Compressed Sensing in Imaging Mass Spectrometry Andreas Bartels 1 - - PowerPoint PPT Presentation

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Compressed Sensing in Imaging Mass Spectrometry

Andreas Bartels1

Joint work with

Patrick D¨ ulk1, Dennis Trede1,2, Theodore Alexandrov1,2,3 and Peter Maaß1,2

1 Center for Industrial Mathematics, University of Bremen, Bremen, Germany 2 Steinbeis Innovation Center SCiLS (Scientific Computing in Life Sciences), Bremen, Germany 3 MALDI Imaging Lab, University of Bremen, Bremen, Germany

Copenhagen 26.-28. March 2014

1 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Outline

1 Imaging mass spectrometry (IMS) 2 Compressed sensing in IMS 3 Numerics: Implementation & Results 4 Conclusion

2 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

Center for Industrial Mathematics Faculty 03 Mathematics/Computer Science

1 Imaging mass spectrometry (IMS) 2 Compressed sensing in IMS 3 Numerics: Implementation & Results 4 Conclusion

3 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Mass spectrometry (MS)

Technique of analytical chemistry that identifies the elemental com- position of a chemical sample based on mass-to-charge ratio of charged particles. What is it used for? drug development detect/identify the use of drugs of abuse (dopings) in athlets identification of explosives and analysis of explosives in postblast residues (puffer machine) study the interaction of two (or more) bacterial cultures detection of disease biomarkers determination of proteins, peptides, metabolites and . . .

4 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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MS methods

Matrix-Assisted Laser Desorption/Ionization (MALDI) Secondary Ion Mass Spectrometry (SIMS) Desorption Electrospray Ionization (DESI) . . .

1 sample is cut and

mounted on glass slide

2 matrix solution is applied

(acid crystalisation)

3 laser desorption of ’spots’

(grid ∼ 20 µm – 200 µm)

4 computer aided analysis

• f m/z-slices

4 (13) Soft laser desorption (SLD) During the 1980s several groups tried to solve the volatilisation/ionisation problem of mass spectrometry using laser light as an energy source. By focusing a light beam onto a small spot

• f a liquid or solid sample, one hoped to be able to vaporise a small part of the sample and still

avoid chemical degradation. V.S. Letokhov in Moscow demonstrated that the method could work for small but polar molecules, like amino acids. This approach was further developed by

• M. Karas and F. Hillenkamp in Münster. In 1985, these scientists showed that an absorbing

matrix could be used to volatilise small analyte molecules, but were without initial success for large molecules. A breakthrough for the laser desorption method in its application to large biomolecules was reported at a symposium in Osaka in 1987, when Koichi Tanaka at the Shimadzu Corp. in Kyoto presented results of a mass spectrometric analysis of an intact protein. In two publica- tions and lectures in 1987-1988, Tanaka presented ionisation of proteins such as chymotrypsi- nogen (25,717 Da), carboxypeptidase-A (34,472 Da) and cytochrome c (12,384 Da) [12-14]. The missing link to make laser desorption work for large macromolecules was a proper com- bination of laser energy and wavelength with the absorbance and heat transfer properties of a chemical/physical matrix plus the molecular structure of the analytes in this matrix. Tanaka showed that gaseous macromolecular ions could be formed using a low-energy (nitrogen) laser,

Soft Laser Desorption Laser

+ +

sample in matrix

+ + 2+ 2+ + + + + 2+ + + + 20000 10000 30000 40000 50000 60000

• Rel. Intensity

m/z + 2+ + Figure 2. The soft laser desorption process.

