Photography
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Photography, literally writing with light, is full of mathematics even though modern auto-exposure and auto-focus cameras may seem to think for themselves. Lens design requires an intimate knowledge of optics and applied mathematics, as does the calculation of correc exposure. When mastered, the mathematics of basic photography allow artists, journals, and scientists to create more compelling and insightful images whether they are using film or digital cameras. The basic concept of using a device to project an image onto a flat surface dates from the camera obscura of ancient times, in which light passed through a small hole that focused the image and projected it in a darkened room. The modern day descendent of the camera obscura is the pinhole camera, which uses a hole without a lens to project an image onto a piece of photographic film. The quality of camera obscura images increased as lenses were developed in the sixteenth century. Still, there as no way to preserve the image except by drawing or painting on the projection screen. The discovery of photosensitive chemicals in the nineteenth century was a major step forward because it allowed images to be preserved without drawing or painting, and many different techniques were invented for creating photographs on paper, glass, and metal sheets. In 1861, Scottish physicist James Clerk-Maxwell invented system of color photography using black and white images taken through red, green, andblue filters and then combined. George Eastman starte his photographic company in 1880, and the first Kodak camera was introduced in 1888. This surge in technolog gave rise to an explosion in the technical, journalistic, and artistic use of photography as mechanical cameras an lenses were continually refined throughout the first half of the twentieth century. The advent of computeraided design in the 1960s and 1970s represented another major step forward, allowing much more sophisticated camera and lens designs, and auto-focus and auto-exposure cameras arrived on the scene shortly thereafter.
Fundamental Mathematical Concepts and Terms THE CAMERA In its simplest form, the camera is a light-tight box containing light sensitive material, either in the form of photographic film or a digital sensor. A lens is used to focus light rays entering the camera and produce a sharp image. The amount of light striking the film and sensor is controlled by shutter, or curtain that quickly opens and exposes the film or sensor to light, and the size of the lens opening, or aperture, through which light can pass.
FILM SPEED The speed of photographic film is a measure of its sensitivity to light, with high speed films being more sensitive to light than low speed films. Film speed is most commonly specified using an arithmetric ISO number that is based on a carefully specified test procedure put forth by the International Organization for Standardization (ISO), for example ISO 200 or ISO 400. Each doubling or halving of the speed represents a doubling or halving of the sensitivity to light. Thus, ISO 400 speed film can be used in light that is half as bright as ISO 200 speed film without otherwise changing camera settings. Some films, particularly those intended
professional photographers or scientific applications, also specify speed using a logarithmic scale that is denoted with a degree symbol ( ). Each logarithmic increment represents an increase or decrease of three units corresponds to a doubling or halving of film speed. ISO 400 film as a logarith-mic speed of 2 but ISO 200 film, which is half as fast, has a logarithmic speed of 24 . Photographic films are coated with grains of light-sensitive silver compounds that form a latent image when exposed to light. Film speed is increased by increasing the size of the silver grains, and the grains in high speed films can be so large that they produce a visible texture, or graininess, in photographs that many people find distracting. Therefore, photographers generally try to usethe slowest possible film for a given situation. In some cases, however, photographers will deliberately choose a high-speed film or use developing methods that increase grain in order to produce an artistic effect. The choice of film speed is also affected by factors such as the desired shutter speed and aperture.
