1

Low-rank Convex/Sparse Thermal MatrixApproximation for Infrared-based Diagnostic

SystemBardia Yousefi †, Clemente Ibarra Castanedo, and Xavier P.V. Maldague

Computer Vision and Systems Laboratory (CVSL), Department of Electrical and Computer Engineering,Laval University, Quebec City (Quebec) G1V 0A6, Canada

† Address: Department of Radiology, University of Pennsylvania, Philadelphia PA 19104

Abstract—Non-negative matrix factorization (NMF) controlsnegative bases in the principal component analysis (PCA) withnon-negative constraints for basis and coefficient matrices. Semi-,convex-, and sparse-NMF modify these constraints to establishdistinct properties for various applications in different fields,particularly in infrared thermography. In this study, we delve intothe applications of semi-, convex-, and sparse-NMF in infrareddiagnostic imaging systems. We applied these approaches toactive and passive thermographic imaging systems to determinethe heterogeneous thermal patterns in these sets. In activethermography, three diverse specimens, carbon fiber-reinforcedpolymer composites (CFRP), poly(methyl methacrylate) (PMMA,also known as Plexiglas), and aluminum plate, were used.Quantitative analyses were performed using the Jaccard index. Inpassive thermography, 55 participants for infrared breast screen-ing selected from the Database for Mastology Research (DMR)dataset with symptomatic and healthy participants. We calculatedfive derived properties of the breast area (contrast, correlation,dissimilarity, homogeneous, and energy) by using thermal levelco-occurrence matrices (TLCMs) and trained a logistic regressionmethod to stratify between healthy and symptomatic patients. Forboth scenarios, we compared the ability of semi-, convex-, andsparse-NMF to state-of-the-art thermographic approaches, suchas principal component analysis/thermography (PCT), candidcovariance-free incremental principal component thermography(CCIPCT), sparse- PCT, and NMF. Measurement of differentdefect depths and sizes indicated significant performance forsparse-NMF (AL(d < 1mm): 90.6%, CFRP (s > 10mm): 42.7%,PLEXI (1mm < d ≤ 3mm): 46.9%, DMR: 74.1%), Semi-NMF(AL (d ≤ 1mm): 86.4%, CFRP(s > 1mm): 81.7%, PLEXI(1mm < d ≤ 3mm): 37.6%, DMR: 75.8%), and Convex-NMF(AL(d ≤ 1mm): 91.2%,CFRP (s > 10mm): 97.4%, PLEXI(d ≤ 1mm): 86.8%, DMR: 77.8%). Moreover, we tested therobustness of all these algorithms against additive Gaussian noise(3% to 20%) through the signal-to-noise-ratio (SNR). The results

This research was conducted by support of the Tier-1 Canadian researchchair in Multipolar Infrared Vision (MIVIM), at Laval University.

B. Yousefi is with the Department of Electrical and Computer Engineering,Computer Vision and Systems Laboratory (CVSL), Laval University. Hiscurrent address is at the Department of Radiology, University of Pennsyl-vania, Philadelphia PA 19104. Corresponding author: B. Yousefi (e-mail:Bardia.Yousefi.1@ulaval.ca).

C. I. Castanedo is with the Department of Electrical and ComputerEngineering, Computer Vision and Systems Laboratory, Laval University (e-mail: IbarraC@gel.ulaval.ca).

X. P. V. Maldague is with the Department of Electrical and ComputerEngineering, Computer Vision and Systems Laboratory, Laval University (e-mail: Xavier.Maldague@gel.ulaval.ca).

This is the authors version of the article published in IEEE Transactionson Instrumentation and Measurement, doi: 10.1109/TIM.2020.3031129.

revealed considerable performance for semi-, convex- and sparse-NMF. Hence, the three methods exhibit promising performance interms of accuracy and robustness for confirmation of the outlinedproperties.

Index Terms—Thermal heterogeneity, Infrared diagnostic sys-tem, Breast Cancer screening, Sparse non-negative matrix fac-torization, Semi -non-negative matrix factorization, Convex- non-negative matrix factorization.