TOF – Time-of-flight [Markides et al.’02] Sample 5 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Matrix-assisted laser desorption/ionization

A) optical image B)

5 mm

nx ny m/z s p e c t r a intensity

4,000 5,000 6,000 7,000 8,000 9,000

[Alexandrov et al.’11] 6 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Matrix-assisted laser desorption/ionization

A) optical image B)

5 mm

nx ny m/z s p e c t r a intensity

4,000 5,000 6,000 7,000 8,000 9,000

[Alexandrov et al.’11] 6 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Matrix-assisted laser desorption/ionization

A) optical image B)

5 mm

nx ny m/z s p e c t r a intensity

4,000 5,000 6,000 7,000 8,000 9,000

[Alexandrov et al.’11] 6 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Matrix-assisted laser desorption/ionization

A) optical image B) C) m/z 4,966 D) m/z 6,717

5 mm

nx ny m/z s p e c t r a intensity

4,000 5,000 6,000 7,000 8,000 9,000 m/z 4,966 m/z 6,717

[Alexandrov et al.’11] 6 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Matrix-assisted laser desorption/ionization

A) optical image B) C) m/z 4,966 D) m/z 6,717

5 mm

nx ny m/z s p e c t r a intensity

4,000 5,000 6,000 7,000 8,000 9,000 m/z 4,966 m/z 6,717

[Alexandrov et al.’11]

= ⇒ IMS data: Hyperspectral data X ∈ R

nx ×ny ×c +

(m/z-spectra and -images)

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The information disaster – data overflow

Data X ∈ Rnx×ny×c

+

typically contains nx · ny = 10,000 − 100,000 pixels c = 10,000 − 100,000 m/z-values 108 − 1010 values, altogether Write X ∈ Rn×c

+

, n = nx · ny. (General) Questions: How to interpret the data? What is the main information? How to compress the data? Where to compress the data?

m/z x y

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Compression perspectives

Mass spectrometry data X ∈ Rn×c

+

is typically large! Nonnegative matrix factorization X ≈ MS, where M ∈ Rn×ρ

+

and S ∈ Rρ×c

+

with ρ ≪ min{n, c}.

• min

M,S αΘ1(M) + βΘ2(S)

s.t. X − MSF ≤ ε M – pseudo m/z-images, S – pseudo spectra Compressed Sensing Y = ΦX ∈ Rm×c

+

, where Φ ∈ Rm×n

+

, m ≪ n.

• min

X

αΘ1(X) + βΘ2(X) s.t. Y − ΦXF ≤ ε

8 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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1 Imaging mass spectrometry (IMS) 2 Compressed sensing in IMS 3 Numerics: Implementation & Results 4 Conclusion

9 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Compressed Sensing in IMS

Problems: MALDI measurements require several hours in time Data interpretation on full data Example: Rat brain dataset ∼ 5 hours Idea: Make use of compressed sensing with the knowledge of sparse m/z-spectra (ℓ1 minimization) and sparse m/z-images (TV minimization)

10 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Compressed Sensing in IMS

Problems: MALDI measurements require several hours in time Data interpretation on full data Example: Rat brain dataset ∼ 5 hours Idea: Make use of compressed sensing with the knowledge of sparse m/z-spectra (ℓ1 minimization) and sparse m/z-images (TV minimization)

• A.B., P. D¨

ulk, D. Trede, T. Alexandrov and P. Maaß, ”Compressed Sensing in Imaging Mass Spectrometry”, Inverse Problems, 29(12), 125015 (24pp), 2013.

10 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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CS-IMS model - The data

IMS data is a hyperspectral data cube consisting of nx · ny (m/z-)spectra of length c (number of channels), whereas nx and ny are the number of pixels in each coordinate direction. Thus, X ∈ Rnx×ny×c

+

. Concatenating each m/z-image as a vector the data X becomes X ∈ Rn×c

+

, where n := nx · ny. Each column corresponds to one m/z-image Each row to one m/z-spectrum.

11 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Sparsity in IMS data - m/z-spectra

detail region (b) 4,000 5,000 6,000 7,000 8,000 9,000 0.1 0.2 0.3 m/z-value

• Rel. intensity (arb.u.)