LENS FOCAL LENGTH The focal length of a simple lens is the distance from the lens to the film when the lens is focused on an object a long distance away (sometimes referred to as infinity, although the distance is always finite), and is related to the size of the image recorded on the film. Given two lenses, the lens with the longer focal length will produce a larger image than the lens with the shorter focal length. Most camera lens focal lengths are given in millimeters. A lens with a focal length of 100 mm (3.9 in) is in theory 100 mm (3.9 in) long, but camera lenses consist of many individual lens elements designed to act together. Therefore, the physical length of a camera lens will not be thesame as the focal length of a simple lens. Zoom lenses have variable focal lengths, for example 80–200 mm (3.1–7.9 in), and also variable physical lengths. The physical lens length will also change as the distance to the object being photographed changes. Lenses are often described as telephoto, normal, and wide angle. Normal lenses cover a range of vision similar to that of the human eye. Wide angle lenses have shorter focal lengths and cover a broader range of vision whereas telephoto lenses have longer focal lengths and cover a narrower range of vision. All of these terms are relative to the physical size of the film being used. A normal lens has a focal length that is about the same as the diagonal size of the film frame. For example, 35 mm (1.4 in) film is 35 mm (1.4 in) wide and each image in a standard 35 mm (1.4 in) camera is 24 mm (0.9 in) by 36 mm (1.4 in) in size. The Pythagorean theorem can be used to calculate that the diagonal size of a standard 35 mm (1.4 in) frames 43 mm (1.7 in). Lenses are usually designed using focal length increments that are multiples of 5 mm (0.2 in) or 10 mm (0.4 in) and 40 mm (1.6 in) lenses are not common so, in practice, the so-called normal lens for a 35mm (1.4 in) camera is a 35 mm (1.4 mm) or 50 mm (2.0in) lens. Manufacturers of cameras with film sizes or digital sensors of different sizes will sometimes describe their lenses using a 35 mm
(1.4 in) equivalent focal length. This means that the photographic effect (wide angle, normal, telephoto) will be the same as that focal length oftens used on a 35 mm (1.4 in) camera. SHUTTER SPEED The amount of light striking the film is controlled by two things: the length of time that the shutter is open (shutter speed) and the lens aperture. Shutter speed is typically expressed as some fraction of a second, for example 1/2 s or 1/500 s, and not as a decimal. Manual cameras allow photographers to choose from a fixed set of mechanically controlled shutter speeds that differ from each other by factors of approximately 2, and the shutter is opened and closed by a series of springs and levers. For example, 1/2, 1/4, 1/8, 1/15, 1/30, 1/60, 1/125, 1/250, and so forth. Note that the factor changes slightly between 1/8 and 1/15, and then again between 1/60 and 1/125. In order to make the best use of limited space on small cameras, film speed dials or indicators in many cases use only their denominator the shutter speed. Thus, a camera dial showing a shutter speed of 250 means that the film will be exposed to light for 1/250 s. Electronic cameras, whether film or digital, contain microprocessors and can offer a continuous range of shutter speeds. The shutter speeds can be set by the photographer or automatically selected by the camera. LENS APERTURE Lens aperture is the diameter of the opening through which light passes on its way to the film. The larger the aperture, the wider the opening and the more light that will pass through the lens to the film. Aperture isexpressed as a socalled f-stop or f-number that is the quotient of the lens focal length divided by the diameter of the aperture, and is controlled by a diaphragm consisting of moving metal blades within the lens. A lens with a focal length of 100 mm (3.9 in) and an opening 50 mm (2.0 in) in diameter is said to have an aperture of 100/50 f/2, but a 400 mm (15.7 in) lens with the same opening would have an aperture of 400/50 f/8. Therefore, the size of the opening must increase proportionately with focal length in order for lenses of two different focal lengths to have the same aperture. This is why the large telephoto lenses used by sports and nature photographers are so long and wide. They must have both long focal lengths and wide openings to transmit enough light to properly expose the film. The term f-stop refers to the fact that photographers have traditionally adjusted the aperture of their lenses by rotating a ring on the lens to choose among several preset apertures. Each pre-set aperture is marked by a sensible and audible click, or stop, hence the name f-stop. The pre-set apertures were chosen so that each stop halved or doubled the radius of the opening, using the same logic as pre-set shutter speeds, thus halving or doubling the amount of light passing through. The result was this progression of f-stops: f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, and f/32. Although many modern lenses have continuously adjustable apertures, and some are electronically controlled with no
