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There is no single universal YUV-to-RGB formula. For digital video, the relevant format is usually Y′CbCr, and the correct conversion depends on its matrix (such as BT.601 or BT.709), range, and bit depth. The formulas below use nonlinear R′G′B′ values and identify where range scaling belongs; they do not, by themselves, convert transfer functions or color primaries.
What “YUV” means in digital video
YUV originally describes an analog color-difference representation. Digital video is generally represented as Y′CbCr: Y′ is nonlinear luma, while Cb and Cr are blue- and red-difference chroma components. People often call digital Y′CbCr “YUV,” but the terms are not interchangeable in every technical context. Microsoft’s overview of YUV video explains the video convention.
The matrix equations relate nonlinear R′G′B′ signals to Y′CbCr. They are not a complete color-management pipeline: transfer-function handling and conversion between RGB primary sets are separate operations, normally performed in linear RGB where required. Khronos describes color conversion as a sequence of distinct operations in its OpenVX specification.
General normalized Y′CbCr equations
For normalized nonlinear red, green, and blue values R′, G′, and B′ in [0,1], choose matrix coefficients Kr and Kb. Set Kg = 1 − Kr − Kb. Then calculate:
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- Y′ = KrR′ + KgG′ + KbB′
- Cb = (B′ − Y′) / (2(1 − Kb))
- Cr = (R′ − Y′) / (2(1 − Kr))
These equations produce normalized components; they are not 8-bit code values. For limited-range 8-bit video, map Y′ to nominal codes 16–235 and chroma to nominal codes 16–240, with neutral chroma at 128. At bit depth M, the nominal offsets and spans scale by 2(M−8). State the matrix, range, bit depth, and whether inputs are normalized or code values before using coefficients. Microsoft documents the code ranges in its YUV video overview and recommended 8-bit YUV formats.
Matrix coefficients: BT.601, BT.709, and BT.2020
The matrix family sets the luma weights. BT.601 and BT.709 coefficients are documented by Microsoft and Linux kernel documentation; libyuv also provides implementation reference coefficients for those families and BT.2020. The values below are normalized coefficients, not a promise that every implementation uses identical fixed-point integers.
| Matrix family | Kr | Kg | Kb |
|---|---|---|---|
| BT.601 | 0.299 | 0.587 | 0.114 |
| BT.709 | 0.2126 | 0.7152 | 0.0722 |
| BT.2020 non-constant-luminance example | 0.2627 | 0.6780 | 0.0593 |
References: Microsoft’s 8-bit YUV format documentation, Linux kernel 4.9 colorspace documentation, and libyuv’s coefficient implementation. The Linux page is version 4.9; use the applicable current standard or implementation documentation for conformance work.
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Using BT.601 coefficients to decode BT.709 material, or vice versa, can shift colors because the luma weights differ. A BT.601-to-BT.709 change may also require converting between their RGB primaries; changing only the matrix coefficients is not necessarily a complete color-space conversion.
BT.601 RGB-to-Y′CbCr for 8-bit limited-range video
Normalized equations with range mapping
For normalized nonlinear R′, G′, and B′, BT.601 uses Kr = 0.299, Kb = 0.114, and Kg = 0.587. Calculate luma and normalized color differences, then map to limited-range code values:
- Y′ = 0.299R′ + 0.587G′ + 0.114B′
- Pb = 0.5(B′ − Y′) / (1 − 0.114)
- Pr = 0.5(R′ − Y′) / (1 − 0.299)
- Ycode = 16 + 219Y′; Cbcode = 128 + 224Pb; Crcode = 128 + 224Pr
The resulting nominal 8-bit ranges are Y′ 16–235 and Cb/Cr 16–240. Values outside nominal ranges can occur as headroom or footroom in formats and workflows that permit them.
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Approximate integer shortcut
For 8-bit computer RGB input and 8-bit BT.601 output, Microsoft gives this approximate fixed-point form. R, G, and B are code values from 0 to 255; the right shift and offsets produce limited-range Y, Cb, and Cr codes:
- Y = ((66R + 129G + 25B + 128) >> 8) + 16
- Cb = ((−38R − 74G + 112B + 128) >> 8) + 128
- Cr = ((112R − 94G − 18B + 128) >> 8) + 128
This is an approximation for that specific input, output, matrix, and range—not a universal RGB-to-video conversion. See Microsoft’s format equations.
BT.601 limited-range Y′CbCr-to-RGB for 8-bit codes
For 8-bit limited-range BT.601 input, let C = Y − 16, D = Cb − 128, and E = Cr − 128. Then compute and clip each output to [0,255]:
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- R = clip(round(1.164383C + 1.596027E))
- G = clip(round(1.164383C − 0.391762D − 0.812968E))
- B = clip(round(1.164383C + 2.017232D))
These inverse coefficients apply to limited-range BT.601 code values. They are not appropriate for BT.709, BT.2020, or full-range input. libyuv gives a closely rounded implementation reference for BT.601 limited range: R=(Y−16)×1.164+V×1.596, G=(Y−16)×1.164−U×0.391−V×0.813, B=(Y−16)×1.164+U×2.018, where U and V are chroma centered at zero. See libyuv’s implementation coefficients and Microsoft’s format documentation.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Full range and limited range are different conversions
For 8-bit limited-range video, nominal luma is 16–235, chroma is 16–240, and neutral chroma is 128. Full-range formats use different offsets and scaling. For example, libyuv labels its BT.601 full-range conversion path “JPEG” and supplies a distinct inverse formula. Do not decode limited-range samples with full-range coefficients or the reverse; use the range signaled by the format or metadata, and match the conversion to it. Nominal ranges do not necessarily exclude values in headroom or footroom.
Chroma subsampling is a separate step
In 4:2:2 or 4:2:0 video, fewer chroma samples are stored than luma samples. The matrix still defines the relationship between components, but RGB reconstruction also needs chroma upsampling and the correct chroma sample placement. Microsoft recommends converting 4:2:0 or 4:2:2 data to 4:4:4 before RGB conversion in its YUV video overview. Chroma subsampling—not the Y′CbCr matrix itself—is what reduces the chroma data volume relative to full-resolution components.
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Choose the right formula before coding
Check the source format or metadata for each of these parameters rather than guessing from the pixel values:
- Matrix: BT.601, BT.709, or BT.2020 (and, for BT.2020, whether the format uses non-constant luminance).
- Range: full or limited/studio.
- Bit depth: code-value scaling changes with depth.
- Direction: RGB to Y′CbCr or Y′CbCr to RGB.
- Signal form: nonlinear R′G′B′ for the matrix equations, not unconverted linear RGB.
- Sampling: 4:4:4 or subsampled chroma requiring reconstruction.
- Color management: whether transfer-function or primary conversion is also needed.
Once these match the source, choose the corresponding forward or inverse equations. A coefficient set alone cannot resolve unknown range, bit depth, or chroma layout.
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