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Protein Image Alignmentvia Quadratic Programming
Florian A. Potra
Department of Mathematics and Statistics
University of Maryland Baltimore County
Baltimore, MD 21250, USA
Research supported by NIH, Grant No. R01GM075298-01
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Proteomics and 2D-PAGE
Proteomics, the study of proteome, is the large-scale analysis of
complex protein mixtures focusing on the qualitative and
quantitative variations of protein expression levels.
By separating, cataloging, and comparing proteins from normal
and diseased cells and tissues we gain invaluable knowledge about
the changes taking place in complex biological systems at the
molecular level, which in turn leads to better diagnostics and
therapeutics.
Two-dimensional polyacrylamide gel electrophoresis (2D-PAGE) is
the core technology to simultaneously separate and quantitate up
to 10000 proteins. With this technique, proteins can be separated
according to their isoelectric point (pl) and relative mass (Mr).
Main challenge of Proteomics: Compare large collections of gels for
proteomics studies. Gels must be aligned before comparison.
Bottleneck: Gel Alignment
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Commercial Software Packages
Delta 2D (DECODON, Greifswald, Germany)
PD Quest (Bio-Rad, Hercules, CA, USA)
Phoretix 2D and Progenesis (Nonlinear Dynamics, Newcastle upon
Tyne, UK)
Image Master 2D and Decyder (Amersham Biosciences)
GELLABII (Scanalytics, Fairfax, VA, USA)
Melanie 3 (GeneBio, Geneva, Switzerland)
Investigator HT Analyzer (Genomic Solutions, Ann Arbor, MI, USA)
KEPLER (Large Scale Biology, Germantown, MA, USA)
Bio Image 2D Investigator (Genomic Solutions)
Z3 (Compugen)
All are deficient on gel alignment
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Gel Alignment
Goal of alignment :
Given I(1), I(2), . . . , I(m) A collection of m 2D gels
Obtain I(1), I(2)
, . . . , I(m)the collection of aligned gels via
geometric transformations.
The transformations are optimally determined to compensate for the
gel deformations and distortions due to differences in gel preparation
and processing in different or even the same laboratory.
Distortions are caused by:
the structure of the polyacrylamide net
the characteristics of the transporting solute
the solvent conditions
the nature of the electric field
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An Example of Gel Alignment
Figure 1: I(1), I(2),I(3), I(4) A collection of 2D gels
Figure 2: I(1),I(2)
, I(3), I(4)
Aligned gels
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Superimposition Before and After Alignment
Figure 3: Superimposition of gels 1 and 2 before and after alignment
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Traditional Approach
A reference gel I(r) is chosen. For each source gel I(s) (s 6= r), a
corrected gel I(s)is obtained either by using a forward transformation
Ts,r : I(s) → I(r), I(s)(x, y) = T−1
s,r (u, v),
or an inverse transformation
T r,s : I(r) → I(s), I(s)(x, y) = T r,s(u, v).
Since I(r) is the chosen reference gel, I(r)
= I(r).
Major Disadvantages:
Manually picked reference gel may be an outlier with substantial
distortion itself, in which case the transformation will be badly scaled
and no future comparison could be done after the alignment.
Since I(r) may not be representative for the whole family. Problems
may arise for assigning intensity values for I(s)with both forward and
inverse transformations.
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Drawback of Forward Transformations
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Drawback of Inverse Transformations
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Assigning the Landmarks
We consider alignment algorithm based on preassigned landmarks
(control points) on the gel images.
Landmarks are a relatively small group of spots present in all gels
being compared.
Landmarks should be selected over the gel image and the number
of the landmarks should be large enough so that they can carry
enough information about the whole image.
Too many landmarks will result in over-fitting or an inefficient
algorithm.
Sizes of the landmarks vary for different types of proteins. Position of
the center of the landmark is considered (instead of the whole
landmark) for alignment.
li1, li2, . . . , lin are the assigned landmarks on gel I(i). There are n
landmarks on each of the m gel images.
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Traditional Approach Based on Forward and In-verse Transformations
Only positions of the assigned landmark centers are used for alignment.
ls1, ls2, . . . , lsn assigned landmarks on the source image.
lr1, lr2, . . . , lrn assigned landmarks on the reference image.
