Expressing Math Equations

Question

I'm confused how to express the following expressions in math equations for publication:

• $x =$ integer part of $y$
• $x =$ fraction part of $y$
• image $x =$ shifted version of image $y$ left with $z$ pixels.
• image $x =$ shifted version of image $y$ up with $z$ pixels.

I thought about this: $x =$ intPart$(y)$ and image$_x =$ shiftLeft$(y, z)$. But they are programming style more than math equations. Any hints ?

Answer 1

$\lfloor y \rfloor $ denotes usually the integer part of $ y $, $\{ y \} $ denotes the fractional part of $ y $ ($\{ y \} =y-\lfloor y \rfloor $). And sorry, but I have no idea what does "image x = shifted version of image y up with z pixels." mean. And I wouldn't use $[y]$ to represent the integer part of $y$, since it can be easily confused (for example in $[5+(6-1)]=x$, $[y]$ doesn't represent the integer part).

Answer 2

Let's assume you represent an image by a matrix, e.g. $ x\in\mathbb {R}^{3x3}$. Then, your upshifted image y is

$y=\left (\begin {matrix}0&1&0\\0&0&1\\0&0&0\end {matrix}\right) x $

However, this assumes padding with zeros, and loss of pixels at the top.

Another solution is simply to define the position of the image's upper left corner by (x, y), and to define $ shiftup ((x, y))=(x, y-1) $...

For the other questions, I refer to hetajr's answer. ..

Edit: A more general notation, to move an abitrary image $x$ up for $n$ pixels: $\operatorname{shiftup}(x, n)=A_nx$, with the square matrix $A_n=(\delta_{i+n,j})$ and $\delta_{k,l}=1$ if $k=l$, and zero otherwise. If you are lazy, you can then define $\operatorname{shiftleft}(x,n)=\operatorname{shiftup}^T(x^T,n)$