I can't really see how a convolution with the 1 pixel box blur + sampling equals the average value inside the pixel. The 1 pixel box blur in the frequency domain is a sinc function so it's not exactly a perfect LPF, so wouldn't some mid-range frequencies not exactly be equal to the average value of f(x, y) in the pixel? I would love to see a more mathematical approach if possible.
Staffyirenng
@stexus Great comment! Let me separate two different issues you raised.
It IS true that the two options here are mathematically identical. Perhaps someone could sketch a proof or give the intuition about why this is so. Hint: what is the sample value at one pixel at the end of Option 1, in terms of the surrounding f(x,y) values?
You are right that the 1-pixel box blur is not a perfect low-pass filter, because it is a sinc function in the frequency domain. Yes, ti will leak some mid-range frequencies as you say. This is shown on the previous slide, where on the right the non-negative frequencies extend far from the origin, beyond the Nyquist rate.
Here are two comments about this situation.
First, in spite of the imperfections you mentioned, the 1-pixel box blur produces good looking results and is quite practical to approximate, as we have you do in the first assignment. From a practical standpoint, it is actually a reasonable low-pass filter for many applications.
Second (this comment goes beyond what students will be responsible for in class), a "perfect" low pass filter would, I argue, look like a box function in the frequency domain centered at the origin with width equal to twice the Nyquist frequency -- do you agree? What is the spatial domain version of this? It is the inverse Fourier transform of a box function, which is a sinc function (appropriately dilated and scaled). So in theory, the perfect low-pass filter in the spatial domain would be a sinc rather than a box. The challenge is the the sinc is not practical to implement because it has infinite extent. But certainly one can approximate it. I make a few comments about this and link to reading in a comment on a later lecture slide.
ChangyiYang
I am confused here in the lecture. In my mind, 1 pixel is the minimum unit in the picture. So how can we use a 1-pixel box blur? Then I realized that the previous sample misled me. In the previous sample, the cute baby picture should not be considered as a 'picture' consisting of many isolated pixels. Instead, it represents some 2D continuous function.
I can't really see how a convolution with the 1 pixel box blur + sampling equals the average value inside the pixel. The 1 pixel box blur in the frequency domain is a sinc function so it's not exactly a perfect LPF, so wouldn't some mid-range frequencies not exactly be equal to the average value of f(x, y) in the pixel? I would love to see a more mathematical approach if possible.
@stexus Great comment! Let me separate two different issues you raised.
It IS true that the two options here are mathematically identical. Perhaps someone could sketch a proof or give the intuition about why this is so. Hint: what is the sample value at one pixel at the end of Option 1, in terms of the surrounding f(x,y) values?
You are right that the 1-pixel box blur is not a perfect low-pass filter, because it is a sinc function in the frequency domain. Yes, ti will leak some mid-range frequencies as you say. This is shown on the previous slide, where on the right the non-negative frequencies extend far from the origin, beyond the Nyquist rate.
Here are two comments about this situation.
First, in spite of the imperfections you mentioned, the 1-pixel box blur produces good looking results and is quite practical to approximate, as we have you do in the first assignment. From a practical standpoint, it is actually a reasonable low-pass filter for many applications. Second (this comment goes beyond what students will be responsible for in class), a "perfect" low pass filter would, I argue, look like a box function in the frequency domain centered at the origin with width equal to twice the Nyquist frequency -- do you agree? What is the spatial domain version of this? It is the inverse Fourier transform of a box function, which is a sinc function (appropriately dilated and scaled). So in theory, the perfect low-pass filter in the spatial domain would be a sinc rather than a box. The challenge is the the sinc is not practical to implement because it has infinite extent. But certainly one can approximate it. I make a few comments about this and link to reading in a comment on a later lecture slide.
I am confused here in the lecture. In my mind, 1 pixel is the minimum unit in the picture. So how can we use a 1-pixel box blur? Then I realized that the previous sample misled me. In the previous sample, the cute baby picture should not be considered as a 'picture' consisting of many isolated pixels. Instead, it represents some 2D continuous function.