AWA: Academic Writing at Auckland

Aim

To perform Fourier transforms both optically and using ImageFFT computer software, and to understand their relevance in real-world contexts.

Computer Fourier Filtering (Procedures 1-10)

Method

The method used for this section of the experiment was as that suggested in the handout.

Results and Analysis

1. Sinusoidal variation along one image dimension.

            The Fourier transform of an image with a pattern of evenly spaced vertical stripes was a horizontal line of evenly-spaced dots. This suggests that the Fourier transform has the same orientation as the direction of variation in the image – in a pattern of vertical stripes, the image intensity in the spatial domain changes along the horizontal axis. Also, the dots in the Fourier transform were evenly spaced, just as the vertical stripes in the image were evenly spaced, suggesting that the constant frequency of the Fourier transform reflects the constant frequency of the image. These findings are in keeping with established understanding of Fourier transforms (Hecht, 2002).

2. Effect of frequency of variation on the Fourier transform.

The Fourier transform of an image with closely spaced stripes (high frequency variation) had widely spaced dots, while the Fourier transform of an image with widely spaced stripes (low frequency variation) had closely spaced dots. This implies that the frequency of the Fourier transform is inversely proportional to the input wave frequency. Again, this finding is what would be expected based on a theoretical understanding of Fourier transforms (Hecht, 2002).

3. Effect of interference on the Fourier transform.

The beating of two or more waves produces a Fourier transform which is symmetrical about the central spot, but which has a non-constant frequency. For example, the Fourier transform of two interfering waves of different frequencies looks like this: 
This pattern reflects the resultant frequency of the two interfering waves. The widely spaced dots correspond to the maximum displacement points of the resultant wave, and the closely spaced dots correspond to the lower amplitude points.

4. Effect of filtering out the DC component of an image.

When you filter out the DC component of an image, you remove the average intensity (brightness) level. This causes the colours to reverse, and the grey areas at the edge of black and white areas to increase in width.

5. Production of different elemental sinusoids through filtering.

Three different elemental sinusoids can be produced by filtering stripes5.png. The resulting images, Fourier transforms and filters are shown in Figure 1 below.

6. Removal of horizontal stripes.

The imposed horizontal strips had a high spatial frequency (they had sharp edges) in the vertical direction, and they were evenly spaced (their spatial frequency was constant). These characteristics made them relatively easy to isolate and remove from the image without significantly altering the original image. Their removal required vertical sinusoidal modulation at the frequency of the stripes to be filtered out. Figure 2, below, details the result.

7. Removal of vertical stripes.

The process for removing evenly-spaced vertical stripes from an image was the same as detailed in Part 6 (above), but horizontal rather than vertical sinusoidal intensity modulation needed to be filtered out. Figure 3, below, shows the result.

8. Creation of a relief type image.

A highpass filter was added to produce a relief image of Einstein (shown below). A relief image only shows the edges of the image, that is, those parts of the image where the change in intensity is most sudden and the spatial frequency is highest. In order create a relief image, therefore, the low spatial frequency components of the image must be filtered out. In this computer program, the ‘highpass’ filter has this effect, filtering out low spatial frequencies but retaining high spatial frequencies to produce a relief image, as shown in Figure 4 below.

9. Descriptions and explanations of phenomena.

• When you apply a lowpass filter to an image, points that are above the threshold spatial frequency are filtered out. This means that contrast in the image is reduced, and the edges appear softened.
• When you apply a highpass filter to an image, points that are below the threshold spatial frequency are filtered out. Only the areas with the steepest intensity gradients (highest frequencies) remain in the image, such as the sharply contrasted right edge of Einstein’s tongue in the above image.
• The edges, where there is most contrast, correspond to the high spatial frequencies.
• The Fourier transform of a periodic feature is three dots, where the distance from the central (DC) point to the point on each side corresponds to the frequency of the periodic feature. The feature with the longer period will have a higher frequency (more closely-spaced) Fourier transform.