Traditional approach based on a forward transformation:
Find transformation Ts,r : I(s) → I(r) that minimizes
nX
j=1
‖Ts,r(lsj)− lrj‖2 .
Traditional approach based on an inverse transformation:
Find transformation Tr,s : I(r) → I(s) that minimizes
nXj=1
‖lsj − Tr,s(lrj)‖2 .
The transformation T is given by (u, v) = T (x, y) = (f(x, y), g(x, y)), where
f and g belong to a certain class of functions.
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Choices of Transformations
general polynomial transformation of order µ
f(x, y) =
µX
κ=0
κX
ι=0
aκ−ι,ιxκ−ιyι , g(x, y) =
µXκ=0
κXι=0
bκ−ι,ιxκ−ιyι .
affine transformations
f(x, y) = a10x + a01y + a00 , g(x, y) = b10x + b01y + b00 .
bilinear transformations
f(x, y) = a10x + a11xy + a01y + a00 , g(x, y) = b10x + b11xy + b01y + b00.
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Our Approach - Forward Transformations
In contrast with traditional approach we align the whole family of gels
simultaneously. We do not choose a reference gel from the family.
Instead, an ideal reference gel I together with the transformations
Ti : I(i) → I are obtained as the solution of a large scale optimization
problem.
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Our Approach: Formulation
Given I(1), . . . , I(m) (family of gels), li1, . . . , lin ∈ I(i) (landmarks);
Find I ideal reference gel with lj ∈ I ideal landmarks,
Ti : I(i) → I continuous transformations
such that an appropriate objective function
ϕ(T1, . . . , Tm, l1, . . . , ln)
is minimized under the following constraints
Ti ∈ C class of admissible transformations,
‖Ti(lij)− lj‖∞ ≤ ε , i = 1, . . . , m ; j = 1, . . . , n ,
‖lj −1
m
mXi=1
lij‖∞ ≤ δ, j = 1, . . . , n .
The objective function is constructed in such a way as to maximize the
smoothness of the transformations.
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Classes of Functions
We look for classes C such that ϕ becomes a quadratic function in
l1, . . . , ln and the parameters defining the class. A quadratic
programming (QP) problem can be solved efficiently by interior-point
method.
Examples of classes C:
piecewise affine transformations
piecewise bilinear transformations
other low order piecewise polynomial transformations
low order piecewise polynomial transformations using hierarchical grid
can avoid the drawbacks and limitations of global transformations
based on higher order polynomials.
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Hierarchical Approach
Piecewise Affine or Bilinear Transformations using Hierarchical Grid
Each gel image is divided into p× p equal rectangles. An affine or a
bilinear transformation is defined locally for each of the rectangle. A
global piecewise affine or bilinear transformation is then constructed
from the local transformations. Due to the fact that any line under an
affine transformation is still a line, and any vertical or horizontal line
under a bilinear transformation is still a line, it is enough to enforce
continuity at the corners of the rectangles (the intersections of the grid).
Global smoothness is optimized by a quadratic objective function. If our
accuracy requirements are not satisfied, then the grid is refined by
dividing each rectangle into four equal sub-rectangles (p← 2p), and a
similar construction of a global transformation is performed.
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Hierarchical grid Using Affine Transformations
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Uniform pi × qi Rectangular Grid on Sample i
x-partition: xi1 < xi2 . . . < xi,pi+1, y-partition: yi1 < yi2 . . . < yi,qi+1,
vertices v(i)jk = (xij , yik).
