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b) Similarly to a), we have: | b) Similarly to a), we have: | ||
| − | <math> p_1(e^{jw}) = X(e^{jw},e^{j\nu}) |_{ | + | <math> p_1(e^{jw}) = X(e^{jw},e^{j\nu}) |_{\nu=0}</math> |
| − | + | ||
| − | + | ||
c) <br> | c) <br> | ||
<math> \sum_{n=-\infty}^{\infty} p_0(n) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} x(m,n) = X(e^{j\mu}, e^{j\nu}) |_{\mu=0, \nu=0} </math> | <math> \sum_{n=-\infty}^{\infty} p_0(n) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} x(m,n) = X(e^{j\mu}, e^{j\nu}) |_{\mu=0, \nu=0} </math> | ||
| + | which is the DC point of the image. | ||
| + | |||
| + | d) No, it can't provide sufficient information. | ||
| + | From the expression in a) and b), we see that <math> p_0(e^{jw}) </math> and <math> p_1(e^{jw}) </math> are only slices of the DSFT. It lost the information when <math> \mu </math> and <math> \nu <math> are not zero. | ||
| + | A simple example would be: | ||
| + | Let <math> | ||
Revision as of 20:38, 10 November 2014
a) Since
$ X(e^{j\mu},e^{j\nu}) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} x(m,n)e^{-j(m\mu+n\nu)} $
and
$ p_0(e^{jw}) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} x(m,n)e^{-jnw} $，
we have:
$ p_0(e^{jw}) = X(e^{j\mu},e^{jw}) |_{\mu=0} $
b) Similarly to a), we have:
$ p_1(e^{jw}) = X(e^{jw},e^{j\nu}) |_{\nu=0} $
c)
$ \sum_{n=-\infty}^{\infty} p_0(n) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} x(m,n) = X(e^{j\mu}, e^{j\nu}) |_{\mu=0, \nu=0} $
which is the DC point of the image.
d) No, it can't provide sufficient information. From the expression in a) and b), we see that $ p_0(e^{jw}) $ and $ p_1(e^{jw}) $ are only slices of the DSFT. It lost the information when $ \mu $ and $ \nu <math> are not zero. A simple example would be: Let <math> $
