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.