(a) m/z-value (Da) 5,475 5,525 0.04 0.07 number of m/z Pixel spectrum

⊸Picked peak

. coefficients Basis function (b) m/z-value (Da)

Sparsity of the spectra in a basis Ψ ∈ Rc×c

+

Only a few peaks arise with high intensities = ⇒ Feature extraction via ℓ1 minimization [Denis et al.’09] Shifted Gaussians: ψk(x) =

1 π1/4σ1/2 exp

• − (x−k)2

2σ2

• ,

k = 1, .., c

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Sparsity in IMS data - m/z-spectra

Let X T

(k,·) ∈ Rc +, k = 1, . . . , n, be the k-th row of X ∈ Rn×c +

, i.e.

• ne spectrum. We assume the spectra to be sparse or compressible

in a (known) basis Ψ ∈ Rc×c

+

, i.e. X T

(k,·) = Ψλ,

λ ∈ Rc

+

(1) at which λ0 ≪ c. With coefficient matrix Λ ∈ Rc×n

+

, Eq. (1) reads X T = ΨΛ. = ⇒ Minimize columns Λ(·,k) of Λ w.r.t. the ℓ0 ’norm’, i.e. Λ(·,k)0 for k = 1, . . . , n.

13 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Sparsity in IMS data - m/z-spectra

Let X T

(k,·) ∈ Rc +, k = 1, . . . , n, be the k-th row of X ∈ Rn×c +

, i.e.

• ne spectrum. We assume the spectra to be sparse or compressible

in a (known) basis Ψ ∈ Rc×c

+

, i.e. X T

(k,·) = Ψλ,

λ ∈ Rc

+

(1) at which λ0 ≪ c. With coefficient matrix Λ ∈ Rc×n

+

, Eq. (1) reads X T = ΨΛ. = ⇒ Minimize columns Λ(·,k) of Λ w.r.t. the ℓ1 ’norm’, i.e. Λ(·,k)1 for k = 1, . . . , n.

13 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Sparsity in IMS data - m/z-images

Sparsity of the m/z-images

• [Alexandrov et al.’10]:

m/z-images of imaging mass spectrometry data usually inherent large variance of noise are piecewise constant = ⇒ Apply TV denoising on each m/z-image.

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Sparsity in IMS data - m/z-images

Recall that Λ ∈ Rc×n

+

is the coefficient matrix in X T = ΨΛ. = ⇒ Minimize rows Λ(k,·) of Λ w.r.t. the TV norm, i.e. Λ(j,·)TV for j = 1, . . . , c.

1, 000 2, 000 3, 000 4, 000 5, 000 6, 000 0.5 1.0 m/z-value (Da)

• Rel. intensity (arb.u.)

Instead of finding a reconstruction ˜ X T = Ψ˜ Λ, we aim to directly recover the features ˜ Λ.

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The compressed sensing process

IMS data X ∈ Rn×c

+

acquisition: Ionizing the given sample on each

• f the n pixels on a predefined grid

.

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The compressed sensing process

IMS data X ∈ Rn×c

+

acquisition: Ionizing the given sample on each

• f the n pixels on a predefined grid Compressed sensing: m ≪ n.

16 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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The compressed sensing process

IMS data X ∈ Rn×c

+

acquisition: Ionizing the given sample on each

• f the n pixels on a predefined grid Compressed sensing: m ≪ n.

Take m measurements yi ∈ Rc

+, i = 1, .., m:

yij = ϕi, X(·, j), j = 1, .., c, ϕi ∈ Rn

+,

ϕi from sub-gaussian distribution Each yi for i = 1, .., m is a measurement-mean spectrum since it is calculated by the mean intensities on each channel: yT

i

= ϕT

i X = n

• k=1

ϕikX(k,·), yT

i

are linear combinations of the original spectra X(k,·), k = 1, .., n.

16 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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CS-IMS model

CS in IMS data acquisition: Ionizing the given sample on randomly selected pixels on a predefined grid. yi T = ϕi TX =

n

• k=1

ϕikX(k,·), X T = ΨΛ.

• ptical image

5 mm

nx ny m/z s p e c t r a intensity

4000 5000 6000 7000 8000 9000

17 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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CS-IMS model

CS in IMS data acquisition: Ionizing the given sample on randomly selected pixels on a predefined grid. y1T = ϕ1TX =

n

• k=1

ϕ1kX(k,·), X T = ΨΛ.