aperture rings at all, the f-stop terminology and progression of f-stops marked on lenses persists. Physical constraints make it difficult to design lenses with large apertures, so the range of most lenses begins above f/1, typically in the range of f/2 or f/2.8. The additional difficulty of designing zoom lenses, especially if they are to be affordable to large numbers of people, sometimes motivates lens designers to use maximum apertures that change according to the focal length. An 18–70 mm (0.7–2.8 in) f/3.5-4.5 zoom lens would have a maximum aperture that ranges from f/3.5 at 18 mm (0.7 in) focal length to f/4.5 at 70 mm (2.8 in) focal length. DEPTH OF FIELD Depth of field refers to the range of distance from the lens, or depth, throughout which objects appear to be in focus. A lens can be focused on objects at only one dis- tance, and objects closer to or farther away from the lens will be out of focus on the plane of the film. In the case of a point of light that is out of focus, the result is a fuzzy circle known as a circle of confusion. Depth of field is increased by decreasing the size of the circles of confusion in an image, which is accomplished by reducing the aperture of the lens, until objects over a wide range of dis- tances appear to be in focus to the human eye. Although a small aperture reduces the sizes of the circles of confusion in an image, it also increases the relative importance of diffraction around the edges of aperture. As light passes through the movable metal blades that control aperture, some of it is scattered or diffracted.When the aperture is large, the effects of diffraction generally go unnoticed. As the aperture decreases, diffracted light becomes an increasingly large proportion of all the light passing through the aperture and image sharpness can decrease. Therefore, setting a lens to its smallest aper- ture will not generally produce the sharpest possible image. The sharpest images will generally be obtained by setting the lens to an aperture in the middle of its range. For a specified aperture, lenses with long focal lengths will always have shallower depths of field than lenses with short focal lengths. This is because the longer lenses must have physically larger openings than the shorter lenses, even if the aperture (f-stop) is the same. A larger opening transmits more light, which in turn pro- duces larger circles of confusion. RECIPROCITY Reciprocity is a mathematical relationship between shutter speed and lens aperture. If a photographer increases the light passing through the lens by opening the aperture one f-stop and then doubles the shutter speed (which will reduce the length of time the shutter is open by one-half), the amount of light reaching the film will not change. An aperture of f/4 and a shutter speed of 1/500 s, for example, will deliver the same amount of light as an aperture of f/5.6 and a
shutter speed of 1/250 s. In other words, aperture and shutter speed share a reciprocal relationship and many different combinations of shutter speed and lens aperture will provide the same amount of light. The reciprocal relationship also extends to film speed. If film speed is doubled, either the shutter speed can be increased (producing a shorter exposure) or aperture can be decreased by the same factor without changing the amount of light that reaches the film. In practice, there are some limitations to reciprocity. Photographs with very slow shutter speeds, for example minutes or hours instead of fractions of a second, can be appear too dark (underexposed) because the reciprocity relationship does not extend to such long exposures. This is known as reciprocity failure and can pose a problem for photographers working at night in situations where artificial lights cannot be used, for example when astronomers are attempting to take photos of the night sky using their very sensitive equipment. Film manufacturers publiss tables that allow photographers to compensate for reciprocity failure in different kinds of film. DIGITAL PHOTOGRAPHY Virtually everything written in this article applies to digital photography as well as film photography. The primary difference is that a digital camera uses an electronic sensor instead of a piece of plastic film coated with silver compounds. In place of the film used in a conventional camera, a digital camera uses an electronic sensor. Two sensor types are commonly used: CCDs, or charged coupled device sensors, and CMOSs, or complementary metal oxide semiconductor sensors. Both kinds are composed of rows and columns of photosites that convert light into an electronic signal. Each photosite is covered with a filter so that it is sensitive to only one of the three components of visible light (red, blue, or green). One widely used configuration, the Bayer array, consists of rows containing red and green filtered photosites alternating with rows containing green and blue photosites. When the image is being processed by the camera, values for the two missing colors are estimated using the math- ematical technique of interpolation. Two primary measures are used to characterize digital images: resolution and size. Resolution refers to the ability of a sensor to represent details, and is generally specified in terms of pixels per inch (ppi). Image size refers to the total number of pixels comprising an image, and is typically given in terms of megapixels. A pixel is the smallest possible discrete component of an image, typically a small square or dot, and one megapixel consists of one million pixels. As of early 2005, the best commercially available digital cameras had resolutions of approximately 20 megapixels and many professional quality digital cameras had resolutions of 5 or 6 megapixels. Digital photographers can adjust the sensitivity of the sensor to light just as film photographers can use films with different ISO speeds. In digital cameras, however, there is done with a switch or button on the camer a and the sensor is not physically removed. Although digital cameras commonly have ISO settings, they vary from manufacturer to manufacturer and do not follow the consistent ISO standard. Instead, they are an
approximate gauge of the sensitivity. The digital equivalent of film grain is electronic noise, which can appear in images as visual static or randomly colored pixels, and is most often a problem using high digital ISO settings. The size of the sensor can also contribute to the amount of noise in a digital image, because the photosites on a small sensor are closer to each other than those on a larger sensor and can interfere with each other.
