Protein Image Alignment via Quadratic Programmingpotra/ProteinPresRev2.pdf · Protein Image Alignment via Quadratic Programming Florian A. Potra [email protected] Department of

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Protein Image Alignmentvia Quadratic Programming

Florian A. Potra

[email protected]

Department of Mathematics and Statistics

University of Maryland Baltimore County

Baltimore, MD 21250, USA

Research supported by NIH, Grant No. R01GM075298-01

Protein Image Alignment via Quadratic Programming – p.1/47

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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


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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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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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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MOLT-4 data, with perfectly matched landmarks (ε = 0):


SM1 SM2 SM3


Affine FT 16× 16 grid 858 0.82 19.30

IT 32× 32 grid 783 0.85 16.77


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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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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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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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:


SM1 SM2 SM3


Affine FT 14× 14 grid 890 0.88 16.63

IT 32× 32 grid 895 0.92 13.42


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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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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Major References

Andrew W. Dowsey, Michael J. Dunn, and Guang-Zhong Yang. The

role of bioinformatics in two-dimensional gel electrophoresis.

Proteomics, 3:1567–1596, 2003.

C. A. Glasbey and Mardia K. V. A review of image warping methods.

Journal of Applied Statistics, 25:155–171, 1998.

A. Goshtasby. Piecewise linear mapping functions for image

registration. Pattern Recognition, 19:459–466, 1986.

A. Goshtasby. Piecewise cubic mapping functions for image

registration. Pattern Recognition, 20:525–533, 1987.

J. S. Gustafsson, A. Blomberg, and M. Rudemo. Warping

two-dimensional electrophoresis gel images to correct for geometric

distortions of the spot pattern. Electrophoresis, 23:1731–1744, 2002.

K. Kaczmarek, B. Walczak, S. de Jong, and B.G.M. Vandeginste.

Matching2d gel electrophoresis images. Journal of Chemical

Information and Computer Sciences, 43:978–986, 2003.


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Major References

J. Klose. Protein mapping by combined isoelectric focusing and

electrophoresis: a two-dimensional technique. Humangenetik,

26:231–234, 1975.

P.H. O’Farrell. High resolution two-dimensional electrophresis of

proteins. J. Biol. Chem., 250:4007–4021, 1975.

F. A. Potra and X. Liu. Aligning families of 2D-gels by a combined

hierarchical forward-inverse transformation approach. Journal of

Computational Biology, 13(7):1384–1395, 2006.

F. A. Potra and X. Liu. Protein image alignment via tensor product

cubic splines. Optimization Methods and Software, 22(1):155–168,

2007.

F. A. Potra, X. Liu, F. Seillier-Moiseiwitsch, A. Roy, Y. Hang, M. R. Marten,

and B. Raman. Protein image alignment via piecewise affine

transformations. Journal of Computational Biology, 13(3):614–630,

2006.


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Major References

Jussi Salmi, Tero Aittokallio, Jan Westerholm, Matias Griese Griese,

Arsi Rosengren, Tuula A. Numan, Riitta Lahesmaa, and Olli

Nevalainen. Hierarchical grid transformation for image warping in

the analysis of two-dimensional electrophoresis gels. Proteomics,

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T. Voss and P. Haberl. Observations on the reproducibility and

matching efficiency of two-dimensional electrophoresis gels:

Consequences for comprehensive data analysis. Electrophoresis,

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Reiner Westermeier and Tom Naven. Proteomics in Practice.

Wiley-VCH, Weinheim, Germany, first edition, 2002.


Documents

Protein Image Alignment via Quadratic Programmingpotra/ProteinPresRev2.pdf · Protein Image Alignment via Quadratic Programming Florian A. Potra [email protected] Department of