xij = τxi + (j − 1)∆xi , j = 1, . . . , pi + 1 , ∆xi =
τxi −τx
i
pi
yik = τyi + (k − 1)∆yi , k = 1, . . . , qi + 1 , ∆yi =
τyi−τ
yi
qi
Ω(i)jk , j = 1, . . . , pi, k = 1, . . . , qi, rectangular patches
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Formulation in Coefficients of the Transformations
T(i)jk : Ω
(i)jk → R2 affine or bilinear
Continuity Constraints (CC): For each i = 1, . . . , m,
continuity at interior grid points:8>><>>: T(i)j−1,k−1(v
(i)jk ) = T
(i)j−1,k(v
(i)jk )
T(i)j−1,k(v
(i)jk ) = T
(i)jk (v
(i)jk ) j = 2, . . . , pi, k = 2, . . . , qi
T(i)jk (v
(i)jk ) = T
(i)j,k−1(v
(i)jk )
continuity at boundary grid points:8>>>>><>>>>>:
T(i)1,k−1(v
(i)1k ) = T
(i)1,k(v
(i)1k ), k = 2, . . . , qi
T(i)pi,k−1(v
(i)pi+1,k) = T
(i)pi,k(v
(i)pi+1,k), k = 2, . . . , qi
T(i)j−1,1(v
(i)j1 ) = T
(i)j,1 (v
(i)j1 ), j = 2, . . . , pi
T(i)j−1,qi
(v(i)j,qi+1) = T
(i)j,qi
(v(i)j,qi+1), j = 2, . . . , pi
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Continuity at Interior Grid Points
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Continuity at Boundary Grid Points
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Quadratic Programming (QP) Problem
minT
(i)jk
, lj
mX
i=1
piX
j=1
qiX
k=2
wD
T(i)jk (v
(i)j,k+1) + T
(i)j,k−1(v
(i)j,k−1)− 2T
(i)jk (v
(i)jk )
∆y2i
2
+
mX
i=1
piX
j=2
qiX
k=1
wD
T(i)jk (v
(i)j+1,k) + T
(i)j−1,k(v
(i)j−1,k)− 2T
(i)jk (v
(i)jk )
∆x2i
2
+
mXi=1
piXj=1
qiXk=1
wI‖T (i)jk − I‖2F
s.t. continuity constraints CC ,
‖T(i)
α(i,j),β(i,j)(lij)− lj ‖∞ ≤ ε, i = 1, . . . , m, j = 1, . . . , n ,
‖ lj − 1m
Pm
i=1 lij ‖∞ ≤ δ, j = 1, . . . , n .
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Notations Used in the QP Problem
wD and wI : constant weights in the objective
I: the identity transformation
(α(i, j), β(i, j)): the index of the rectangle that lij lies in:
lij ∈ Ω(i)α(i,j),β(i,j).
ε: a certain admissible tolerance between lj and the transformed
landmarks
δ: distance allowed between 1m
Pm
i=1 lij and lj
‖ ∗ ‖ is the l2 norm, ‖ ∗ ‖∞ is the l∞ norm and ‖ ∗ ‖F is the Frobenius
norm given by
‖A‖F =
MXi=1
NXj=1
a2ij ,
where A = (aij) is a M ×N matrix.
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Feasibility
When ε = δ = 0, we have only equality constraints. A solution for this
problem is clearly feasible when ε ≥ 0 or δ ≥ 0 or both.
Total number of equality constraints
6
mX
i=1
piqi − 2
mX
i=1
(pi + qi)− 2m + 2n + 2mn
Total number of unknowns
6
Pm
i=1 piqi + 2n for piecewise affine,
8
Pm
i=1 piqi + 2n for piecewise bilinear.
In case of pi = qi = p, it is sufficient to have
p ≥ n/2 for piecewise affine,
p ≥ √n for piecewise bilinear,
p ≥ n/8 for piecewise cubic.
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Computational Complexity
Assume final grid is obtained after k refinements
q: number of variables on the initial grid
4q: number of variables after one refinement
O(qγ): number of arithmetic operations for solving a QP problem
with q variables using interior-point method (2.5 ≤ γ ≤ 4).
Total work required using a hierarchical approach
Cqγ + C(4q)γ + . . . + C(4kq)γ = Cqγ (4γ)(k+1) − 1
4γ − 1≈ C(4kq)γ .
Total work required for the hierarchical method is only slightly higher
than the work in the finest grid. The coarseness of the final grid cannot
be estimated in advance and may differ from gel to gel.
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Formulation in Transformed Vertices
Bilinear transformations are completely determined by the coordinates
of the four vertices.
z(i)jk = (u
(i)jk , r
(i)jk )T
z = Tjk(v) = (1−λ)(1−µ) zjk +λ(1−µ) zj+1,k +(1−λ)µ zj,k+1+λµ zj+1,k+1 .