10. Removal of noise from a two-dimensional Gaussian.

The noise was removed from the two-dimensional Gaussian by applying a lowpass filter. A smooth Gaussian has no sharp edges, so it should not have any high spatial frequency components. Thus, any high spatial frequency components in the image can be assumed to be noise. The application of a lowpass filter filtered out these high spatial frequencies, removing the noise and leaving a smooth Gaussian (as shown in Figure 5).

Free space optics requires the transmission of signals over long distances without accruing noise that could obscure the signal. A Fourier transform that removes high frequencies (equivalent to the lowpass function on ImageFFT) would be useful as it would remove this noise, allowing clear transmission. This could be implemented by inserting a high frequency filter at the focal point of the lens.

Optical Fourier Transform (Procedures 11-14)

Method

            The beam produced by the HeNe laser has a diameter of 0.5-1mm, which is too small to encode the images used for this experiment. As such, the beam needed to be expanded. This expansion was performed using a short focal length lens (f_s = 10mm) and a long focal length lens (f_l = 200mm). The expansion ratio and lens distances were calculated as follows:

            Expansion ratio = f_l / f_s

= 200 / 10

= 20

            Lens distance = f_s + f_l

                                   = 200 + 10

                                   = 210 mm

            Two further lenses of even greater focal length (f = 750mm) were used to perform the initial Fourier transform on the image and to then perform an inverse Fourier transform after the beam had passed through the filter. The final experimental setup is detailed in Figure 6.

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Figure 6. Sketch of experimental setup.

A significant difficulty faced in the setup of this experiment was the condition of the lenses. Marks and smudging on the lenses made projecting a clear image impossible, which in turn limited the success of the experiment as a whole. Choosing widely spaced grid patterns meant that the effect of the Fourier transform and filter could still be seen, albeit blurrily, in Procedures 11 and 12, but the fine detail on the image of Einstein was lost. Future experiments of this kind would do well to utilise pristine lenses, or, in the absence of such lenses, image stimuli that lack fine detail.

            Another difficulty faced in this experiment was determining the focal length of the lenses. The fluorescent laboratory lights were not sufficient to find the focal light, nor was the weak winter sunlight.  Prior knowledge that the lenses had focal lengths of 10mm, 200mm and 750mm meant that a desk lamp could be used to determine which lenses were 10mm and 200mm, then the two leftover lenses were assumed to have focal lengths of 750mm. The use of lenses which have their focal lengths marked on them would improve the accuracy of this part of the experiment, and improve efficiency.

Results and Analysis

11. Elimination of vertical lines from a grid pattern.

            A widely spaced grid pattern was chosen and projected onto the screen. In order to eliminate the vertical lines, a filter, which was opaque except for a transparent vertical line down the centre, was placed between the two f=750 lenses, in the location shown in Figure 6. The image projected onto the screen was of parallel horizontal lines; the vertical lines of the grid had been eliminated. The grid, filter and resulting image are shown in Figure 7, below.

12. Elimination of horizontal lines from a grid pattern.

            A more closely spaced grid pattern was chosen and projected onto the screen. In order to eliminate the horizontal lines, a filter, which was opaque except for a transparent horizontal line down the centre, was placed between the two f=750 lenses, in the location shown in Figure 6. The image projected onto the screen was of parallel vertical lines; the horizontal lines of the grid had been eliminated. The grid, filter and resulting image are shown in Figure 8, below. There was unfortunately some blurring of the output image due to smudges on the lens, but it was nevertheless apparent that the chosen filter had had the expected effect.

13. Creation of a relief-type image.

            The stimulus slide prescribed for this procedure was an image of Albert Einstein. As discussed previously, attempts to produce a Fourier transform with this slide were unsuccessful as smudges on the lenses caused blurring of the image that entirely distorted it, and made it impossible to tell the difference between the filtered and unfiltered image. However, it is hypothesised that an optical equivalent of the ‘highpass’ filter used in the computerised equivalent of this procedure would produce a relief image of Einstein, if the lenses were in better condition. Such a filter would be transparent except for a small opaque circle in the centre of the filter, as shown in Figure 9.

