Combine and Simplify Equation

**dhany19** · January 8th 2010, 01:19 AM

Hi Guys

,

I'm newbie in this forum. Pardon me if this thread not in appropriate sub forum.
I have SSIM algorithm. I want to derive this algorithm.
$\mu_x$ for average x
$\sigma_x$ for variance x
$\sigma_{xy}$ for covariance x

$l(x,y) = \frac{2\mu_x\mu_y + c_1}{\mu_x^2 + \mu_y^2 + c_1}<br />$

$r(x,y) = \frac{2\sigma_x\sigma_y + c_2}{\sigma_x^2 + \sigma_y^2 + c_2}$

$r(x,y) = \frac{2\sigma_{xy} + c_2}{2\sigma_x\sigma_y + c_2}$

To be like this

$SSIM(x,y) = \frac{(2\mu_x\mu_y + c_1)(2\sigma_{xy} + c_2)}{(\mu_x^2 + \mu_y^2 + c_1)(\sigma_x^2 + \sigma_y^2 + c_2)}$

Please help me to solve step by step.

Thank you.

**novice** · January 9th 2010, 12:10 PM

Originally Posted by dhany19

Hi Guys

,

I'm newbie in this forum. Pardon me if this thread not in appropriate sub forum.
I have SSIM algorithm. I want to derive this algorithm.
$\mu_x$ for average x
$\sigma_x$ for variance x
$\sigma_{xy}$ for covariance x

$l(x,y) = \frac{2\mu_x\mu_y + c_1}{\mu_x^2 + \mu_y^2 + c_1}<br />$

$r(x,y) = \frac{2\sigma_x\sigma_y + c_2}{\sigma_x^2 + \sigma_y^2 + c_2}$

$r(x,y) = \frac{2\sigma_{xy} + c_2}{2\sigma_x\sigma_y + c_2}$

To be like this

$SSIM(x,y) = \frac{(2\mu_x\mu_y + c_1)(2\sigma_{xy} + c_2)}{(\mu_x^2 + \mu_y^2 + c_1)(\sigma_x^2 + \sigma_y^2 + c_2)}$

Please help me to solve step by step.

Thank you.

I am not an optic engineer, but I found the definition of SSIM.

Definition says SSIM(x,y) = L(x,y)C(x,y)S(x,y) where

$L(x,y) = \frac{2\mu_x\mu_y + c_1}{\mu_x^2 + \mu_y^2 + c_1}<br />$

$C(x,y) = \frac{2\sigma_x\sigma_y + c_2}{\sigma_x^2 + \sigma_y^2 + c_2}$

$S(x,y) = \frac{2\sigma_{xy} + c_2}{2\sigma_x\sigma_y + c_2}$
$SSIM(x,y) = \frac{(2\mu_x\mu_y + c_1)<2\sigma_{x}\sigma_y + c_2>}{(\mu_x^2 + \mu_y^2 + c_1)(\sigma_x^2 + \sigma_y^2 + c_2)}\frac{(2\sigma_{xy} + c_2)}{<2\sigma_x\sigma_y + c_2>}$

where the <> canceled out. Only you would know how the Stabilizer Constant and Dynamic Range come about.

You can post your question at the physics forum. There are people there who can help you.

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