_x \bar{u}^{k}_{x,y} + I_y\bar{v}^{k}_{x,y} 
+ I_t]}{\alpha^2 + I_x^2 + I_y^2}\\
v^{k+1}_{x,y} &= \bar{v}^{k}_{x,y} - 
\frac{I_y[I_x\bar{u}^{k}_{x,y} + I_y\bar{v}^{k}_{x,y} 
+ I_t]}{\alpha^2 + I_x^2 + I_y^2}
\end{align*}
\end{document}

Answer 2

Here is an attempt to get something sensible. Note that you have a command for double integrals, that is \iint. The amsmath package (loaded by mathtools) defined a number of multiline equations environment, of which I use align*. The nccmath package is here to have medium-sized fractions in in-line formulae, rather than text-style, which is too small in my opinion.

As Mico explained the meaning of the \stackrel commands, I solved using the mathabx package that has a widebar command, that's better-looking in my opinion than plain \bar:

     \documentclass[a4paper]{article}
    \usepackage[utf8]{inputenc}
    \usepackage[T1]{fontenc}
    \usepackage{mathtools, nccmath, mathabx}%

     \begin{document}

    The Horn-Schunk method, by assuming that the optical flow is smooth over the entire image, computes an estimate of the velocity field, that minimizes this equation:
    \[
    E = \iint \bigl(I_x u + I_y v + I_t\bigr)^2 dxdy + \alpha \iint\left(\frac{\partial \textit{u}}{\partial x}^2 + \frac{\partial \textit{u}}{\partial y}^2 + \frac{\partial \textit{v}}{\partial x}^2 + \frac{\partial \textit{v}}{\partial y}^2\right)dxdy,
    \]
    where $ \alpha $ is the smoothness term of the velocity field, $ \mfrac{\partial \textit{u}}{\partial x} $ and $ \mfrac{\partial \textit{v}}{\partial x} $ are the spatial derivatives of the optical velocity component \textit{u}. The $ \alpha $ regularization parameter controls the strength of the smoothness constraint and is usually selected heuristically. The Horn-Schunck method minimizes the previous equation to obtain the velocity field, $ [u, v] $, for each pixel in the image, which is given by the following equations:
    \begin{align*}
      \widebar u^ {k+1}_{x,y} & = \widebar u^{k}_{x,y} - \frac{I_x[I_x \widebar u^{k}_{x,y} + I_y \widebar v^{k}_{x,y} + I_t]}{\alpha^2 + I_x^2 + I_y^2} \\
    \widebar v^{k+1}_{x,y} & = \widebar v^{k}_{x,y} - \frac{I_y[I_x \widebar u^{k}_{x,y} + I_y \widebar v^{k}_{x,y} + I_t]}{\alpha^2 + I_x^2 + I_y^2}
    \end{align*}

    \end{document} 

Answer 3

You're quite close but you are over working things.

First, inside a displayed equations you do no need to enclose mathematics within dollar signs, so you can drop then $...$.

Secondly, your \stackrel's are unnecessary and completely confused me: subscripts and superscripts are done with _ and ^, respectively.

Here is a cleaned up working version of your code:

\documentclass{amsart}

\usepackage{amsmath}

\begin{document}


\begin{equation*}
E = \iint (I_xu + I_yv + I_t)^2\,dx\,dy + \alpha \iint
 (\frac{\partial u}{\partial x}^2 + \frac{\partial
u}{\partial y}^2 + \frac{\partial v}{\partial x}^2 +
\frac{\partial v}{\partial y}^2)\,dx\,dy
\end{equation*}

where $\alpha$ is the smoothness term of the velocity field,
    \[
    \frac{\partial u}{\partial x}
\]
 and
     \[
    \frac{\partial v}{\partial x}
 \]
 are the spatial derivatives of the optical velocity component $u$. The $\alpha$ regularization parameter controls the strength of the smoothness constraint and is usually selected heuristically. The Horn-Schunck method minimizes the previous equation to obtain the velocity field, $[u v]$, for each pixel in the image, which is given by the following equations:

    \[
    u^{k+1}_{x,y} = u^{-k}_{x,y} - \frac{I_x[I_xu^{-k}_{x,y} + I_yv^{-k}_{x,y} + I_t]}{\alpha^2 + I_x^2 + I_y^2}
    \]

    \[
    v^{k+1}_{x,y} = v^{-k}_{x,y} - \frac{I_y[I_xu^{-k}_{x,y} + I_yv^{-k}_{x,y} + I_t]}{\alpha^2 + I_x^2 + I_y^2}
    \]

\end{document}

Answer 4

\documentclass{article} \begin{document}

[ E = \int \int (\frac{\partial I}{\partial x} {\cdot} u +\frac{ \partial I}{\partial y} \cdot v + \frac{\partial I}{\partial t}) =0 ]

[ E+\int \int (I_u+I_v+I_t)^2 dx dy + \alpha \int \int \frac{\partial u^2}{\partial x} +\frac{\partial u^2}{\partial y}+\frac{\partial u^2}{\partial x} +\frac{\partial v^2}{\partial y} ) dx dy ]

where $\alpha $ is the smoothness term of the velocity field, [ \frac{\partial u}{\partial x} ] and [ \frac{\partial v}{\partial x} ] are the spatial derivatives of the optical velocity component \textit{u}. The $\alpha$ regularization parameter controls the strength of the smoothness constraint and is usually selected heuristically. The Horn-Schunck method minimizes the previous equation to obtain the velocity field, [u v], for each pixel in the image, which is given by the following equations:

\[
    \stackrel u{k+1}{x,y} = \stackrel u{-k}{x,y} - \frac{I_x [ I_x \stackrel u{-k}{x,y} + I_y \stackrel v{-k}{x,y} + I_t]}{\alpha^2 + I_x^2 + I_y^2}
\]

\[
    \stackrel v{k+1}{x,y} = \stackrel v{-k}{x,y} - \frac{I_y[I_x \stackrel u{-k}{x,y} + I_y \stackrel v{-k}{x,y} + I_t ]}{\alpha^2 + I_x^2 + I_y^2}
\]
\end{document}

I have edited your file. Please compile and see the output.