0.3 Signal processing in processing: sampling and quantization  (Page 3/4)

2-d: images

Let us assume we have a continuous distribution, on a plane, of values of luminance or, more simply stated, animage. In order to process it using a computer we have to reduce it to a sequence of numbers by means ofsampling. There are several ways to sample an image, or read its values of luminance at discrete points. Thesimplest way is to use a regular grid, with spatial steps $X$ e $Y$ . Similarly to what we did for sounds, we define the spatial sampling rates $F_{\mathrm{X}}=\frac{1}{X}$ and $F_{\mathrm{Y}}=\frac{1}{Y}$ . As in the one-dimensional case, also for two-dimensional signals, or images, sampling can bedescribed by three facts and a theorem.

• Fact 1 The Fourier Transform of a discrete-space signal is a function (called spectrum ) of two continuous variables ${\omega }_{\mathrm{X}}$ and ${\omega }_{\mathrm{Y}}$ , and it is periodic in two dimensions with periods $2\pi$ . Given a couple of values ${\omega }_{\mathrm{X}}$ and ${\omega }_{\mathrm{Y}}$ , the Fourier transform gives back a complex number that can be interpreted as magnitude andphase (translation in space) of the sinusoidal component at such spatial frequencies.
• Fact 2 Sampling the continuous-space signal $s(x, y)$ with the regular grid of steps $X$ , $Y$ , gives a discrete-space signal $s(m, n)=s(mX, nY)$ , which is a function of the discrete variables $m$ and $n$ .
• Fact 3 Sampling a continuous-space signal with spatial frequencies $F_{\mathrm{X}}$ and $F_{\mathrm{Y}}$ gives a discrete-space signal whose spectrum is the periodic replication along the grid of steps $F_{\mathrm{X}}$ and $F_{\mathrm{Y}}$ of the original signal spectrum. The Fourier variables ${\omega }_{\mathrm{X}}$ and ${\omega }_{\mathrm{Y}}$ correspond to the frequencies (in cycles per meter) represented by the variables $f_{\mathrm{X}}=\frac{{\omega }_{\mathrm{X}}}{2\pi X}$ and $f_{\mathrm{Y}}=\frac{{\omega }_{\mathrm{Y}}}{2\pi Y}$ .

The [link] shows an example of spectrum of a two-dimensional sampled signal. There, thecontinuous-space signal had all and only the frequency components included in the central hexagon. The hexagonalshape of the spectral support (region of non-null spectral energy) is merely illustrative. The replicas of the originalspectrum are often called spectral images .

Spectrum of a sampled image

Given the above facts , we can have an intuitive understanding of the Sampling Theorem.

Sampling theorem (in 2d)

A continuous-space signal $s(x, y)$ , whose spectral content is limited to spatial frequencies belonging to the rectangle having semi-edges $F_{\mathrm{bX}}$ and $F_{\mathrm{bY}}$ (i.e., bandlimited) can be recovered from its sampled version $s(m, n)$ if the spatial sampling rates are larger than twice the respective bandwidths (i.e., if $F_{\mathrm{X}}> 2F_{\mathrm{bX}}$ and $F_{\mathrm{Y}}> 2F_{\mathrm{bY}}$ )

In practice, the spatial sampling step can not be larger than the semi-period of the finest spatial frequency (or thefinest detail) that is represented in the image. The reconstruction can only be done through a filter thateliminates all the spectral images but the one coming directly from the original continuous-space signal. In otherwords, the filter will cut all images whose frequency components are higher than the Nyquist frequency defined as $\frac{F_{\mathrm{X}}}{2}$ and $\frac{F_{\mathrm{Y}}}{2}$ along the two axes. The condition required by the sampling theorem is equivalent to requiring that there are no overlaps between spectral images. If there were suchoverlaps, it wouldn't be possible to eliminate the copies of the original signal spectrum by means of filtering. In case of overlapping, a filtercutting all frequency components higher than the Nyquist frequency would give back a signal that is affected byaliasing.