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Formulation in Transformed Vertices
minz, l
MX
i=1
piX
j=2
qi+1X
k=1
‖z(i)j+1,k + z
(i)j−1,k − 2z
(i)jk
(∆x(i))2‖2 +
pi+1Xj=1
qiXk=2
‖z(i)j,k+1 + z
(i)j,k−1 − 2z
(i)jk
(∆y(i))2‖2
!
+w1
MX
i=1
piX
j=1
qi+1X
k=1
‖r(i)j+1,k − r
(i)jk
(∆x(i))2‖2 +
pi+1Xj=1
qiXk=1
‖u
(i)j,k+1 − u
(i)jk
(∆y(i))2‖2
!
+w2
MXi=1
pi+1Xj=1
qi+1Xk=1
‖z(i)jk − v
(i)jk ‖
2
s.t. |u(i)jk − u
(i)j+1,k| ≤ ηx∆x(i) , i = 1, . . . , m, j = 1, . . . , pi, k = 1, . . . , qi + 1 ,
|r(i)jk − r
(i)j,k+1| ≤ ηy∆y(i) , i = 1, . . . , m, j = 1, . . . , pi + 1, k = 1, . . . , qi ,
‖T(i)
α(i,j),β(i,j)(lij)− lj ‖∞ ≤ ε, i = 1, . . . , m, j = 1, . . . , n ,
‖ lj − 1m
Pm
i=1 lij ‖∞ ≤ δ, j = 1, . . . , n .
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Drawbacks
Curse of Dimensionality: number of unknowns proportional with the
number of gels in the collection.
Difficulty in assigning intensity values for all pixels in the ideal gel: it is
not guaranteed that each pixel of the ideal gel is in the range of the
forward transformation.
Note : Inverse transformation cannot be used for determining ideal
landmarks since the corresponding optimization problem is no longer
convex.
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Solution
The forward transformation (FT) is carried only on a coarse grid to
obtain the ideal landmarks and the approximate size of I.
Intensity of aligned image I(i)is assigned by using an inverse
transformation (IT) T(i)
: I → I(i) for each i. T(i)
’s are piecewise
affine or bilinear transformations obtained using hierarchical grids.
Since the ideal landmarks are obtained by FT, the T(i)
’s are
obtained by solving m independent QPs.
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Assign Intensities Using IT
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Gel Obtained From FT
Figure 4: The figure shows an image before and after a forward piecewise
affine transformation based on a 16 × 16 grid. Landmarks are shown as
red crosses.
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Further Refinement Using IT
Figure 5: The left figure shows the grid on the target image under an
inverse piecewise affine transformation after one more refinement, the
right figure shows the target image constructed from the inverse transfor-
mation (on a 32× 32 grid). Landmarks are shown as red crosses.
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Numerical Experiments - Data sets
Data set 1 : Gel images were obtained from an experiment in which 12
rats (6M, 6F) were treated either with nicotine (3M, 3F) or with buffer
control (3M, 3F). Subsequently the animals were sacrificed and their
spleen cells cultured (one spleen cell culture for each animal) in vitro in
the presence of the T-cell mitogen concanavalin A. After stimulation,
cells from each individual culture were harvested and proteins
solubilized for 2D gel analysis. There are 12 gel samples in this data set
and 20 landmarks were pick on each of them. Each gel on average has
about 500× 1100 pixels.
Gel images were generated by our collaborator Carol Whisnant (RTI
International) with landmarks picked by our colleague Yaming Hang
(UMBC).
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Numerical Experiments - Data sets
Data set 2 Molt-4 data: Data set contains 4 gel images (512× 512
pixels, 8-bit, 250 microns/pixel). 22 landmarks are picked on each gel.
(http://www.lecb.ncifcrf.gov/2DgelDataSets/index.html#MOLT-4)
Data set 3 : Human leukemia data: Data set contains 170 gel images
(512× 512 pixels, 8-bit, 250 microns/pixel). Most of the gel images
contain 22 landmarks, while some images contain different numbers of
landmarks. For our numerical experiments, we picked all the images
containing 22 landmarks, so that we have 123 gel images with 22
landmarks on each.