I. INTRODUCTION

Matrix factorization techniques can effectively detect de-fects in sequence of thermal images [1], [2], [3], [4], [5],[6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Somewell-known matrix factorization methods, such as principalcomponent analysis/thermography (PCA/PCT) [1] and non-negative matrix factorization (NMF) [16], [11], [15], havebeen used for past several years in different fields. However,the major challenge of dealing with high-dimensional infraredimaging sequences in different applications persists. In thispaper, we propose a comparison assessment of low-rankmatrix approximation by applying semi-, convex-, and sparse-NMF to two different scenarios through active and passivethermographical imaging.After using PCT to detect thermal defects [16], several otheralternative approaches have been used to modify the PCT toimprove its performance by using fixed eigenvector analysismethod [17], incremental PCT [18], or candid covariance-freeincremental principal component thermography (CCIPCT) [2],[10]; these approaches resulted in better computational load bya fixed set of previously generated eigenvectors and covariancefree approach. Sparse- PCT [19], [3], [4] (or even presentedas sparse dictionary matrix decomposition [20]) considersregularization terms to increase the sparsity in the analyses andstrengthening the important bases to detect defective patterns.

Nevertheless, all these methods successfully followed thefundamental principle of thermal non-destructive testing(NDT) signals and considered the initial eigenvector as the pre-dominant bases of the entire data set. A spatial-transient-stagetensor mathematical model of the Tucker decomposition algo-rithm motivated by eigendecomposition is used in an inductivethermography system to track and characterize variations inproperties [21]. This method has also been expanded to show

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𝑩 = 𝑿𝑾 𝑨 ≥ 𝟎 ,𝑾 ≥ 𝟎

Fig. 1. Workflow of matrix factorization methods to obtain low-rank estimation of input thermal imaging stream.

spatial-sparse thermal patterns and involves electromagneticenergy along with thermography [22]; the method has consid-erable similarity to a non-negative pattern separation model byusing sparse greedy-based principal component analysis (SG-PCA) [20]. The reconstruction of input thermal sequence in-volves a combination of negative and positive bases due to lackof restriction in extracting bases and coefficients in PCT, whichmight cause overlapping among the distinct basis in low-rank matrix approximation. Non-negative matrix factorization(NMF) is a matrix factorization technique [23] similar to PCAand has additive constraints in basis and coefficient matrices;this method decomposes an input matrix into low-rank non-negative basis [23] to solve the aforementioned issue. NMF isused to estimate geometrical properties along with PCT andarchetypal analysis (AA) [5]. An approach for dimensionalityreduction, which is not an IRNDT method, is performed usingNMF for visible and infrared fusion [24]. NMF is appliedin IRNDT through two ways of computations using gradient

descend (GD) [25] and non-negative least square (NNLS) [26],[27] to evaluate cultural heritage objects and buildings; thismethod exhibits considerable performance for detecting sub-surface defects [28]. An ensemble joint sparse low-rank matrixdecomposition is presented for detecting thermal defects inCFRP specimens by using the optical pulse thermography(OPT) diagnosis system [29]. Semi-NMF and Sparse-NMFfor thermography very briefly discussed an application ofthese approaches without detailed analyses [30], [6], withoutdiscussing Convex-NMF. Despite considerable developmentsin detecting subsurface spatio-thermal defect patterns, semi-, convex-, and sparse- NMF have not been discussed yet.Moreover, the effects of non-negative constraints have not beeninvestigated in thermography applications Changing the basesin the NMF method to convert the problem to a segmentationproblem while selecting a predominant basis remains chal-lenging. This study shows an application of low-rank semi-,convex-, and sparse-NMF in thermography and breast cancer

3

screening to detect defects and identify symptomatic patients,respectively. The rest of the paper is organized as follows.In the next section, thermal transfer for imaging system inactive and passive thermography is summarized. In SectionIII, the methodology of the approach will be briefly describedby applying semi-, convex-, and sparse-NMF analysis forthermography. The experimental and computational results arepresented in Section IV, and discussion is shown in SectionV. The conclusions are summarized in Section VI.

