I understand the math behind what's happening here but I'm still trying to develop the intuition for why the output image looks this way.
I assume that the band-pass filter "masks" the frequencies of the image. I've played around with this in the google collab notebook but I'm still having a hard time relating what it means to restrict the frequency in the end image. It seems that restricting the frequencies leads to a blur as there's less differentiation between different pixels. This makes sense in the sense "frequencies $\approx$ details" but I still don't understand what frequencies visually reflects in the source image, if anything.
mpotoole
The frequencies represent how pixel values change spatially across the image. The position of every Fourier coefficient (complex value in the bottom-left image) determines the period and orientation of the corresponding sinusoidal-like pattern, as shown on this slide. (Really, it's a complex exponential, but I'll simplify for now and just call it a sinusoid).
As we covered earlier in this lecture, an image can be modeled as the sum of these sinusoidal patterns. This includes both high-frequency and low-frequency patterns. By applying a band-pass filter to an image, we control which frequencies will be used to reconstruct that image.
A low-band-pass filter allows only low-frequencies through. Regions in the image where pixel values change slowly can be represented as a sum of low-frequency sinusoid patterns. However, regions in the image where pixel values change quickly (e.g., around edges or corners) cannot be reproduced with low-frequency sinusoids. These regions will therefore look smoothed out.
A high-band-pass filter allows only high-frequencies through, and so you get the opposite effect. Regions in the image where pixel values change quickly can be captured accurately (e.g., edges). However, high-frequencies cannot accurately capture flat or slowly-varying regions in the image. Instead, you will get high-frequency patterns emerge from your image.
I understand the math behind what's happening here but I'm still trying to develop the intuition for why the output image looks this way.
I assume that the band-pass filter "masks" the frequencies of the image. I've played around with this in the google collab notebook but I'm still having a hard time relating what it means to restrict the frequency in the end image. It seems that restricting the frequencies leads to a blur as there's less differentiation between different pixels. This makes sense in the sense "frequencies $\approx$ details" but I still don't understand what frequencies visually reflects in the source image, if anything.
The frequencies represent how pixel values change spatially across the image. The position of every Fourier coefficient (complex value in the bottom-left image) determines the period and orientation of the corresponding sinusoidal-like pattern, as shown on this slide. (Really, it's a complex exponential, but I'll simplify for now and just call it a sinusoid).
As we covered earlier in this lecture, an image can be modeled as the sum of these sinusoidal patterns. This includes both high-frequency and low-frequency patterns. By applying a band-pass filter to an image, we control which frequencies will be used to reconstruct that image.
A low-band-pass filter allows only low-frequencies through. Regions in the image where pixel values change slowly can be represented as a sum of low-frequency sinusoid patterns. However, regions in the image where pixel values change quickly (e.g., around edges or corners) cannot be reproduced with low-frequency sinusoids. These regions will therefore look smoothed out.
A high-band-pass filter allows only high-frequencies through, and so you get the opposite effect. Regions in the image where pixel values change quickly can be captured accurately (e.g., edges). However, high-frequencies cannot accurately capture flat or slowly-varying regions in the image. Instead, you will get high-frequency patterns emerge from your image.