sRGB↔XYZ conversionBy Michał ‘mina86’ NazarewiczUpdated on 21st of March 2021Share on BlueskyIn an earlier post, I’ve shown how to calculate an RGB↔XYZ conversion matrix. It’s only natural to follow up with a code for converting between sRGB and XYZ colour spaces. While the matrix is a significant portion of the algorithm, there is one more step necessary: gamma correction.What is gamma correction?Human perception of light’s brightness approximates a power function of its intensity. This can be expressed as \(P = S^\alpha\) where \(P\) is the perceived brightness and \(S\) is linear intensity. \(\alpha\) has been experimentally measured to be less than one which means that people are more sensitive to changes to dark colours rather than to bright ones.Based on that observation, colour space’s encoding can be made more efficient by using higher precision when encoding dark colours and lower when encoding bright ones. This is akin to precision of floating-point numbers scaling with value’s magnitude. In RGB systems, the role of precision scaling is done by gamma correction. When colour is captured (for example from a digital camera) it goes through gamma compression which spaces dark colours apart and packs lighter colours more densely. When displaying an image, the opposite happens and encoded value goes through gamma expansion.1.00.90.80.70.60.50.40.30.20.10.0EncodedIntensityMany RGB systems use a simple \(S = E^\gamma\) expansion formula, where \(E\) is the encoded (or non-linear) value. With decoding \(\gamma\) approximating \(1/\alpha\), equal steps in encoding space correspond roughly to equal steps in perceived brightness. Image on the right demonstrates this by comparing two colour gradients. The first one has been generated by increasing encoded value in equal steps and the second one has been created by doing the same to light intensity. The former includes many dark colours while the latter contains a sudden jump in brightness from black to the next colour.sRGB uses slightly more complicated formula stitching together two functions: $$ \begin{align} E &= \begin{cases} 12.92 × S & \text{if } S ≤ S_0 \\ 1.055 × S^{1/2.4} - 0.055 & \text{otherwise} \end{cases} \\[.5em] S &= \begin{cases} {E \over 12.92} & \text{if } E ≤ E_0 \\ \left({E + 0.055 \over 1.055}\right)^{2.4} & \text{otherwise} \end{cases} \\[.5em] S_0 &= 0.00313066844250060782371 \\ E_0 &= 12.92 × S_0 \\ &= 0.04044823627710785308233 \end{align} $$The formulæ assume values are normalised to [0, 1] range. This is not always how they are expressed so a scaling step might be necessary.sRGB encodingMost common sRGB encoding uses eight bits per channel which introduces a scaling step: \(E_8 = ⌊E × 255⌉\). In an actual implementation, to increase efficiency and accuracy of gamma operations, it’s best to fuse the multiplication into aforementioned formulæ. With that arguably obvious optimisation, the equations become: $$ \begin{align} E_8 &= \begin{cases} ⌊3294.6 × S⌉ & \text{if } S ≤ S_0 \\ ⌊269.025 × S^{1/2.4} - 14.025⌉ & \text{otherwise} \end{cases} \\[.5em] S &= \begin{cases} {E_8 \over 3294.6} & \text{if } E_8 ≤ 10 \\ \left({E_8 + 14.025 \over 269.025}\right)^{2.4} & \text{otherwise} \end{cases} \\[.5em] S_0 &= 0.00313066844250060782371 \\ \end{align} $$This isn’t the only way to represent colours of course. For example, 10-bit colour depth changes the scaling factor to 1024; 16-bit high colour uses five bits for red and blue channels while five or six for green producing different scaling factors for different primaries; and HDTV caps the range to [16, 235]. Needless to say, correct formulæ need to be chosen based on the standard in question.The implementationAnd that’s it. Encoding, gamma correction and the conversion matrix are all the necessary pieces to get the conversion implemented. Like before, Rust programmers can take advantage of the srgb crate which implemented full conversion. However, to keep things interesting, in addition, here’s the conversion code written in TypeScript:type Tripple = [number, number, number];
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2024年，我成为马来西亚“花踪文学奖”的评委。那年10月11日，我到马来西亚新山参加颁奖典礼。在从北京飞往吉隆坡的航班上，我捕捉到这样一帧画面：偌大的机舱里乘客并不多，大多数昏昏欲睡。我余光一瞥，发现就在我座位的右后方，一位沉静的姑娘十分专注地在读一本书。她穿着蓝色的套头衫，深色的裤子上随意披散着一条纱巾。她那么投入地读着的书是阿来的《尘埃落定》。她看的还是人民文学出版社1998年出版的“中国当代名家长篇小说代表作”的版本，颇有些年头了。这位姑娘家在吉隆坡，是一位钢琴教师。在征得她同意后，我拍下了这个动人的瞬间，并在第一时间发给阿来。阿来很快发过来一张图片，图片上是高山草甸上挺立盛开的两朵蓝玉簪龙胆花。蓝色的高原之花与姑娘的蓝色上衣似乎在遥相呼应。阿来还特别附上了一句话：的确是美丽瞬间。
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