II. THERMAL TRANSFER FOR IMAGING SYSTEM

Thermal camera captures the spatial variation of temperatureon the targeted region of interest (ROI) on specimen overthe time. This heat transient can be through active or pas-sive thermography procedure. In general, thermal transfer/heatconduction equation of a specimen can be summarized byfollowing equation:

∂T

∂t=

k

ρCρ

(∂2T

∂x2+∂2T

∂y2+∂2T

∂z2

)+ q̇ (1)

where T = T (x, y, z, t) is a temperature field, k is thermalconductivity constant from the material (W/m.K). ρ is thedensity (kg/m3), Cp is specific heat (J/kg.K), q̇(x, y, z, t) isthe internal heat generation function per unit volume, in thepassive thermography (q̇ ∼= 0 in active thermography). Otherthermal transfer (i.e. convection) through the object and thefluid were neglected here. Applying infrared thermography onbiological organs and tissues composed of fat, blood vessels,parenchymal tissues, and nerves increase the complexity ofthe previous equations. This complexity aggravates with somedegree of uncertainty regarding the rate of blood perfusionand metabolic activity. Pennes’ bioheat equation [31], [32]provides accurate thermal computations and states as follows:

ρCρ∂T

∂t= 5.(k5 T ) + ωbCb(Ta − T ) + q̇ (2)

Where ω represents the flow rate of blood, q̇ is the metabolicrate (heat generation), and b, and a in ωbCb(Ta − T ) additiveterm stand for blood, and arteries (in targeted tissue-breast),respectively.

III. METHOD

A. Related works

Let I represents a sequence with τ thermal images whereeach image has spatial dimension of N ×M . The X is theinput data constructed by appending the vectorized thermalimages, X = {x1, x2, . . . , xτ} and x1 = vec[I1]. The PCT[16] decomposes the input data, X , to UΣV T , where U and Σrepresent the eigenvector (basis) and eigenvalue (coefficient)matrices. The bases corresponding to 80% highest variancecapture the maximum thermal patterns (maximum thermalheterogeneity) among the bases, called low-rank matrix ap-proximation. However, there is no guarantee of independencyand non-overlapping property in the decomposed bases, whichincreases collinearity among the bases. This leads to generate

many similar components and difficult selecting the represen-tative basis. This considers to be a drawback of PCT despite asubstantial contrast respond of the algorithm in low-rank ap-proximation. Sparse-PCT [19], [3], [4] optimizes the selectionof the bases by adding penalty terms, that increases the sparsityin the bases (from the basis matrix B - Figure 1) and moderatesthe collinearity by restricting the domain of solution. Thermalpatterns measured by low-rank matrix approximation usingSparse-PCT show more separability among the bases compareto PCT, despite more computation load for its calculations[19], [3], [4]. This subsequently lead to ambiguity among thebases while extracting the components. Similar to PCT, theNMF algorithm can be presented by a linear combination ofk basis vectors to reconstruct the data, X , whereas limited theoverlapping bases by having additive non-negative constraintsfor coefficient and basis matrices. NMF is comparable tothe PCA decomposition yet the PCA’s basis vectors can benegative. Input data X can be shown by a linear combinationof τ bases, B = [β1, β2, . . . , βτ ] and A coefficient. This isshowed as follows:

X = BA s.t. A ≥ 0 B ≥ 0 (3)

where X ∈ R+∆×τ , ∆ = MN , A ∈ R+

τ×τ , and B ∈ R+∆×τ or

with `2 equation that provides Euclidean distance [23] (Figure1):

minA,Bf(A,B) = ‖X −BA‖2 s.t. A ≥ 0 B ≥ 0 (4)

NMF assumes that matrices X , B, and A are not negative.This property significantly increases the chance for havingbases with unique characteristic of thermal patterns as eachbasis corresponds to specific thermal patterns captured throughselecting the direction of the highest thermal variance. Hence,the NMF directly associates with clustering due to non-negative constraints of bases [23]. NMF used in thermographyand k first bases could represent thermal patterns using gradi-ent descent and non-negative least square error algorithms.Restricting both coefficients and bases in the NMF mightlimit the selection of the bases, and lead to discarding usefulinformation. This study challenges this problem by usingsemi- and convex- NMF, where the constraints are loosen todetermine their effects on selected bases.

