The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
FinalRadifcol, row)- [l + Pj (col, row)] ■ InitialRadifcol, row) 
J i 
+Qi(col,row) 
i Pjm g i (col, row)-IniRadioi m gI (col, row)/a PI = 0 
ölm gl (pol, row)/CTqj = 0 
(3) 
where Pj and Qi are polynomials of varying degrees. With 
constant polynomials, this formula becomes a simple look-up 
table (LUT) and with polynomials of degree one and above, 
image radiometry correction according to pixel location. 
2.2 Building a grid of radiometric values 
First step of radiometric block adjustment consists in building a 
regular grid of radiometric values, extracted from the block of 
overlapping images to harmonize. Each useful grid node 
contains one or several radiometric values, depending of how 
many images it covers. These values are calculated by 
interpolation in initial image, sub-sampled at a given resolution. 
A radiometric initial threshold is applied so that too bright 
points, supposed to be part of clouds, are invalidated. A global 
mask, for example a water mask, can also be used to reduce the 
number of invalid points. Finally, these radiometric values are 
going to fuel a single large equation system. 
2.3 Fuelling the system with observation equations 
Observation equations are set up to equalize final radiometry of 
images on overlapping areas. They are written for each grid 
node containing at least two values, as in the following example 
for a node over images 1 and 2 (eq. 2). 
FinalRadio [mgX (col 1, rowi) FinalRadic\ m g2 (col2, row 2) 
a Obs 
a Obs 
[l + Pjm g i (col 1, row l)] • IniRadio lm g x (col 1, row l) 
+Q\ mg \(coll,row\) \j 
[l + E Im g 2 (col2, row 2)]- IniRadioi mg2 (col2, row2) 
+Qimg2(col2,row2) 
t-^Obs 
where: 
• IniRadio Img i (coll,rowl) is the radiometric value 
extracted from image 1 at (coll,rowl) position, 
• IniRadiO] mg2 (col2,row2) is the radiometric value 
extracted from image2 at (col2,row2) position, 
• Oobs is used to weight the equation 
With only observation equations, this system has a set of 
obvious solutions: P=-l and Q=constant value for any image. 
All images become uniform and information is totally lost. That 
is why we have to add constraint equations to the system to 
avoid this unacceptable solution. 
2.4 Adding constraint equations to the system 
2.4.1 Constraints on initial radiometry invariance: These 
constraint equations are set up to maintain final radiometry 
same as initial radiometry for all valid grid nodes, those on 
overlapping areas but also those covering only one image. In 
practice, we write two equations for a given image I (eq. 3): 
where: 
• IniRadio ImgI (col,row) is the radiometric value 
extracted from image I at (col,row) position, 
• dpi and ctqj are used to weight equations. These values 
are set image by image. 
Relative weighting of constraint equations compared to 
observation equations lets image radiometry to change more or 
less. With a very constrained system, final radiometry will be 
very close to initial one. On the contrary, radiometry can 
change a lot with a less constrained system, with the risk of 
reducing dynamic of images due to influence of observation 
equations. As weighting can be done image by image, we have 
here a way to fix radiometry of an image by giving very low 
values to its c P j and ct qi . 
2.4.2 Constraints on global average invariance by image: 
These constraints are used to modify global radiometric average 
value of each image. Precisely, they try to make final 
radiometry average of all valid grid points concerning an image 
equal to valid initial average of the whole grid. We write as 
much equations (eq. 4) as there are images in the dataset. 
FinalRadio_Average_ Im g I 
crAvlmgl 
InitialRadio_Average_ Grid 
crAvlmgl 
(4) 
Weighting can be done image by image to control how average 
radiometry of an image gets closer to global average of the 
dataset. 
2.4.3 Constraints on global average invariance: As an 
alternative to global constraints by image defined in the 
previous paragraph, we can just add one single global constraint 
on average invariance. We write one equation (eq. 5) that tries 
to make final radiometry average of all valid grid points equal 
to valid initial average. 
FinalRadio_Average_ Grid 
oAvGrid 
InitialRadio_ A verage_ Grid 
oAvGrid 
(5) 
Weighting is global and helps to control average radiometry of 
the whole dataset. 
2.5 System solving and validity updating 
All equations are weighted differently in order to favour 
observations or a type of constraints. Weighting can also be 
tuned image per image and some images can be considered as 
invariant. Polynomial coefficients for each image are calculated 
globally by least-square resolution of this linear system 
composed by observation and constraints equations. 
Once the system is solved, validity of all grid nodes is updated. 
A node covering only one image is always considered as valid 
whereas a node over at least two images can be invalidated if its 
final radiometries are too different. A previously invalidated 
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