# The difference of downsampling an image and smoothing an image?

I am, at the moment, trying to read up on some simple computer vision elements, in which i have become a bit comfused on the terms downsampling and smoothing, and whether there is a difference between those two terms.

I know that downsampling means to reduce the resolution (DPI) of an image to a lower value. Which will result in a smaller image compared to the original.

But what about smoothing? Or convolution? Would it not result in the same thing? Or am i misunderstanding something?

• Your question has beeen answered. Do not hesitate to vote for the useful ones and accept the most suitable – Laurent Duval Feb 9 '17 at 17:08

## 2 Answers

Downsampling is the process of properly discarding every $(M-1)$ samples out of $M$ to reduce the signal sample rate by $M$. To avoid aliasing the signal being downsampled is initially lowpass filtered at a proper cutoff frequency before compression. The effect is a change of sample rate. Also the spectral positions of low frequencies are stretched to higher frequencies towards $\omega = \pi$. As a consequence, any spectra that passes beyond $\omega = \pi$ becomes aliased into mistreated bands, hence the requirement of a pre-processing lowpass anti-aliasing filter.

Smoothing is a more general thing. Whereas it may naturally refer to lowpass filtering in general, it may also mean more complicated things such as Kalman estimators which are also defined as smoothing operators. In the former case smoothing refers to getting rid of high frequency signal components, whereas in the latter case it refers to getting rid of randomness (usually of high frequency).

• Wanting to add that this applies to any signal, be it 1D (e.g. time-series data) or 2D (e.g. images). – Marcus Müller Jun 16 '16 at 20:25

Smoothing consists in reducing irregularities in an image, so that it becomes... smoother. Smoothing is not a very precise term, as irregularity. It is generally performed by replacing each pixel value $y_i$ by a function on a subset $L_i$ (that may depend on $i$) of other pixels $\hat{y}_i=f(x_l| l\in L_i)$. Generally, $\min x_l \le \hat{y}_i \le \max x_l$, but this is not a sufficient characterization. In standard smoothing, $L_i$ surrounds $y_i$. In nonlocal smoothing, you can take pixels everywhere, and weight them according to the similarity of these patches with respect to the patch around $y_i$.

If you have a measure of irregularities, for instance based on a norm of derivatives computed on pixels, this measure should be reduced after applying smoothing.

Downsampling consists in replacing each subset $L_i$ of pixels by only one value $\hat{y}_i$ taken in the values of the pixels in $L_i$, where all the subsets are a "partition" of the image: the union of the $L_i$ give the whole image, and their two-by-two union is empty.

Globally, smoothing yields an image with the same number of pixels and less variations. Downsampling yields an image with less pixels, and generally more variations.