(http://www.lecb.ncifcrf.gov/2DgelDataSets/index.html#HEME-MALIG)
Data sets 2 and 3 were made available to the public by Peter Lemkin
from the National Cancer Institute.
(http://binkley.ncifcrf.gov/users/lemkin/)
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Similarity Measures
SM1: covariance between the intensity values of I and J .
SM1 =
P
i(Ii − I)(Ji − J)
N − 1.
N : the number of elements in I,
Ii: the intensity value of the ith pixel of I,
I: the mean value of I.
SM2: the cross-correlation coefficient:
SM2 =SM1
σ(I)σ(J),
where σ denotes the unbiased standard deviation.
SM3: normalized l2-norm difference:
SM3 =
rPi(Ii − Ji)2
N.
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Results for Data Set 1
Comparison of piecewise affine and bilinear transformations for the
mice spleen cells data, with perfectly matched landmarks (ε = 0):
Image 1 v.s. Image 4
SM1 SM2 SM3
Source Images 413 0.29 46.98
Affine FT 16× 16 grid 1631 0.92 17.27
IT 32× 32 grid 1620 0.97 10.48
Source Images 413 0.29 46.98
Bilinear FT 16× 16 grid 1635 0.91 18.04
IT 32× 32 grid 1641 0.97 11.30
Table 1: A sample comparison of the mice spleen cell data.
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Results for Data Set 1
Figure 6: The left figure shows a superimposition of two source images.
The center part of the left figure is enlarged into the right figure.
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Results for Data Set 1
Figure 7: The left figure shows the superimposition of the two images af-
ter performing a forward transformation based on a 16 × 16 grid. The
right figure shows the superimposition of the two images after performing
another inverse transformation based on a 32× 32 grid.
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Results for Data Set 2
Comparison of piecewise affine and bilinear transformations for the
MOLT-4 data, with perfectly matched landmarks (ε = 0):
Image 1 v.s. Image 2
SM1 SM2 SM3
Source Images 1816 0.68 41.17
Affine FT 16× 16 grid 858 0.82 19.30
IT 32× 32 grid 783 0.85 16.77
Source Images 1816 0.68 41.17
Bilinear FT 16× 16 grid 806 0.77 21.70
IT 32× 32 grid 694 0.79 19.06
Table 2: A sample comparison of the MOLT-4 data.
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Results for Data Set 2
Figure 8: Alignment result of a pair of images in the MOLT-4 data, using
piecewise affine transformations. Left figure shows the superimposition
of the source images. Right figure shows the enlargement of the super-
imposition after performing forward transformations followed by inverse
transformations.
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Results for Data Set 2
Figure 9: Alignment result of a pair of images in the MOLT-4 data, using
piecewise bilinear (left) and piecewise cubic (right) transformations. In
each case forward transformations were followed by inverse transforma-
tions.
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Results for Data Set 3
Comparison of piecewise affine and bilinear transformations for the
human leukemia data, with ε = 5 (5 pixels tolerance is considered very
good in proteomics studies). Due to the huge computational cost, the
forward transformation meet its limit at a 14× 14 grid:
Image 1 v.s. Image 2
SM1 SM2 SM3
Source Images 2778 0.93 23.04
Affine FT 14× 14 grid 890 0.88 16.63
IT 32× 32 grid 895 0.92 13.42
Source Images 2778 0.93 23.04
Bilinear FT 14× 14 grid 893 0.87 17.05
IT 32× 32 grid 879 0.93 13.04
Table 3: A sample comparison of the human leukemia data.
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Results for Data Set 3
Figure 10: Alignment result of a pair of images in the human leukemia
data, using piecewise bilinear transformations. Left figure shows the su-
perimposition of the source images. Right figure shows the enlargement
of the superimposition after performing forward transformations followed
by inverse transformations.
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Future Work
Work with statisticians and biologists to derive better similarity
measures.
Use other classes of transformations (cubic splines, thin plate splines,
etc.)
Develop alignment algorithms that do not require preassigned
landmarks.
In the human leukemia data, some landmarks were not picked
accurately, and landmarks were not well distributed over the images.
Preassigning landmarks requires extensive human intervention, which
prevents automated analysis of large collection of gels.
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