4.7 Java1491-2d fourier transforms using java, part 2  (Page 15/21)

A different plotting color scheme

Note that the plotting color scheme that I used for Figure 7 is different from any of the plots previously shown in this module. This color scheme is whatI refer to as the Color Contour scheme in the module titled Plotting 3D Surfaces using Java .

This scheme quantizes the range from the lowest to the highest elevation into 23 levels, coloring the lowest elevation black, the highest elevation white, andassigning very specific colors to the 21 levels in between. The colors and the levels that they represent are shown in the calibration scales under the plotsin Figure 7 . The lowest elevation is on the left end of the calibration scale. The highest elevation is on the right end of the calibration scale.

The amplitude spectrum

As before, the wavenumber amplitude spectrum with the origin in the top-left corner is shown in the top-right image in Figure 7 . The amplitude spectrum with the origin shifted to the center is shown in the lower right image in Figure 7 .

If you were to use the calibration scale to convert the colors along the horizontal and vertical axes in the lower right image into numeric values, youwould find that they approximate a rectified sin(x)/x shape as expected.

The output surface

The output surface produced by performing an inverse Fourier transform on the complex wavenumber spectrum is shown in the lower-left image in Figure 7 . This surface appears to match the input surface shown in the top-left image in Figure 7 .

The overall results

The numeric output for this case isn't very useful because none of the samples for which numeric data is provided fall within the square. However,because the real and imaginary parts exhibit the correct symmetry , , the shape of the amplitude spectrum is generally what we expect, and the output from the inverse Fouriertransform appears to match the original input causes us to conclude that the program is working properly in this case.

Case 3

This case places a raised box at the top near the origin in the space domain, but the box is not on the diagonal as it was in Case 2.

(See the top-left image in Figure 8 for the new location of the box.)

Figure 8. Graphic output for Case 3.

Amplitude spectrum should not change

As long as the size and the orientation of the box doesn't change, the wavenumber amplitude spectrum should be the same as Case 2 regardless of thelocation of the box in space. Since the size and orientation of this box is the same as in Case 2, the amplitude spectrum for this case should be the same asfor Case 2.

The real and imaginary parts of the spectrum may change

However, the real and imaginary parts (or the phase) change as the location of the box changes relative to the origin in space.

A hypothetical example

The purpose of this case is to illustrate a hypothetical example. If two different photographic images contain a picture of the same object in the samesize and the same orientation in space, that object will contribute the same values to the amplitude spectrum of both images regardless of where the objectis located in the different images.