In question 15 I want to make a picture how $url1u%28x%2Cy%29%3A%3D%5Clangle%20x%2Cy%5Crangle$url2u(x,y):=\langle x,y\rangle$url3u(x,y):=\langle x,y\rangle$url4 looks like for $url1x%2Cy%5Cin%5CbbR%5Ed$url2x,y\in\bbR^d$url3x,y\in\bbR^d$url4. I want to take at least two-dimensional vectors, as the inner product is too trivial otherwise. But then $url1u$url2u$url3u$url4 is a function on $url1%5CbbR%5E4$url2\bbR^4$url3\bbR^4$url4, which is rather difficult to plot. So I want to fix $url1x%3Da$url2x=a$url3x=a$url4 and plot $url1y%5Cmapsto%20u_a%28y%29%3A%3D%5Clangle%20a%2Cy%5Crangle$url2y\mapsto u_a(y):=\langle a,y\rangle$url3y\mapsto u_a(y):=\langle a,y\rangle$url4, which is then a function on $url1%5CbbR%5E2$url2\bbR^2$url3\bbR^2$url4 with values in $url1%5CbbR$url2\bbR$url3\bbR$url4 and therefore easy to plot. As the inner product is $url1%5Csum_k%20x_k%20y_k$url2\sum_k x_k y_k$url3\sum_k x_k y_k$url4, it is in the 2-dimensional case a simple plane-equation $url1z%3Da_1y_1%2Ba_2y_2$url2z=a_1y_1+a_2y_2$url3z=a_1y_1+a_2y_2$url4; of course, the inner product is linear and goes through the origin! It cuts the zero-plane $url1z%3D0$url2z=0$url3z=0$url4 in an angle which is dependent on the orientation of the fixed $url1x%3Da$url2x=a$url3x=a$url4: The zero-line evolves where the vectors $url1y$url2y$url3y$url4 are orthogonal to the fixed $url1x$url2x$url3x$url4, i.e., $url1%5Clangle%20x%2Cy%5Crangle%3D0%5CLeftrightarrow%20x%5Cperp%20y$url2\langle x,y\rangle=0\Leftrightarrow x\perp y$url3\langle x,y\rangle=0\Leftrightarrow x\perp y$url4.

This zero-line will now change its angle on the zero-plane while varying the fixed vector $url1x$url2x$url3x$url4. This gives a picture how it behaves as a whole. In Octave, I set

`[xx yy]=meshgrid(linspace(-2,2,50));`

and then made a plot corresponding to $url1x%3D%5Cleft%28%5Cbegin%7Bsmallmatrix%7D1%5C%5C1%5Cend%7Bsmallmatrix%7D%5Cright%29$url2x=\left(\begin{smallmatrix}1\\1\end{smallmatrix}\right)$url3x=\left(\begin{smallmatrix}1\\1\end{smallmatrix}\right)$url4 by

`mesh(xx+yy)`

The resulting 3D-plot shows a boring inclined plane. To better see the location and orientation of the zero-line, I plot the absolute values:

`mesh(abs(xx+yy))`

To see the orientation even better, I chose

`imagesc(abs(xx+yy))`

`imagesc(abs(xx+2*yy))`

`imagesc(abs(-xx+4*yy))`

corresponding to the further vectors $url1%5Cleft%28%5Cbegin%7Bsmallmatrix%7D1%5C%5C2%5Cend%7Bsmallmatrix%7D%5Cright%29$url2\left(\begin{smallmatrix}1\\2\end{smallmatrix}\right)$url3\left(\begin{smallmatrix}1\\2\end{smallmatrix}\right)$url4 and $url1%5Cleft%28%5Cbegin%7Bsmallmatrix%7D-1%5C%5C4%5Cend%7Bsmallmatrix%7D%5Cright%29$url2\left(\begin{smallmatrix}-1\\4\end{smallmatrix}\right)$url3\left(\begin{smallmatrix}-1\\4\end{smallmatrix}\right)$url4. This is what I get:

The turning around of that plane now corresponds to the behavior of the mentioned inner product function on $url1%5CbbR%5E%7B2d%7D$url2\bbR^{2d}$url3\bbR^{2d}$url4. (Beware that the absolute values used in the plots hide the asymmetry of the inner product function!) The values are always real and go into the exponent of the exponential function. $url1e%5E%7B%5Ci%5Clangle%20x%2Cy%5Crangle%7D$url2e^{\i\langle x,y\rangle}$url3e^{\i\langle x,y\rangle}$url4 then “rotates fastest” as $url1y$url2y$url3y$url4 runs along to a fixed $url1x$url2x$url3x$url4, and is constant = 1 if it runs orthogonally; Beware that there’s a difference whether it is *constant* = 1 or if it reaches that value again and again. So, in the integral of the FT, the exponential function simply covers all frequencies via exhausting all values of $url1x$url2x$url3x$url4, and the FT is then a function in $url1%5Comega$url2\omega$url3\omega$url4, which takes out a single orientation of those frequencies; As already mentioned some time ago, the term “frequency” isn’t useful anymore in higher dimensions. For two-dimensional signals, i.e. images, a “frequency” can be interpreted as a pattern of lines: The narrower the pattern, the higher the frequency is. The difference to one-dimensional signals is, that those frequencies may now have an angle. It is not enough to just look at a single frequency of that line-pattern, but also to all their orientations! Only with using all orientations of patterns, an image can be analyzed according to the contained frequencies. The 2D-FT of an image therefore tells you what patterns occur in what orientation. To be more precise, you select the frequency by the length $url1%7C%5Comega%7C$url2|\omega|$url3|\omega|$url4 and the orientation by its angle $url1%5Carg%5Comega$url2\arg\omega$url3\arg\omega$url4, because the inner product as a function in $url1x$url2x$url3x$url4 grows faster with growing length of $url1%5Comega$url2\omega$url3\omega$url4. For me, this is now a milestone in understanding the 2D-FT.