# What is $F(0)$ is “dc” component in the context of image processing?

It has always been said that $F(0)$ is the "DC component" in fourier transform. However, I don't get what it means to say that $F(0)$ is "DC" in the context of image processing.

The zero in this case just meant zero frequency, and hence no changes isn't it? Then in the context of image processing, how should I imagine the picture to be like at $F(0)$? What exactly is that "DC" component?

• In the context of an image the "DC" component is just the average (arithmetic mean) of the entire image. – Paul R Mar 12 '13 at 13:58
• The term "DC" comes from the field of electrical engineering, where "DC" refers to direct current, or current flow that doesn't cycle periodically like alternating current (AC). Thus, DC is stated to have zero frequency, hence the association with the Fourier transform component at $f=0$. – Jason R Mar 12 '13 at 14:17
• An image that is entirely black has all frequency components 0, including DC. An entirely white image has value 1.0 at DC, and 0 for all other frequencies. A completely gray image has 0.5 at DC, and 0 at all other frequencies. – endolith Mar 12 '13 at 16:26

The zero frequency of an image DCT is the mean gray value of the pixels of the input image (for graylevel images). $F(0)$ is not an image: it is a single coefficient.
In the Fourier series, $x(t) = \sum\limits_{-\infty}^{\infty} C_n e^{jnwt}$ where $C_n =\frac{1}{T} \int\limits_0^T x(\lambda)e^{-jnwt} d\lambda$. $T$ is the period, $w$ the frequency, $j = \sqrt{-1}$.
So if we plug in $n=0$, we get the following: $x(t) = \sum\limits_{-\infty}^{\infty} C_n e^{0}$, with $C_n =\frac{1}{T} \int\limits_0^T x(\lambda)e^{0} d\lambda$. Any number raised to the $0$th power is $1$, so $C_n =\frac{1}{T} \int\limits_0^T x(\lambda) d\lambda$. This is exactly how we define the average value. So in image processing, $F(0)$ corresponds to the average value of all the pixels.