Fourier Transform, Discrete Forier Transform image processing

In summary, the conversation discusses the concept of the Fourier Transform and its applications in physics and image processing. The speaker is struggling to understand the continuous FT and its interpretation in terms of linear coefficients. They also question the meaning of frequency in an image and how it relates to matrix algebra. The conversation ends with a discussion about the usefulness of frequency in the context of sampled data and the different scales that can be used to present an image of a 2-D Fourier transform.
  • #1
joshthekid
46
1
Hi all,

Now naturally after completing a physics degree I am very familiar with the form and function of the Fourier Transform (FT) but never have grasped it quite conceptually. I understand that given a function f(x) I can express every functional value as a linear combination of complex sinusoidal functions as in the typical illustration of a square wave and you keep adding sine waves of higher and higher frequencies to get an approximate square wave (that would approach the exact with an infinite number of frequencies). Here is what I struggle with in terms of the continuous FT. When I compute a FT of function I get another function F(k), where k is frequency. I can plot this like any other function but what am I plotting? I think, but I am not positive, that the value of F(k) is the summation of all the linear coefficients C(k) for a particular frequency over all values f(x)? Is this correct?

Second, the real reason I am trying to get everything straight in my head is because I am doing image processing for my research and when I look at the FT, as an image, I am not sure what I am looking at. I know that it represents spatial frequencies in the image, i.e if you make a interference band you get a dot in the center and two dots spaced equidistant from the center. Now, image processing basically comes down to matrix algebra because an image in nothing more than a matrix of intensity values, I(x,y). So when I do the FT I am doing matrix algebra of the image with a matrix composed of several different sinusoidal complex functions. Here is where I am confused, what constitutes a frequency in an image? On a pixel basis, say I have a white dot followed by a black dot pattern that would constitute a certain frequency, I think. However, how does it look from a linear algebra standpoint?

If I have a i by j image A, how do I interpret the image of the FT? So if I follow my logic from the 1D continuous case, assuming it is the correct interpretation, looking at pixels (1,1:j) in A and taking the FT, B=TA, where T is the FT transformation matrix the first matrix element is going to be the summation of all the pixels intensities in that row with a corresponding T element. But when all is said and done and I pull up the image of the Fourier Transform on MATLAB I don't think I am looking at matrix B but instead each pixel (x,y) corresponds to frequency in the x and y directions, i.e if I look at pixel (1,1) in the FT, the intensity corresponds to "how much" a white pixel is followed by a black pixel in the image?

If you have read all this, thanks, I am hoping my interpretation is correct but just want to make sure.
 
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  • #2
If you are working with digital sampled data, you might as well measure "time" in units one sampling interval apart. You can then forget about the actual sample rate (number of samples per second) for time dependent data, and the number of pixels per inch (or per mm) in an image.

In other words, "frequency" doesn't mean very much for the maths of sampled data. "Wavelength" (measured by the number of samples) is a more useful idea.
 
  • #3
joshthekid said:
i.e if I look at pixel (1,1) in the FT, the intensity corresponds to "how much" a white pixel is followed by a black pixel in the image?

Which MATLAB function are you using? I think an image of a 2-D Fourier transform (or anything else) can be presented using various scales.
 
  • #4
Stephen,

I am using the MATLAB function FFT2() which is the fast Fourier transform which is computationally different from what I described, I am just trying to get and idea for the actual mathematics.
 
  • #5


Hello,

As a fellow scientist, I can understand your struggle with grasping the concept of Fourier Transform (FT) and its application in image processing. Let me try to provide some clarification and answer your questions.

Firstly, your understanding of the continuous FT is correct. The value of F(k) in the FT of a function f(x) represents the linear combination of complex sinusoidal functions with frequency k. In other words, it shows how much of each frequency component is present in the original function f(x). The higher the frequency, the more "oscillations" there are in the function.

In image processing, the concept is similar, except now we are dealing with a 2D function (the image) instead of a 1D function. The FT of an image represents the spatial frequencies present in the image. Just like in the 1D case, lower frequencies correspond to smoother variations in intensity, while higher frequencies correspond to more rapid changes in intensity (such as edges or textures).

Now, regarding your question about what constitutes a frequency in an image, the answer is not as straightforward. In the 1D case, a frequency can be defined as the number of oscillations per unit length. In the 2D case, it can be thought of as the number of oscillations per unit area. However, in an image, there are no physical units like length or area, so the concept of frequency becomes more abstract. In this case, we can think of frequency as a measure of how rapidly the intensity of the image changes in a particular direction (x or y).

To interpret the image of the FT, you are correct in thinking that each pixel (x,y) corresponds to a frequency in the x and y directions. The intensity of the pixel represents the strength of that frequency component in the image. So, if you see a bright spot at a particular pixel, it means that there is a strong presence of that frequency in the image. This can help in identifying important features in the image, such as edges or textures.

I hope this helps in clarifying your understanding of Fourier Transform and its application in image processing. Keep exploring and asking questions, as it is through understanding the concepts that we can truly excel in our research. Best of luck!
 

What is Fourier Transform?

Fourier Transform is a mathematical technique used to analyze the frequency components of a continuous signal or function. It decomposes a signal into its constituent frequencies, allowing for easier analysis and manipulation of the signal.

What is Discrete Fourier Transform?

Discrete Fourier Transform (DFT) is a discretized version of Fourier Transform used to analyze the frequency components of a discrete signal or function. It takes a finite sequence of data points as input and produces a finite sequence of complex numbers as output, representing the frequency components of the signal.

What is the difference between Fourier Transform and Discrete Fourier Transform?

The main difference between the two is that Fourier Transform is used for continuous signals while DFT is used for discrete signals. Fourier Transform produces a continuous spectrum of frequencies, while DFT produces a discrete spectrum.

How is Fourier Transform used in image processing?

In image processing, Fourier Transform is used to analyze the frequency components of an image. It allows for the separation of an image into its high and low-frequency components, which can then be manipulated and processed to enhance the image quality or extract specific features.

What are some common applications of Fourier Transform in image processing?

Some common applications of Fourier Transform in image processing include image denoising, image compression, and image restoration. It is also widely used in medical imaging for analysis and diagnosis, as well as in computer vision for object detection and recognition.

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