Digital Cleaning of Old Paintings
Adam Gully
University of Utah
Department of Mathematics
July 9 2004
Problem:
• Many old paintings have become darkened or faded due
to centuries of exposure to harmful chemicals and/or
sunlight
Goal:
• To digitally recover original color of old
paintings using inverse techniques
Problem Formulation:
• In 1995, Dr. Cortelazzo, Dr. Geremia and Dr. Mian
of the University of Padova, Italy suggested that an
appropriate model would follow this form...




X ⇒  D ⇒ Y ⇒  V  ⇒ Xnew
X is the original painting
Y is the dirty painting
Xnew is the digitally cleaned painting
Different Approaches:
• Linear Model
• Quadratic Model
• “Bleeding” Pixel Model
Linear Model:
Forward model is:
Xr Xg Xb


Crr Crg Crb


 Cgr Cgg Cgb  = Yr Yg Yb
Cbr Cbg Cbb
Backward model is:
Xnew = Y ∗ C T
−1
T
C∗C
Quadratic Model:
Forward model is:






Xr Xg Xb Xr2 Xg2 Xb2 




Crr
Cgr
Cbr
2
Crr
2
Cgr
2
Cbr
Crg
Cgg
Cbg
2
Crg
2
Cgg
2
Cbg
Crb
Cgb
Cbb
2
Crb
2
Cgb
2
Cbb
Backward model is:
Xnew = Y ∗ C T
−1
T
C∗C






 = Yr Yg Yb




“Bleeding” Pixel Model:
Forward model is:









c
c
c
u
u
u
Xr Xg Xb Xr Xg Xb . . . 








c
Crr
c
Cgr
c
Cbr
u
Crr
u
Cgr
u
Cbr
..
..
..
c
Crg
c
Cgg
c
Cbg
u
Crg
u
Cgg
u
Cbg
..
..
..
c
Crb
c
Cgb
c
Cbb
u
Crb
u
Cgb
u
Cbb
..
..
..









 = Yr Yg Y
b








Backward model is:
Xnew = Y ∗ C T
−1
T
C∗C
Results:
• Linear Model
• Quadratic Model
• “Bleeding” Pixel Model
Results From Linear Model:
Before Chemically Cleaned ... After Chemically Cleaned
After Digitally Cleaned (Xnew)
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Problems With Linear Model:
Sample Size...???
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Results From Quadratic Model:
Before Chemically Cleaned ... After Chemically Cleaned
After Digitally Cleaned (Xnew)
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Problems With Quadratic Model:
Sample Size...???...Again...
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Results From “Bleeding” Pixel Model:
NO SUCCESSFULL RESULTS
• Forward Model Was Successful, Indicating Correct “C”
matrix values
•
Condition Number of C ∗ C T Was Ridiculous
•
Taking R1 and Attempting to Project into R9 is a Bad Thing
Possible Future Questions/Objectives:
1. Find a better way to recover Xnew values.
2. In particular, can the quadratic method produce better
results.
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