In order to create a relief image, the low spatial frequency components of the image must be filtered out. These components are centred around the DC point of the Fourier transform, that is, in the centre. Therefore, an opaque dot in the centre of the filter will prevent the transmission of the low spatial frequency components, leaving an image that has only high spatial frequency components. Such an image will be comprised of the edges of the original image, ie. it will be a relief type image.

14. Elimination of horizontal lines from an image.

            As in Part 13, the resulting image was too blurry to tell if the filter chosen was the correct one. However, it is expected that a filter similar to that used in Part 12 (ie. a filter that would remove periodic components changing vertically) would be effective.

Comparison of Computerised and Optical Fourier Filtering

15. Comparison of the two methods.

            Optical Fourier filtering is advantageous in the study of Fourier transforms as it allows the student to physically manipulate the filters and hence to gain a better understanding of how Fourier filtering works. It is also useful in free-space optics, where it can be used to reduce noise, allowing more efficient transmission of information. In medical imaging, the major advantage of optical Fourier filtering is its speed – images can be processed in real time, which is often important in a diagnostic setting (Li, Chen, Zhang Qian, & Liu, 2002). However, optical filtering requires a reasonable amount of equipment (lenses, appropriate filters, a collimated laser…) and the quality of the resulting image is limited by the quality of the equipment and the accuracy of the setup. Lens aberrations, in particular, can introduce errors.

Computerised Fourier filtering may be less comprehensible to the student, but it removes all the problems associated with poor equipment and indeed can be performed on any computer equipped with the correct program. The fact that the output is on a computer rather than a screen is also a practical advantage, as if one is to go to the trouble of filtering an image then one presumably wants to use the image, and having it on a computer would be useful in many cases, particularly for graphics applications. However, computerised Fourier transformation does not occur in real-time; though modern computer programs can rapidly perform Fourier transforms, medical applications require such high resolution that computer transforms cannot be performed at speeds close enough to real-time. This requirement for high resolution also provides image storage problems, whereby the resulting files may be so large as to be impractical. Also, computerised Fourier transforms may be inaccurate due to warping and aliasing effects (Li et al., 2002).

16. Uses of computer based Fourier transformations.

            Fourier transformations are built in to many photo editing programs, such as Adobe Photoshop. These computer based Fourier transformations can be used to alter digital images, for example to remove shadow lines from a photo or to create a striking effect by leaving only the glowing edges of an image. Computer based Fourier transformations can also be used to remove unwanted frequencies from audio recordings, for example to filter out a constant sound caused by a stuck key on a pipe organ in a live recording. In the field of finance, computer based Fourier transforms (specifically fast Fourier transforms) are used in the valuation of stock options (Carr & Madan, 1999).

17. Uses of optical Fourier transformations.

The greatest practical advantage of optical Fourier transformations over digital is speed. Optical transformations are instantaneous and, assuming good equipment and setup, provide high accuracy; as such it would be advantageous to use the optical setup rather than a computerised version in situations where it is important that a highly accurate result is obtained immediately. Many medical imaging techniques, such as fluoroscopy, require both accuracy and speed, and it is these characteristics of optical Fourier transforms that ensures that they are still used today, despite the availability of digital alternatives (Li et al., 2002).

Conclusion

            Both optical and computerised Fourier transforms have important applications in a wide range of fields. In this experiment, the computer-based procedures were more successful due to issues with the components of the optical setup, but it is clear from the theory and the surrounding literature that optical procedures can be very effective.

References

Carr, P., &  Madan, D. (1999). Option valuation using the fast Fourier transform. Journal of Computation Finance, 2, 61-73.

Hecht, E. (2002). Optics (4^th ed.). San Francisco, CA: Addison Wesley.

Li, Y., Chen, W., Zhang, Y., Qian, W., & Liu, H. (2002). Comparison of analog and digital Fourier transforms in medical image analysis. Journal of Biomedical Optics, 7(2), 255-261.