There is no single universal YUV-to-RGB formula. For digital video, the conversion is usually between nonlinear R′G′B′ and Y′CbCr, and its coefficients depend on the matrix (such as BT.601 or BT.709) and whether the samples use full or limited range. Identify those settings before applying a formula; otherwise, the result can have incorrect brightness or color.
What “YUV to RGB” means in digital video
YUV is the historical analog color-difference representation. Digital video commonly uses Y′CbCr, which represents luma and color differences using scaled and offset values. People often say “YUV” when they mean digital Y′CbCr, but the distinction matters when choosing equations.
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The prime in Y′ indicates a nonlinear signal. The usual video matrix equations use nonlinear R′G′B′ values—not linear-light RGB. A matrix conversion alone also does not change color primaries or perform transfer-function conversion. If those changes are required, they are separate processing steps. See Microsoft’s overview of YUV video and the Khronos YUV conversion description.
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Choose the matrix and range first
The normalized equations use matrix coefficients Kr and Kb, with Kg = 1 − Kr − Kb:
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- BT.601: Kr = 0.299, Kg = 0.587, Kb = 0.114.
- BT.709: Kr = 0.2126, Kg = 0.7152, Kb = 0.0722.
- BT.2020 non-constant-luminance: example coefficients are Kr = 0.2627, Kg = 0.6780, Kb = 0.0593.
These values are documented by Microsoft and the Linux 4.9 colorspace documentation; implementation references such as libyuv’s format documentation also provide matrix and range-specific coefficients. Integer constants in software may be fixed-point approximations rather than universal exact values.
For 8-bit limited-range video, nominal luma uses codes 16–235, chroma uses 16–240, and neutral chroma is 128. Full-range formats use different offsets and scaling. At bit depth M, nominal studio-range offsets and spans scale by 2(M−8). State the bit depth and range when sharing code values; normalized components and integer code values cannot be substituted for one another without scaling.
Normalized RGB-to-Y′CbCr equations
For nonlinear R′, G′, and B′ normalized to [0,1], calculate luma and normalized chroma differences as follows:
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- Y′ = KrR′ + KgG′ + KbB′
- Cb = (B′ − Y′) / (2(1 − Kb))
- Cr = (R′ − Y′) / (2(1 − Kr))
For 8-bit limited-range code values, map normalized luma and chroma to their nominal code ranges:
- Ycode = 16 + 219Y′
- Cbcode = 128 + 224Cb
- Crcode = 128 + 224Cr
Use the coefficients for the selected matrix. The BT.601 equations below are one specific case, not a generic RGB-to-video conversion.
8-bit limited-range BT.601: RGB to Y′CbCr
For normalized nonlinear RGB, use BT.601 weights Kr = 0.299, Kg = 0.587, and Kb = 0.114 in the normalized equations above, then scale to limited-range code values.
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For 8-bit computer RGB input, Microsoft also gives this approximate fixed-point form:
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- Cb = ((−38R − 74G + 112B + 128) >> 8) + 128
- Cr = ((112R − 94G − 18B + 128) >> 8) + 128
Here R, G, and B are 8-bit input values. This shortcut is an approximation specifically for 8-bit BT.601 limited-range output, not a replacement for other matrices, ranges, or bit depths. The equations and code-value conventions are described in Microsoft’s YUV overview and recommended 8-bit YUV formats.
8-bit limited-range BT.601: Y′CbCr to RGB
For 8-bit limited-range BT.601 input, first center the chroma values and remove the luma offset:
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- C = Y − 16
- D = Cb − 128
- E = Cr − 128
Then calculate each output channel, round, and clip to the 8-bit range [0,255]:
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- B = clip(round(1.164383C + 2.017232D))
These coefficients apply to this matrix and range. A full-range input or a different matrix needs its corresponding inverse coefficients. For example, libyuv documents a distinct BT.601 full-range path labeled “JPEG,” as well as limited-range implementations; do not decode limited-range samples with full-range coefficients or vice versa. See Microsoft’s format guidance and libyuv’s implementation reference.
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BT.601 and BT.709 use different luma weights, so applying the wrong matrix changes the relationship between luma and the two color-difference channels. Range mismatches cause a different problem: limited-range values interpreted as full range, or the reverse, produce incorrect levels. Check the input format’s metadata or documentation for its matrix and range instead of selecting coefficients by habit.
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Nominal ranges describe the usual coding intervals, not a guarantee that every valid signal sample stays inside them; formats and formulas may allow headroom or footroom values. Avoid automatically clamping input components to nominal limits unless the relevant format or processing step calls for it.
Chroma subsampling is a separate conversion step
Formats such as 4:2:2 and 4:2:0 store fewer chroma samples than luma samples. The conversion matrix still relates component values, but the missing chroma positions must be reconstructed with the correct sample placement before or during RGB conversion. Microsoft recommends converting 4:2:0 or 4:2:2 to 4:4:4 and then to RGB in its YUV video guidance.
RGB-to-Y′CbCr conversion does not itself reduce data size. Space savings come from chroma subsampling. Likewise, a BT.601-to-BT.709 change may require more than changing luma coefficients: if the source and destination primaries differ, a separate primary-conversion operation may be needed.
Checklist before applying a formula
- Confirm whether the format is digital Y′CbCr, even if software or documentation casually calls it YUV.
- Identify the matrix family: BT.601, BT.709, BT.2020, or another specified matrix.
- Identify full versus limited range and the bit depth.
- Confirm the direction of conversion and whether the input values are normalized components or integer code values.
- Use nonlinear R′G′B′ for the matrix equations; handle transfer functions and primary changes separately when required.
- For subsampled input, account for chroma reconstruction and sample placement.
For conformance work, consult the applicable current standard and format specification. The Linux reference linked above is specifically version 4.9, and implementation formulas should be treated as implementation guidance rather than proof of conformance to a complete standard.
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