Fourier transforms in image processing (Maths Relevance)

Hi my name is Michelle and I’m an
academic at Swinburne. A lot of my research is in image processing which is
using computers and computer programs to interpret images. Computers don’t have
the same kind of knowledge about the world that you and I do.
For instance, unless they are specifically programmed to do so, they
couldn’t identify what is in this picture. Whereas you and I can
immediately identify it as an aeroplane however computers can pick up on a lot
of details and nuances that we as humans might miss. For instance a computer
program can identify the differences between these images, whereas we would really struggle with it. One of the tools that we use in image processing is the
Fourier transform. The Fourier transform is an extension of the Fourier series
which we use for periodic continuous signals. The Fourier transform can take
care of non periodic signals like this white noise. One of the things that we
are often interested in with image processing is reading text. The fact is
that while humans are very good at reading handwritten and printer texts,
computers really struggle with it. This is the reason that some websites use
these capture images. These allow the website to make sure that it is dealing
with a real human instead of a computer or a robot. Having a computer read text
is useful for the post office, for transferring books to electronic copies,
for number plate recognition and for automating the input of handwritten
forms. One of the methods that is used for recognising characters is to use the
Fourier transform. When we apply a transform to something, we retain the
information that is captured in the original data but we display it in a
different way. It’s the same as taking these waveforms which are displayed in
Cartesian space and displaying them using a log scale instead. All the
information is still there it is just easier to interpret it in this new form.
If we take the Fourier transform of a signal, we move it from the time domain
to the frequency domain a step function in the time domain becomes a sync
function in the frequency domain. If we move back to the time domain, you
would still get back to the original data. As a quick aside, the sync function
in the time domain becomes a step function in the frequency domain. But a
word of warning it doesn’t work like that for all signals. We can also move
images into the frequency domain. This image of parallel graduated lines
becomes two white dots on a black background in the frequency domain. This
white circle becomes a pattern that looks like ripples from a stone thrown
in the water. And this white square on a black background becomes a cross. The
more you look at the patterns in the frequency domain caused by these simple shapes the more you can predict what might happen if you change the design.
For instance if I change the spacing between the graduated lines, the dots in
the frequency domain move apart. And if I rotate the lines the dots rotate as well.
We can also take the Fourier transform of more complex images . This image is
very famous in the image processing community and here it is in the
frequency domain. The lines in the frequency domain image correspond to the legs of the tripod and the line of the horizon. What researchers have found is
that we can actually exploit the lines and circles and their corresponding
patterns in the frequency domain to use it to recognize characters. Let’s look at
the letter A for example. Pretty much however you draw it it has to upright or
close to upright lines and a crossbar at the bottom. When we transfer these to the frequency domain, they all end up looking quite similar. And they definitely look
more similar to one another than the B or a C or a . In fact, we can use the
representation of written numbers and letters in the frequency domain to
identify what letter or number is in the time domain. And so we use the Fourier
transform to identify letters and numbers. We can also use the frequency
domain to remove noise from an image for instance in this case the image has been
corrupted with diagonal lines. These lines appear as very bright spots and
circles in the frequency domain. So we can remove them using a filter. This is
the filter applied and this is the image with the noise
removed. We can also remove white noise or speckle by removing the low frequency
components of the signal in the frequency domain. These sit at the
extremities of the image. So we only keep the signals in the middle. This is the
filter applied. And this is the result. While it may look a bit blurry to you
and I, there is now no noise so the computer can make better judgments about the image. I’ve only shown you a few examples, but you can see from this short
presentation just how useful the Fourier is for image processing and analysis. And
once you understand the basics, it isn’t too difficult to apply.

18 Replies to “Fourier transforms in image processing (Maths Relevance)

  1. Amazing presentation. I wish you also had explained how periodic behavior can be seen in the frequency domain because that is what I want to learn now but well, we don't always get what we want. 😛

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