• ptical image

5 mm

nx ny m/z s p e c t r a intensity

4000 5000 6000 7000 8000 9000

17 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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CS-IMS model

CS in IMS data acquisition: Ionizing the given sample on randomly selected pixels on a predefined grid. yT

2/3 = ϕT 2/3X = n

• k=1

ϕ2/3kX(k,·), X T = ΨΛ.

• ptical image

5 mm

nx ny m/z s p e c t r a intensity

4000 5000 6000 7000 8000 9000

17 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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CS-IMS model

CS in IMS data acquisition: Ionizing the given sample on randomly selected pixels on a predefined grid. yT

2/3 = ϕT 2/3X = n

• k=1

ϕ2/3kX(k,·), X T = ΨΛ. In matrix form this becomes Rm×c

+

∋ Y = ΦX = ΦΛTΨT Y = ΦX + Z, ZF ≤ ε Ψ ∈ Rc×c

+

• Dictionary,

Λ ∈ Rc×n

+

• Coefficients
• ptical image

5 mm

nx ny m/z s p e c t r a intensity

4000 5000 6000 7000 8000 9000

17 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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CS-IMS model

CS in IMS data acquisition: Ionizing the given sample on randomly selected pixels on a predefined grid. yT

2/3 = ϕT 2/3X = n

• k=1

ϕ2/3kX(k,·), X T = ΨΛ. In matrix form this becomes Rm×c

+

∋ Y = ΦX = ΦΛTΨT Y = ΦX + Z, ZF ≤ ε Ψ ∈ Rc×c

+

• Dictionary,

Λ ∈ Rc×n

+

• Coefficients
• ptical image

5 mm

nx ny m/z s p e c t r a intensity

4000 5000 6000 7000 8000 9000

min

Λ∈Rc×nα c

• j=1

Λ(j,·)TV + βΛ1, s.t. Y − ΦΛTΨTF ≤ ε, Λ ≥ 0

17 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Theorem: Robustness

A.B., P.D¨ ulk, D.Trede, T.Alexandrov and P.Maaß, ”CS in IMS”, Inverse Problems, 29(12), 125015 (24pp), 2013.

Let M : Rc×n → R4cm1 × Rm2×c be the linear operator with components M(Λ) =

• A0Λ1, A0Λ1, A′0Λ1, A′

0Λ1, . . . , A0Λc, A0Λc, A′0Λc, A′ 0Λc, ΦΛTΨT

• .

If noisy measurements Y = M(Λ) + Z are observed with noise level ZF ≤ ε, then Λ⋄ = argmin

W ∈Rc×n W 1 + c

• i=1

WiTV s.t. M(W ) − Y F ≤ ε, satisfies both Λ − Λ⋄F +

c

• i=1

∇Λi − ∇Λ⋄

i F

1 √ K

• Λ − ΛS01 +

c

• i=1
• ∇Λi − (∇Λi)Si
• 1
• + ε,

and Λ − Λ⋄1 +

c

• i=1

Λi − Λ⋄

i TV Λ − ΛS01 + c

• i=1
• ∇Λi − (∇Λi)Si
• 1 +

√ Kε.

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Theorem: Robustness - The tools

• 1. RIP

The linear operator A : Rnx×ny → Rm×p has the restricted isometry property of order s and level δ ∈ (0, 1) if (1 − δ)X2

F ≤ A(X)2 F ≤ (1 + δ)X2 F

for all s-sparse X ∈ Rnx×ny .

• 2. D-RIP (extends the RIP to matrices adapted to a dictionary)

A linear operator A : Rnx×ny → Rm×p has the D-RIP of order s and level δ∗ ∈ (0, 1), adapted to a dictionary D, if for all s-sparse X ∈ Rnx×ny it holds (1 − δ∗)DX2

F ≤ A(DX)2 F ≤ (1 + δ∗)DX2 F.

