Gain-and-bias and level

Linear transformations can be applied by gain and bias, or two parameters to "-level", or two parameters to "+level". These methods are equivalent. If we know the parameters for one method, we can calculate the parameters for the other two methods.

ASIDE. This page is one of a series that address the question: "What colour transformation makes this image look like that image?"

The pages in that series, roughly from simple to complex, are:

Setting the mean: make the mean each channel match those of a second image, by gain, bias, power or negative gain methods.
Gain and bias: make the mean and standard deviation of each channel match those of a second image, by gain and bias method.
What level?: calculate level, or gain and bias, from two pixels or linear regression of all pixels.
Gain-and-bias and level: convert between parameters for -level, +level and gain-and-bias.
Process modules: matching histograms: apply an inverse cumulative normalised histogram as a clut. Precise, but can give bad results.
Blending pyramids: matching gain and bias: build Laplacian pyramids from the images, adjust the gain and bias of the grids, and collapse the resulting pyramid.
Colours to matrix and polynomials: a process module.
Sparse hald cluts: from a Nx2 image of "from" and "to" colours, a process module makes a (3D) hald clut.

The method

The diagram illustrates the situation. From the origin (0,0), the x-axis is input image values, and the y-axis is output image values. The central square is input and output values in the range 0 to 100%. The blue line is a transformation from input to output, which can be implemented by "-level L1,L2%" or "+level L3p,L4p%" or "-function Polynomial gain,bias".

Six scripts are provided, for conversion from any of the three methods to any of the others. The scripts are concerned only with the six parameters; they do no image processing.

Examples

The "-level" and "+level" methods are identical, and virtually identical to the gain-and-bias kludge method.

Scripts

For convenience, .bat scripts are also available in a single zip file. See Zipped BAT files.

gbToL1L2.bat

gbToL3pL4p.bat

L1L2toGb.bat

L3pL4ptoGb.bat

L1L2toL3pL4p.bat

L3pL4ptoL1L2.bat

All images on this page were created by the commands shown, using:

%IMG7%magick identify -version

Version: ImageMagick 7.1.1-15 Q16-HDRI x64 a0a5f3d:20230730 https://imagemagick.org
Copyright: (C) 1999 ImageMagick Studio LLC
License: https://imagemagick.org/script/license.php
Features: Cipher DPC HDRI OpenCL OpenMP(2.0) 
Delegates (built-in): bzlib cairo freetype gslib heic jng jp2 jpeg jxl lcms lqr lzma openexr pangocairo png ps raqm raw rsvg tiff webp xml zip zlib
Compiler: Visual Studio 2022 (193532217)

To improve internet download speeds, some images may have been automatically converted (by ImageMagick, of course) from PNG or TIFF or MIFF to JPG.

Source file for this web page is gblevel.h1. To re-create this web page, execute "procH1 gblevel".

This page, including the images, is my copyright. Anyone is permitted to use or adapt any of the code, scripts or images for any purpose, including commercial use.

Anyone is permitted to re-publish this page, but only for non-commercial use.

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With "`-function Polynomial`". %IM7DEV%magick ^ toes.png ^ -channel R ^ -function Polynomial %gbl_params_gain_R%,%gbl_params_bias_R% ^ -channel G ^ -function Polynomial %gbl_params_gain_G%,%gbl_params_bias_G% ^ -channel B ^ -function Polynomial %gbl_params_gain_B%,%gbl_params_bias_B% ^ +channel ^ gbl_distgb.png
With "`-function Polynomial`", kludge fix. %IMG7%magick ^ toes.png ^ -channel R ^ -evaluate Multiply %gbl_params_gain_R% ^ -evaluate Add %%[fx:%gbl_params_bias_R%QuantumRange] ^ -channel G ^ -evaluate Multiply %gbl_params_gain_G% ^ -evaluate Add %%[fx:%gbl_params_bias_G%QuantumRange] ^ -channel B ^ -evaluate Multiply %gbl_params_gain_B% ^ -evaluate Add %%[fx:%gbl_params_bias_B%*QuantumRange] ^ +channel ^ gbl_distgb_k.png
With "`-level`". %IMG7%magick ^ toes.png ^ -channel R ^ -level %gbl_params_L1_R%,%gbl_params_L2_R%%% ^ -channel G ^ -level %gbl_params_L1_G%,%gbl_params_L2_G%%% ^ -channel B ^ -level %gbl_params_L1_B%,%gbl_params_L2_B%%% ^ +channel ^ gbl_levm.png
With "`+level`". %IMG7%magick ^ toes.png ^ -channel R ^ +level %gbl_params_L3p_R%,%gbl_params_L4p_R%%% ^ -channel G ^ +level %gbl_params_L3p_G%,%gbl_params_L4p_G%%% ^ -channel B ^ +level %gbl_params_L3p_B%,%gbl_params_L4p_B%%% ^ +channel ^ gbl_levp.png