02.1-color-representation.pdf

mca101 : computeR gRaphics coloR RepResentation Raghav B. Venkataramaiyer CSED TIET Patiala India. August 22, 2024 outline 1 colouRs — intRoduction 2 colouR tables 3 colouR spaces 4 baRycentRic inteRpolation what aRe colouRs Colour is the visual pe...

mca101 : computeR gRaphics coloR RepResentation Raghav B. Venkataramaiyer CSED TIET Patiala India. August 22, 2024 outline 1 colouRs — intRoduction 2 colouR tables 3 colouR spaces 4 baRycentRic inteRpolation what aRe colouRs Colour is the visual perception based on the electromagnetic spectrum. Though color is not an inherent prop- erty of matter, color perception is re- lated to an object’s light absorption, reflection, emission spectra, and in- terference — Wikipedia FiguRe: Image Courtesy: Wikipedia Rgb colouRs Perception of color originates from different light wavelength or spec- tral sensitivity in cone cell types, which is then processed by the brain. For most humans, colors are per- ceived in the visible light spec- trum with three types of cone cells (trichromacy) — Wikipedia FiguRe: Image Courtesy: Wikipedia visible spectRum FiguRe: Visible Spectrum. Courtesy: Wikipedia Rgb/a RepResentation A colour in machine representation is a 3-vector for RGB, i.e. c ∈ R3[0,1] in floating-point representation; c ∈ Z3[0,255] in 8-bit fixed-point representation. a 4-vector for RGBA, i.e. c ∈ R4[0,1] in floating-point representation; c ∈ Z4[0,255] in 8-bit fixed-point representation. outline 1 colouRs — intRoduction Image Representation 2 colouR tables 3 colouR spaces 4 baRycentRic inteRpolation an image as composition of pixels An image is represented in pixels; Conceptually, an image may be thought of split into a rectangular grid of height H and width W; Each cell of such a grid, is called a pixel. A pixel is the smallest unit of area patch in an image and contains a uniform colour throughout; H × W is called the resolution of such an image. an image as a bounded space An optical camera sensor receives coloured signals within the spatial opening from its shutter; The sensor registers the signals as energy/ intensity corresponding to each of the R,G,B frequencies of visible spectrum. The signals are physically continuous in space. In practice, samples are discrete in space. The space is 2-dimensional and bound on top, bottom, left and right. Hence an image may also be interpreted as a quantised and bounded spatial signal. image channels Each of R, G, B, A etc. are known as channels. If a camera may sample more frequencies, including ultra- and infra-optical spectrum, the resulting image is called a hyper-spectral image. The RGB image has 3-channels The RGBA image has 4-channels A hyper-spectral image may have many channels, say C channels. image in floating-point RepResentation A floating-point may be 16-bit, 32-bit or 64-bit per channel, generally used internally within graphic frameworks like OpenGL/ WebGL etc. An image with resolution H × W, containing C channels is represented as x ∈ RC×H×W [0,1]. An RGB image with resolution H × W, is represented as x ∈ R3×H×W [0,1]. An RGBA image with resolution H × W, is represented as x ∈ R4×H×W [0,1]. image in 8-bit fixed-point A fixed-point may be 8-bit, 16-bit, 32-bit or 64-bit per channel, generally 8-bit is most commonly used. An image with resolution H × W, containing C channels is represented as x ∈ ZC×H×W [0,255]. An RGB image with resolution H × W, is represented as x ∈ Z[0,255] 3×H×W. An RGBA image with resolution H × W, is represented as x ∈ Z4×H×W [0,255]. question 1 How is a grayscale image with resolution H × W with floating-point representation described mathematically? 2 How is a grayscale image with resolution H × W with 8-bit fixed-point representation described mathematically? 3 How is a grayscale image with resolution H × W with 16-bit fixed-point representation described mathematically? question 1 What is the minimum size (in bytes) required to store an uncompressed RGB image in FHD resolution. Assume the image header to be H bytes. 2 What is the minimum size (in bytes) required to store an uncompressed RGBA image in FHD resolution. Assume the image header to be H bytes. outline 1 colouRs — intRoduction 2 colouR tables 3 colouR spaces 4 baRycentRic inteRpolation colouR tables In order to improve storage efficiency of images, Included colours are listed, the count n < 256 generally. This is called a colour table. For each pixel, the colour is approximated and assigned to be one of the colours in the table, and the colour index is stored instead of the complete colour information. This reduces the storage requirement by order of magnitude. The only additional storage requirement is that of the colour table, n × 3 bytes stoRage efficiency For example, RGB image requires 8 × 3 = 24 bits per pixel; whereas Colour table with upto 2 colours would require only 1-bit per pixel; Colour table with 3-4 colours would require only 2-bit per pixel; Colour table with 5-8 colours would require only 3-bit per pixel; stoRage efficiency Num Bits Max Num Colours 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 question What is the minimum size (in bytes) of storage required to store an image with colour table with 100 RGB-colours? Assume the image header to be H bytes. outline 1 colouRs — intRoduction 2 colouR tables 3 colouR spaces 4 baRycentRic inteRpolation hsl and hsv FiguRe: Image Courtesy: Wikipedia outline 1 colouRs — intRoduction 2 colouR tables 3 colouR spaces 4 baRycentRic inteRpolation outline 1 colouRs — intRoduction 2 colouR tables 3 colouR spaces 4 baRycentRic inteRpolation Linear Interpolation lineaR inteRpolation Solution Point on straight line connecting a, b, and in between, may be given parameterised by t ∈ R[0,1] as, Question Given two points f(t) = (1 − t)a + t b a, b ∈ R2 , find the point on straight line Here if the Euclidean distance ∆E (a, b) = 1 unit, then connecting them, and in ∆E (a, f(t)) = 1 − t units, and ∆E (f(t), b) = t units. between, so that it is twice Hence, as far away from b as it is from a. 2 t= ∵ t = 2(1 − t) 3 2 1 2 p = f(t = ) = a + b 3 3 3 lineaR inteRpolation Solution Point on hypothetical straight line in RGB colour space connecting a, b, and in between, may be given parameterised by t ∈ R[0,1] as, Question Given two RGB colours f(t) = (1 − t)a + t b a, b ∈ R3 , find the colour c, so that it is twice as far Here if the Euclidean distance ∆E (a, b) = 1 unit, then away from b as it is from ∆E (a, f(t)) = 1 − t units, and ∆E (f(t), b) = t units. a. Hence, 2 t= ∵ t = 2(1 − t) 3 2 1 2 p = f(t = ) = a + b 3 3 3 baRycentRic cooRdinates FiguRe: Image Courtesy: Wikipedia