Imaging and Signalling

Imaging

Converging lenses

Light can be modelled using rays or wavefronts. which are perpendicular to the direction of wave motion. Wavefronts have curvature, unless they come from a very distant source, in which case they are plane wavefronts.

Curvature is given by:

$$\text{curvature}=\frac{1}{\text{radius}}$$

As the radius approaches infinity, the curvature approaches zero.

Converging lenses focus light onto a point behind them, called the principal focus. The principal focus lies a distance $f$, the focal length, away from the lens. A converging lens has a power (measured in Dioptres, $\text{D}$) of $\frac{1}{f}$, which means the lens adds a curvature of $\frac{1}{f}$ to wavefronts passing through it.

The Lensmaker’s Equation shows how a converging lens works:

$$\begin{array}{rl}\text{curvature of waves leaving lens}& =\text{curvature of waves entering}+\text{power of lens}\\ \frac{1}{v}& =\frac{1}{u}+\frac{1}{f}\end{array}$$

Where $v$ is image distance, $u$ is object distance, and $f$ is the focal length. By the Cartesian sign convention¹, $u$ is negative, while $v$ and $f$ are positive.

Lenses can also magnify objects. Linear magnification is given by:

$$\text{linear magnification}=\frac{\text{image height}}{\text{object height}}=\frac{\text{image distance}}{\text{object distance}}=\frac{v}{u}$$

Images

Digital cameras contain a microchip called a CCD (charge-coupled device). The CCD has a screen with millions of pixels, which store charge when light falls on them, storing an amount of charge proportional to the intensity of incident light. A pixel is the smallest unit of a single image, with a single colour. The charges are converted to an array of numbers which represents an image.

Information is stored in binary, using bits with one of two values: 0 and 1. The number of bits, $b$ needed to represent $N$ different alternatives are related by:

$$N=2^{b}b={\log }_{2}N$$

The resolution of an image is the smallest distance between two points which can be distinguished:

$$\text{resolution}=\frac{\text{width / height of image}}{\text{number of pixels}}$$

The amount of information in an image (in bits) is given by:

$$\text{amount of information in image}=\text{no. of pixels}\times \text{bits per pixel}$$

Manipulating images

Computer images are stored as an array of numbers. They can be manipulated in the following ways to enhance the image:

• Varying brightness: adding a fixed number to the value of each pixel.
• Increasing contrast: multiplying the value of each pixel by a fixed number so the range of pixel values is greater.
• Removing noise: replacing the value of each pixel with the mean / median of the 8 pixels immediately around it.
• Edge detection: subtracting the average value of the 8 pixels around a pixel from the pixel value.
• False colour: assigning colours different from the true colours, commonly used for displaying images captured with parts of the electromagnetic spectrum outside visible light.

Digital and analogue signals

Analogue signals continuously vary between values, whereas digital signals can only take discrete values. Analogue signals are converted to digital signals by sampling. The signal is sampled at regular intervals, with a measurement of the amplitude taken and rounded to the nearest quantisation level. This produces a quantisation error equal to the difference between the actual value and the quantisation level.

Advantages of digital signals:

• More resistant to noise
• Easier to send / store / receive with digital systems
• Faster transmission
• Can be compressed easily.

Disadvantages of digital signals:

• Quantisation error causes loss of detail.

The maximum number of useful quantisation levels is given by:

$$\text{Maximum no. of useful quantisation levels}=\frac{V_{\text{total}}}{V_{\text{noise}}}$$

The number of bits required is given by:

$$b={\log }_2\left(\frac{V_{\text{total}}}{V_{\text{noise}}}\right)$$

When sampling, the minimum sampling rate must be at least twice the highest frequency in the signal (Nyquist’s Theorem), otherwise aliasing will occur, producing spurious low frequency signals.

Bit rate is the rate of transmission of digital information. It is given by:

$$\text{Rate of transmission of information}=\text{samples per second}\times \text{bits per sample}$$

Polarisation

Electromagnetic waves are transverse waves, consisting of oscillating magnetic and electric fields at right angles to each other. Unpolarised waves have oscillations in all perpendicular directions². A polarising filter can be used to polarise transverse waves, which makes them only oscillate in one plane.

To test if a wave is polarised, it can be observed through a polarising filter. For an unpolarised wave, the intensity should remain constant as the polariser is rotated, whereas for a polarised wave, the intensity will vary sinusoidally. There is $90°$ between the maximum and minimum intensity.

NIS: The received intensity is given by $I=I_0{\cos }^2\theta$, where $I_0$ is the incident intensity and $\theta$ is the angle between the polariser and the incident polarisation direction. This means $I$ is at a maximum for $\theta =0°,180°$, and at a minimum for $\theta =90°,270°$. Intuitively, this can be thought of as the bars of the polarising ‘blocking’ the oscillations.

Footnotes

1. Footnote on the Cartesian sign convention This is mentioned explicitly in the specification. ↩

2. Footnote on wording of polarisation This wording is mildly confusing. In a polarised EM wave, each field of the wave oscillates in only one plane; the magnetic field oscillates in only one plane and the electric field oscillates in only one other plane (at right angles still) ↩