B. Semi-, Convex- and Sparse- NMF in thermography

Semi-NMF performs matrix decomposition while A isrestricted to be non-negative but there is no restriction formatrix B. This provides freedom to the basis matrix (similarto PCT) which is restricted by coefficient matrix (like NMF).Having constrained coefficients control the bases avoidingover-expression of collinear components. Semi-NMF has noconstraint for basis vectors, which provides limited freedomto the selected bases to capture all the possible heteroge-neous thermal patterns. Similarly, convex-NMF imposes theconstraint of basis vectors lie within the column space of X:

β` = w1`x1 +wn`x2 + . . .+wMN`xMN = Xw`, B = XW(5)

4

Where there is a convex combination of the columns ofX . This restriction has the benefit that we could take thecolumns βl as weighted summations of particular data points.In fact, these columns would motivate a concept of centroidsand this restricted form of B factor refers as Convex-NMF[33]. Low-rank matrix approximation using semi- and convex-NMF provide more freedom to bases as compare to NMFwhile controlled by non-negative coefficients. It means despitepotentially negative bases, they do not overlap. This propertymakes these algorithms to perform grouping of bases driven bytheir thermal variability (similar to K-means clustering [33])on input thermal data, X . The aforementioned minimizationproblem (3) does not have concurrent convex property forboth matrices, A, B, whereas the problem is convex foreach matrix discretely. Many researches have proposed waysand optimization solution such as: GD algorithm and NNLSby Paatero and Tapper [34], multiplicative algorithm by Leeand Seung [23], a computational improved GD algorithm byLin [25], and modified alternating non-negative least squares(ANNLS) by Liu et al. [26]. To compensate the uniquenessof the decomposition and enforcing a representation of basis,sparseness constraints are proposed for NMF [27]. A `1norm penalty term has been introduced by implementing thefollowing equation:

cSparse−NMF = 1/2‖X −AB‖2F + λ‖B‖1 (6)

Let ‖B‖p is the `p-norm of B given by Lp =(∑d,m ‖Bd,m‖p)1/p similar to usual `1 penalty term to im-

itate the `0 behavior [35] to calculate B for convex A.It is the unconstrained least squares minimization with `1-norm constraint, also referred as least absolute shrinkageand selection operator (LASSO) [36]. The solution for allvalues of λ achieves through applications of the least angleregression and selection algorithm (LARS) [37]. Similar tosparse-PCT but with non-negative constraints, low rank sparse-NMF obtains by selecting k bases correspond to the highestcoefficients (as showed in Figure 1). Sparse constraint in thecalculation alleviates noisy thermal signal in the bases, whichprovides more robust thermal patterns in the region of interest(ROI). Sparse-NMF theoretically more suitable for extractionof thermal patterns as it does have minimal collinearity amongbases, which alleviates the ambiguity in selecting targetedcomponents. Moreover, sparsity in the calculation of bases,which decreases noise effect in measurement of low-rankmatrix approximation.

IV. RESULTS

The proposed procedures for thermal pattern detection wereexamined. The methods were applied for analysis of threedifferent surface panels with diverse material and thermalbreast cancer screening datasets. The resulting semi-, convex-and sparse- NMF were then compared with other state-of-the-art data post-processing algorithms. The results were assessedin the presence of additive Gaussian noise to determine thestability of methods facing noise by calculating signal-to-noiseratio (SNR).

TABLE ICLINICAL INFORMATION AND DEMOGRAPHICS OF THE DATABASE FOR

MASTOLOGY RESEARCH.

DMR - Database for Mastology Research

Age Median (±IQR) 61 (27,87)

Race Caucasian African Pardo Mulatto

28 (51%) 15 (27.3%) 11 (20%) 1 (1.8%)

Diagnosis1

Healthy2

With symptoms Without symptoms

Sick3

Unknown

24 (43.6%)

9 (16.4%) 15 (27.3%)

28 (51%)

3 (5.4%)

Family history Diabetes Leukemia None

18 (32.7%) 1 (1.8%) 36 (65.5%)

Hormone therapy (HT) Hormone replacement None

9 (16.4%) 46 (83.6%)

1 This diagnosis performed with mammography as ground truth in this Dataset. 2

Healthy term is used as non-cancerous patients, but might have symptoms 3

We use the term “sick”, which includes different types of breast cancer patients.

