Use of Mathematics in Photography
Introduction
Photography is an art or a process through which still or moving pictures are created by recording radiation on a radiation-sensitive object such as electronic sensors and photographic films. Radiation is used extensively in the form of ultra violet, visible and near-IR rays of the spectrum. The lay man’s term for radiation is light such that the light that is emitted by a real object is lands on a sensitive film on the camera or a FPA pixel array sensor via a lens and it forms a real image after being exposed to the object for some time. Certain mathematical figures and descriptive terms are used to define the process through which light moves to convert the object into an image hence the essence of this paper. This depicts a close relationship between mathematics and photography as mathematical figures, as well as, formulas is used to explain the processes involved in the art.
Discussion
Components of the camera
There are different types of cameras all of which are used for the same purpose of recoding still and motion pictures which are later on processed to the most appropriate form. A camera as an image forming device uses photographic film or a silicon electronic image sensor as a sensing medium to record the intended details (Stroebel, 2000, et.al).
The aperture (given by the f-number) is a component of the camera which is useful at adjusting the lens opening hence controlling the amount of light that reaches the sensor. The diameter of the opening is determined by the amount of light that is required by the photograph such that the less the opening the greater the depth of the field and the more the light results into blurriness. The lens opening is measured as f-number where dividing the f-number by the focal length yields the aperture diameter more precisely. The focus in this case is the most accurate adjustment that allows the photographer to generate the sharpest focus on the desired object (Ang, 2002).
The shutter speed is the next important part of the camera which adjusts the speed of the shutter in an attempt to control the amount of time taken for the sensor to absorb the light when exposed. This speed is expressed in terms of fractions of seconds or at times as an angle of the mechanical shutter. The speed of the shutter is used to control the amount of light striking the imagining medium such that faster shutters reduces the amount of exposed light thus results in more clearer images being formed (Ang, 2002).
Digital cameras have white balances which provide electronic compensation for color temperature thus ensuring that the white light is recorded on the imaging chip hence making the colors on the frame natural. Film-based cameras have a film stock which is determined by the operator who adjusts color as deemed fit (Stroebel, 2000).
Metering is the other component of a camera which is used to measure exposure of highlights and shadows such that these two vital elements of the image are exposed as per the wishes of the photographer. However, in order for the camera to translate the amount of light to appropriate shutter and aperture speed, the meter has to undergo adjustments for sensitivity of the sensor light. This adjustment is carried out by the film speed whose numerical capacity affects the sensitivity of the sensor to light prior to recording (Lenman, 2005).
Applying mathematics to the photography
The aperture
There are wide varieties of apertures in photography based on their normal sizes ranging from f/2-f/4 for widest lens and f/11-f/22 being the smallest. Specifications of lens usually depict maximum and minimum apertures such that for the most popular aperture is used in description of the lens as the lens speed which has direct implications on the exposure time is connected to the aperture. This makes the size of the aperture being directly proportional to the square root of accepted light and at the same time it is inversely proportional to the square root of exposed time. Therefore, faster lens are those wider than f/2.8 thus reducing the amount of light that gets to the film sensor (Lenman, 2005).
When calculating the area of the aperture, the following formula is used;
Where f is the focal length while N is the number designated to f such that a lower f-number will lead to greater aperture area hence more light is allowed into the image sensor. Similarly, a larger f-number will result into a smaller aperture area and less light will be illuminated to the sensor resulting into blurred images.
Consequently, control of the aperture involves switching from working aperture and full aperture. Sometimes a preset aperture is used to ensure that the camera can revert from full to working aperture immediately before exposure even without the operator’s interventions. However, cameras without the aperture are preset using a button which puts on the full aperture after capturing an image (Stroebel, 2000).
Shutter speed and exposure in photography
Shutter speed is used in photography to measure the amount of exposure time and it the total exposure time is proportional to the duration of light reaching the image sensor. The shutter speed in collaboration with the aperture (f-number) is used to determine the amount of light that reaches the image sensor. The units of measurement used for measuring the exposure are called exposure value and are indicated by “EV” also called stops. When the aperture is doubled and the shutter speed is halved increases the exposure factor which is explained thus; 1/125s and f/8 are the same as 1/500s and f/5.6 respectively. Hence EV is dependent on the degree of aperture opening and speed of the shutters at the time of exposure. This leads to the next vital component of photography without which objects could not be recorded as light is illuminated from objects hence making them visible (Lenman, 2005).
Color in photography
Color is a characteristic of visible light that allows photographers to record objects in their cameras regardless of the nature of hue. The most important feature of color is its temperature which is stated in units of absolute temperature as Kelvin (K). Colors with high temperatures such as blue and green have a temperature of 5,000K thus they are referred to as cool colors. Those with lower color temperatures such as yellow have 2,700-3,000K and are referred to as warm colors. The sun is the most common source of illumination and daylight color temperature goes up to a maximum of 6,000K making it the most conspicuous with a wider spectrum of visible light (Ang, 2002).
The luminous exposure of objects is determined by the accumulated physical quantity of visible light energy which is applied to a surface during the defined exposure time. The formula used to relate H=Et
Where ‘H’ is given as the luminous exposure measured in lux seconds, ‘E’ is the image-plane illuminance measured in lux and‘t’ is the time of exposure measured in seconds.
The quantity of light measured is dependent on the spectrum of light while others are sensitive to “invisible” light such as infrared and ultra violet rays. However, the amount of time and light required to determine exposure is dependent on the sensitivity of the medium used in the camera. For non-digital cameras, sensitivity is referred to film speed and is measured on scales prepared by the International Organization for Standardization (ISO), such that faster films require less exposure time as their ISO rating is higher. Consequently, slower shutter periods ensures that the medium is exposed for a longer period of time used together with greater lens apertures which admit more light result into substantial exposures which are mandatory for quality photography (Stroebel, 2000).
An example of applying the illumination is exemplified whereby the correct exposure on a sunny day using the ISO is given as thus: using a 100film, an aperture of f/16 and a shutter speed of 1/100th of a second gives the sunny 16 rule which states that an aperture of f/16 on a sunny day and a suitable shutter speed yields one over the film speed (1/100).
Film speed
Film speed is a measure of photographic film’s sensitivity to light such that insensitive films with lower speed indices require more exposure time to light to produce same results as highly sensitive film. Fast films which are practically sensitive result in reduced image quality such that film grain is made finer while the image noise is lowered considerably. Currently, the ISO standard 12232:2006 is used to measure the film speed in digital cameras (Lenman, 2005).
The formula used to measure standard film speed is give by:
Where ‘H’ is the illumination exposure, ‘t’ is time of exposure, ‘q’ is a constant given as 0.65, ‘N’ is the f-number of the aperture and ‘L’ is the speed of the film (Lenman, 2005).
Conclusion
Mathematics is used widely by photographers especially those working towards keener understanding of the inner operations of the camera. This includes a broad array of elements such as formulas, figures and S.I units which are part of the entire photographing stage. Various aspects of the entire preparations and recording of images are dependent on the photographer’s capacity to apply mathematical knowledge on the process to ensure that high quality still, as well as, motion pictures is obtained.
References
Ang Tom . Dictionary of Photography and Digital Imaging: The Essential Reference for the Modern Photographer. Watson-Guptill (2002)
Lenman Robin, The Oxford Companion to the Photograph, OxfordUniversity Press 2005
Stroebel Leslie, Compton John, Current Ira, and Zakia Richard. Basic Photographic Materials and Processes, second edition. Boston: Focal Press, 2000.
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