Fundamental Mathematical Concepts and Terms THE CAMERA In its simplest form, the camera is a light-tight box containing light sensitive material, either in the form of photographic film or a digital sensor. A lens is used to focus light rays entering the camera and produce a sharp image. The amount of light striking the film and sensor is controlled by shutter, or curtain that quickly opens and exposes the film or sensor to light, and the size of the lens opening, or aperture, through which light can pass.
FILM SPEED The speed of photographic film is a measure of its sensitivity to light, with high speed films being more sensitive to light than low speed films. Film speed is most commonly specified using an arithmetric ISO number that is based on a carefully specified test procedure put forth by the International Organization for Standardization (ISO), for example ISO 200 or ISO 400. Each doubling or halving of the speed represents a doubling or halving of the sensitivity to light. Thus, ISO 400 speed film can be used in light that is half as bright as ISO 200 speed film without otherwise changing camera settings. Some films, particularly those intended
professional photographers or scientific applications, also specify speed using a logarithmic scale that is denoted with a degree symbol ( ). Each logarithmic increment represents an increase or decrease of three units corresponds to a doubling or halving of film speed. ISO 400 film as a logarith-mic speed of 2 but ISO 200 film, which is half as fast, has a logarithmic speed of 24 . Photographic films are coated with grains of light-sensitive silver compounds that form a latent image when exposed to light. Film speed is increased by increasing the size of the silver grains, and the grains in high speed films can be so large that they produce a visible texture, or graininess, in photographs that many people find distracting. Therefore, photographers generally try to usethe slowest possible film for a given situation. In some cases, however, photographers will deliberately choose a high-speed film or use developing methods that increase grain in order to produce an artistic effect. The choice of film speed is also affected by factors such as the desired shutter speed and aperture.
LENS FOCAL LENGTH The focal length of a simple lens is the distance from the lens to the film when the lens is focused on an object a long distance away (sometimes referred to as infinity, although the distance is always finite), and is related to the size of the image recorded on the film. Given two lenses, the lens with the longer focal length will produce a larger image than the lens with the shorter focal length. Most camera lens focal lengths are given in millimeters. A lens with a focal length of 100 mm (3.9 in) is in theory 100 mm (3.9 in) long, but camera lenses consist of many individual lens elements designed to act together. Therefore, the physical length of a camera lens will not be thesame as the focal length of a simple lens. Zoom lenses have variable focal lengths, for example 80–200 mm (3.1–7.9 in), and also variable physical lengths. The physical lens length will also change as the distance to the object being photographed changes. Lenses are often described as telephoto, normal, and wide angle. Normal lenses cover a range of vision similar to that of the human eye. Wide angle lenses have shorter focal lengths and cover a broader range of vision whereas telephoto lenses have longer focal lengths and cover a narrower range of vision. All of these terms are relative to the physical size of the film being used. A normal lens has a focal length that is about the same as the diagonal size of the film frame. For example, 35 mm (1.4 in) film is 35 mm (1.4 in) wide and each image in a standard 35 mm (1.4 in) camera is 24 mm (0.9 in) by 36 mm (1.4 in) in size. The Pythagorean theorem can be used to calculate that the diagonal size of a standard 35 mm (1.4 in) frames 43 mm (1.7 in). Lenses are usually designed using focal length increments that are multiples of 5 mm (0.2 in) or 10 mm (0.4 in) and 40 mm (1.6 in) lenses are not common so, in practice, the so-called normal lens for a 35mm (1.4 in) camera is a 35 mm (1.4 mm) or 50 mm (2.0in) lens. Manufacturers of cameras with film sizes or digital sensors of different sizes will sometimes describe their lenses using a 35 mm