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Theorem: Robustness - The tools

• 3. A-RIP

A matrix D ∈ Rnx×nx satisfies the asymmetric restricted isometry property (A-RIP), if for all s-sparse X ∈ Rnx×ny the following inequalities hold: L(D)XF ≤ DXF ≤ U(D)XF, where L(D) and U(D) are the largest and the smallest constants for which the above inequalities hold. The restricted condition number of D is defined as ξ(D) = U(D) L(D) ≤ maxXF =1 DXF minXF =1 DXF = κ(D).

20 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Theorem: Robustness - The tools

min

Λ∈Rc×n c

• j=1

Λ(j,·)TV + Λ1, s.t. Y − ΦΛTΨT

• =:DΦ,ΨΛ

F ≤ ε, Λ ≥ 0 Argue that DΦ,Ψ fulfils the D-RIP Argument via Kronecker product and blockdiagonal RIP results [Eftekhari, A, et al.’12] Ψ fulfils the A-RIP Ψ will consist of shifted Gaussians Ψ is invertible, i.e. ξ(Ψ) is bounded by κ(Ψ) B = [A A′, . . . , A A′] (operator with artificial gradient measurements) fulfils the RIP Argument similar as for DΦ,Ψ.

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1 Imaging mass spectrometry (IMS) 2 Compressed sensing in IMS 3 Numerics: Implementation & Results 4 Conclusion

22 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Algorithm (PPXA)

We aim to solve the following problem min

Λ∈Rc×nα c

• j=1

Λ(j,·)TV + βΛ1, s.t. Y − ΦΛTΨTF ≤ ε, Λ ≥ 0. We use the parallel proximal splitting algorithm (PPXA) [Combettes&Pesquet’08] which solves problems of the kind: min

x∈H ℓ

• i=1

fi(x), where H is a Hilbert space and (fi)1≤i≤ℓ are proper lower semicontinuous convex functions fi : H → ] − ∞, +∞]

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Algorithm (PPXA)

Here: H = Rc×n, ℓ = 4 and f1(Λ) = α

c

• i=1

ΛiTV , f2(Λ) = βΛ1, f3(Λ) = ιBε

2(Λ),

f4(Λ) = ιB+(Λ). ιC is the indicator function, ιC(Λ) =

• if Λ ∈ C

+∞

• therwise ,

applied to the convex sets Bε

2 = {A ∈ Rc×n : Y − DΦ,ΨAF ≤ ε} (Fidelity constraint),

B+ = {A ∈ Rc×n : A ≥ 0} (Positive orthant).

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The proximity operator. . .

. . . is defined as proxf : H → H. proxf (X) is the unique point in H that satisfies proxf (X) = argmin

Y ∈H

1 2X − Y 2

F + f (Y ).

For f1 (TV-norm): Via an implementation from [Beck’09]. For f2 (1-norm): proxγ·1(Z) =

• max
• 0,
• 1 −

γ |Zi,j|

• Zi,j
• 1≤i≤c

1≤j≤n

For f3 (Bε

2): Via a Douglas-Rachford splitting scheme [Fadili’09]

For f4 (B+): proxγιB+(·)(Z) =

• max{0, Zi,j}
• 1≤i≤c

1≤j≤n 25 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Parallel proximal splitting algorithm

Algorithm 1: PPXA Input: Y , Ψ, Φ, α, β, ε, γ > 0 Initializations: k = 0; Λ0 = Γ1,0 = Γ2,0 = Γ3,0 = Γ4,0 ∈ Rc×n repeat for j = 1 : 4 do Pj,k = proxγfj(Γj,k) Λk+1 = (P1,k + P2,k + P3,k + P4,k)/4 for j = 1 : 4 do Γj,k+1 = Γj,k + 2Θk+1 − Θk − Pj,k until convergence Implementations given in the UNLocBoX [Perraudin’14].

