We are
familiar with one dimensional trigonometric functions like sine
and cosine functions
etc. There is another set of functions called two dimensional
trigonometric functions, which most of us are not familiar with.
These functions are very important in many image processing
applications. This exercise helps us to create such functions and to
visualize them. A big difference from the one dimensional
trigonometric function is that, it has two frequencies, one along the
X-axis and the other along the Y-axis. In addition we can have sine
and cosine waves in the same 2-D trigonometric function, i.e. in one
axis we may see 3 sine waves and 2 cosine waves along the other.
We are
creating such functions by using the concept of outer product. Here
we multiply two vectors of size and we get a matrix of size .
This matrix produces the required trigonometric function. A plot of
these values are given in each sheet for visualization purpose.
Excel
Implementation.
We
use the following inbuilt functions to find the sine
and cosine values.
SIN():This
function returns the sine value.
COS():Similarly
returns the cosine value.
RADIANS():Returns
the angle in radians for the input is in degrees.
ROW():Returns
the row number of the cell.
COLUMN():Returns
the column number of the cell.
The steps
involved in the excel implementation are given below. We use the
first sheet as example.
Step
1: Click on cell A5
and type the following expression
=COS(RADIANS((ROW()-5)*10)).
Step
2: Similarly type the expression
=COS(RADIANS((COLUMN()-2)*10))
in the cell B4.
The
expressions return the cosine values for the 0 degree angle in both
the cases.
Step
3: Click on the cell A5
and drag down till A41.
This produces one cycle of cosine values.
Step
4: Click on the cell B4
and drag right till AL4.
This produces the second set of values.
If we
want to produce two cycles replace 10 by 20 gives the effect of
multiplying by 2.
Step
5: Click on cell B5
and type the expression =$A5*B$4.
Click on the small black box on the right-bottom of the cell and
drag right and downwards to create the 37 by
37 matrix.
The
above step performs the outer product operation.
Step
6: Now click on cell B5
and drag till cell AL41
to select the 37 by 37
matrix. Now go to the insert tab
and from other charts
choose 3-D Surface to
get the visualization of trigonometric function.
The 37
by 37 matrix has the required function
values.
Step
7: You can repeat this for different
combinations of sines and cosines and different number of cycles.
Explain
How to view it
Figures
Here
Draw a
dependancy diagram
Create
Two diensional Exponential functions
Take
functions from Matlab(by rudrapratap) and Plot
Plot also
parametric functions
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