(1.4 in) equivalent focal length. This means that the photographic effect (wide angle, normal, telephoto) will be the same as that focal length oftens used on a 35 mm (1.4 in) camera. SHUTTER SPEED The amount of light striking the film is controlled by two things: the length of time that the shutter is open (shutter speed) and the lens aperture. Shutter speed is typically expressed as some fraction of a second, for example 1/2 s or 1/500 s, and not as a decimal. Manual cameras allow photographers to choose from a fixed set of mechanically controlled shutter speeds that differ from each other by factors of approximately 2, and the shutter is opened and closed by a series of springs and levers. For example, 1/2, 1/4, 1/8, 1/15, 1/30, 1/60, 1/125, 1/250, and so forth. Note that the factor changes slightly between 1/8 and 1/15, and then again between 1/60 and 1/125. In order to make the best use of limited space on small cameras, film speed dials or indicators in many cases use only their denominator the shutter speed. Thus, a camera dial showing a shutter speed of 250 means that the film will be exposed to light for 1/250 s. Electronic cameras, whether film or digital, contain microprocessors and can offer a continuous range of shutter speeds. The shutter speeds can be set by the photographer or automatically selected by the camera. LENS APERTURE Lens aperture is the diameter of the opening through which light passes on its way to the film. The larger the aperture, the wider the opening and the more light that will pass through the lens to the film. Aperture isexpressed as a socalled f-stop or f-number that is the quotient of the lens focal length divided by the diameter of the aperture, and is controlled by a diaphragm consisting of moving metal blades within the lens. A lens with a focal length of 100 mm (3.9 in) and an opening 50 mm (2.0 in) in diameter is said to have an aperture of 100/50 f/2, but a 400 mm (15.7 in) lens with the same opening would have an aperture of 400/50 f/8. Therefore, the size of the opening must increase proportionately with focal length in order for lenses of two different focal lengths to have the same aperture. This is why the large telephoto lenses used by sports and nature photographers are so long and wide. They must have both long focal lengths and wide openings to transmit enough light to properly expose the film. The term f-stop refers to the fact that photographers have traditionally adjusted the aperture of their lenses by rotating a ring on the lens to choose among several preset apertures. Each pre-set aperture is marked by a sensible and audible click, or stop, hence the name f-stop. The pre-set apertures were chosen so that each stop halved or doubled the radius of the opening, using the same logic as pre-set shutter speeds, thus halving or doubling the amount of light passing through. The result was this progression of f-stops: f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, and f/32. Although many modern lenses have continuously adjustable apertures, and some are electronically controlled with no
aperture rings at all, the f-stop terminology and progression of f-stops marked on lenses persists. Physical constraints make it difficult to design lenses with large apertures, so the range of most lenses begins above f/1, typically in the range of f/2 or f/2.8. The additional difficulty of designing zoom lenses, especially if they are to be affordable to large numbers of people, sometimes motivates lens designers to use maximum apertures that change according to the focal length. An 18–70 mm (0.7–2.8 in) f/3.5-4.5 zoom lens would have a maximum aperture that ranges from f/3.5 at 18 mm (0.7 in) focal length to f/4.5 at 70 mm (2.8 in) focal length. DEPTH OF FIELD Depth of field refers to the range of distance from the lens, or depth, throughout which objects appear to be in focus. A lens can be focused on objects at only one dis- tance, and objects closer to or farther away from the lens will be out of focus on the plane of the film. In the case of a point of light that is out of focus, the result is a fuzzy circle known as a circle of confusion. Depth of field is increased by decreasing the size of the circles of confusion in an image, which is accomplished by reducing the aperture of the lens, until objects over a wide range of dis- tances appear to be in focus to the human eye. Although a small aperture reduces the sizes of the circles of confusion in an image, it also increases the relative importance of diffraction around the edges of aperture. As light passes through the movable metal blades that control aperture, some of it is scattered or diffracted.When the aperture is large, the effects of diffraction generally go unnoticed. As the aperture decreases, diffracted light becomes an increasingly large proportion of all the light passing through the aperture and image sharpness can decrease. Therefore, setting a lens to its smallest aper- ture will not generally produce the sharpest possible image. The