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Test-Setting - Rat brain dataset

Part of a rat brain dataset: X ∈ Rn×c

+

, n = 121 · 202, c = 2, 000. Assume mass spectra to be sparse in a basis Ψ consisting of shifted Gaussians [Denis et al.’09] ψk(x) = 1 π1/4σ1/2 exp

• −(x − k)2

2σ2

• Choose std. deviation of Ψk(x)

consistent with the data and such that the condition κ(Ψ) is small

• σ = 0.75.

Elements of measurement matrix Φ ∈ Rr×n (r ≪ n) chosen at random from an i.i.d. Gaussian distribution. Noise level ε = 3.75 × 103. Parameters α, β set by hand.

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Reconstructed mean spectrum

0.3 20% 0.3 40%

• Rel. intensity (arb.u.)

4, 000 5, 000 6, 000 7, 000 8, 000 9, 000 0.3 60% m/z-value (Da) 28 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Reconstructed m/z-image

Original 20 % 40 % 60 % 80 % 100 % 29 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Reconstructed m/z-image - Cont.

Original 20 % 40 % 60 % 80 % 100 % 30 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Kidney - X ∈ Rn×c

+

, n = 113 · 71, c = 10,000, ε = 7 × 104

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Reconstructed m/z-image

40 % Original 60 % 80 % 100 % 40 % Original 60 % 80 % 100 % 32 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Others?

Are there similar results for other MS systems?

33 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Others?

Are there similar results for other MS systems? Yes! Gao, Y., Zhu, L., Norton, I., Agar, N. Y. R., Tannenbaum, A. Reconstruction and feature selection for desorption electrospray ionization mass spectroscopy imagery Proc. SPIE 9036, Medical Imaging 2014, March 12, 2014. (DESI) From the abstract: “. . . time it takes for imaging and data analysis becomes a critical

• factor. Therefore, [. . . ] we utilize compressive sensing to perform

the sparse sampling of the tissue, which halves the scanning time.”

33 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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1 Imaging mass spectrometry (IMS) 2 Compressed sensing in IMS 3 Numerics: Implementation & Results 4 Conclusion

34 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Conclusions

First model for compressed sensing in MALDI-IMS Reconstruction of whole dataset w.r.t. its features Robustness of reconstruction with respect to noise

combines ℓ1 and TV includes two sparsity aspects

Future Prospects

How to chose regularization parameters α and β? Noise models (e.g. Poisson noise, etc.) Numerics (alternating minimization, surrogate functionals) Include sparse representation X ≈ MS, M ∈ Rm×ρ

+

, S ∈ Rρ×n

+

M – Matrix with m/z-images, S – Matrix with pseudo spectra

35 Imaging mass spectrometry Compressed sensing in IMS Numerics Conclusion

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Outlook - Nonnegative matrix factorization

minM,S X − MS2

F + α ρ i=1 MiTV + βS1

s.t. M ≥ 0, S ≥ 0.

Pham, A.B., P.M. - in progress

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Working group

Thank you! bartels@math.uni-bremen.de

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Bibliography

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spectrometry data with edge-preserving image denoising and clustering, J Proteome Research, 6, pp. 6535–6546, 2010

[2] Alexandrov, T. and Kobarg, J.H.: Efficient spatial segmentation of

large imaging mass spectrometry datasets with spatially aware clustering, Bioinformatics, 27, pp. i230–i238, 2011

[3] Denis, L., et al.: Greedy Solution of Ill-Posed Problems: Error

Bounds and Exact Inversion, Inverse Problems, 25:11, pp. 1–14, 2009

[4] Markides, K. and Gr¨

aslund, A.: Mass spectrometry (MS) and nuclear magnetic resonance (NMR) applied to biological macromolecules, Adv. inf. on the Nobel Prize in Chemistry 2002, The Royal Swedish Academy of Sciences

38

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Bibliography - Continued

[5] Golbabaee, M. and Arberet, S. and Vandergheynst, P., Distributed

compressed sensing of hyperspectral images via blind source separation, Presentation given at Asilomar conf. on signals, systems, and computers, Pacific Groove, CA, USA, November 7-10, 2010

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