sharpest images will generally be obtained by setting the lens to an aperture in the middle of its range. For a specified aperture, lenses with long focal lengths will always have shallower depths of field than lenses with short focal lengths. This is because the longer lenses must have physically larger openings than the shorter lenses, even if the aperture (f-stop) is the same. A larger opening transmits more light, which in turn pro- duces larger circles of confusion. RECIPROCITY Reciprocity is a mathematical relationship between shutter speed and lens aperture. If a photographer increases the light passing through the lens by opening the aperture one f-stop and then doubles the shutter speed (which will reduce the length of time the shutter is open by one-half), the amount of light reaching the film will not change. An aperture of f/4 and a shutter speed of 1/500 s, for example, will deliver the same amount of light as an aperture of f/5.6 and a
shutter speed of 1/250 s. In other words, aperture and shutter speed share a reciprocal relationship and many different combinations of shutter speed and lens aperture will provide the same amount of light. The reciprocal relationship also extends to film speed. If film speed is doubled, either the shutter speed can be increased (producing a shorter exposure) or aperture can be decreased by the same factor without changing the amount of light that reaches the film. In practice, there are some limitations to reciprocity. Photographs with very slow shutter speeds, for example minutes or hours instead of fractions of a second, can be appear too dark (underexposed) because the reciprocity relationship does not extend to such long exposures. This is known as reciprocity failure and can pose a problem for photographers working at night in situations where artificial lights cannot be used, for example when astronomers are attempting to take photos of the night sky using their very sensitive equipment. Film manufacturers publiss tables that allow photographers to compensate for reciprocity failure in different kinds of film. DIGITAL PHOTOGRAPHY Virtually everything written in this article applies to digital photography as well as film photography. The primary difference is that a digital camera uses an electronic sensor instead of a piece of plastic film coated with silver compounds. In place of the film used in a conventional camera, a digital camera uses an electronic sensor. Two sensor types are commonly used: CCDs, or charged coupled device sensors, and CMOSs, or complementary metal oxide semiconductor sensors. Both kinds are composed of rows and columns of photosites that convert light into an electronic signal. Each photosite is covered with a filter so that it is sensitive to only one of the three components of visible light (red, blue, or green). One widely used configuration, the Bayer array, consists of rows containing red and green filtered photosites alternating with rows containing green and blue photosites. When the image is being processed by the camera, values for the two missing colors are estimated using the math- ematical technique of interpolation. Two primary measures are used to characterize digital images: resolution and size. Resolution refers to the ability of a sensor to represent details, and is generally specified in terms of pixels per inch (ppi). Image size refers to the total number of pixels comprising an image, and is typically given in terms of megapixels. A pixel is the smallest possible discrete component of an image, typically a small square or dot, and one megapixel consists of one million pixels. As of early 2005, the best commercially available digital cameras had resolutions of approximately 20 megapixels and many professional quality digital cameras had resolutions of 5 or 6 megapixels. Digital photographers can adjust the sensitivity of the sensor to light just as film photographers can use films with different ISO speeds. In digital cameras, however, there is done with a switch or button on the camer a and the sensor is not physically removed. Although digital cameras commonly have ISO settings, they vary from manufacturer to manufacturer and do not follow the consistent ISO standard. Instead, they are an
approximate gauge of the sensitivity. The digital equivalent of film grain is electronic noise, which can appear in images as visual static or randomly colored pixels, and is most often a problem using high digital ISO settings. The size of the sensor can also contribute to the amount of noise in a digital image, because the photosites on a small sensor are closer to each other than those on a larger sensor and can interfere with each other.