A. Infrared image sets

1) IR-NDT: The inspection results were obtained from thefront side of the specimens, namely, carbon fiber-reinforcedpolymer composites (CFRP), Plexiglas (PLEXI), and alu-minum (AL) plates. The inspected plates had some defectsin different depth ranges: 0.2–1 mm for CFRP plate, 1–3.5mm for Plexiglas, and 3.5–4.5 mm for aluminum. Twophotographic flashes were used: 5 ms Balcar FX 60 and6.4 kJ/flash thermal pulse. The infrared camera used wasSanta Barbara Focal plane (MWIR, nitrogen cooled, InSb,320 × 256 pixels). The acquisition parameters were set asfollows: sampling rate, fs =157 Hz; Duration, tacq = 6.37s;time step, Dt = 0.025s; truncation window, w(t)s= 6.37 s; andtotal number of frames of 250. The sampling rate was 157 Hz,and 1000 images were recorded. In two other image sets, theexperimental setup was similar to the previous experiment ofinspecting photographic flashes and infrared camera [38]. ALand PLEXI specimens were tested for defect detection andestimation of the depth of each defect. The Plexiglas plate hasa thickness of 4mm and contains six artificial defects (flat-bottom holes) located at depths of 3.0, 4.0, 3.5, and 4.5 mm(Figure 2).

2) Infrared breast cancer dataset: Fifty-five participantswho were healthy (with/without symptoms) or sick (diag-nosed by mammography) were employed for breast screening.The median age in our study sample was 61 years, andthe participants comprised 28 Caucasian (51%), 15 African

5

Fig. 2. Schematic of the specimens used for benchmarking of theproposed approach. The picture in the first column represents carbon fiber-reinforced plastic (CFRP) specimen with 25 defects of different depths andsizes. The sizes and depth of each defect is clearly indicated. The second andthird columns present PLEXI and AL specimens with six and four defects,respectively, where each defect is of a different size and depth.

(27.3%), 11 Pardo (20%), and 1 Mulatto (1.8%) women.Among the participants, 18 had history of diabetes in theirfamilies (32.7%), and 9 were undergoing hormone replacement(16.4%). All patients had IR images obtained by the followingacquisition protocol: images have spatial resolution of 640× 480 pixels and were captured by a FLIR thermal camera(model SC620) with sensitivity of less than 0.04◦C range andcapture standard of -40◦C to 500◦C [39]. Table 1 shows theclinical information and demography of the cohort.

TABLE IIJACCARD ACCURACY INDEX OF EACH ALGORITHM FOR ALL SPECIMENS

IN DIFFERENT DEPTHS AND SIZES OF DEFECTS.

Accuracy of the methods in different depths of defects

Method d 1 mm mm d 3 mm

CFRP AL PLEXI PLEXI AL PLEXI

CCIPCT 17.6 94.8 <10 25.37 90.0 12.1

PCT 39.6 87.3 69.8 64.73 37.2 54.9

NMFgd 15.9 91.7 22.9 54.7 84.2 <10

NMFnnls <10 91.8 81.3 58.2 80.1 <10

Sparse-PCT 51.4 80.6 31.1 22.6 55.3 62.6

Sparse-NMF <10 90.6 <10 46.9 83.7 21.9

Semi NMF 12.3 86.4 13.9 37.6 65.4 <10

Convex-NMF <10 91.2 86.8 28.7 32.7 <10

Accuracy of the methods in different size of defects

Method s 10 mm s 10 mm

PLEXI CFRP CFRP AL

CCIPCT 28.9 16.4 >95 90.8

PCT 45.5 37.3 >95 48.1

NMFgd 40.8 15.3 43.1 84.8

NMFnnls 54.1 <10 47.6 81.2

Sparse-PCT 30.2 47.3 >95 58.5

Sparse-NMF 33.9 <10 42.7 85.1

Semi NMF 26.8 17.9 81.7 70.8

Convex-NMF 31.4 <10 97.4 42.8

B. Outcome of thermal defect patterns in IRNDT

To calculate the quantitative accuracy, we employed a binaryimage as ground truth (GT) and as a reference. The GTswere labeled the pixels in defects and background of thespecimens by 1 and 0 intensity levels, respectively. A metriccalled Jaccard index was used to quantify the percent overlapbetween the GT and our predicted output and was measuredby the number of pixels shared between the GT and resultingmasks divided by the overall number of pixels across bothmasks (or area of union). Given that the binary results werecompared to the binary reference, a threshold would be usedto convert the results to binary images. In this regard, weempirically selected the optimum threshold based on thecorrectness of subsurface detection (as the location of defectsare known).

J(W,GT ) =|W ∩GT ||W ∪GT |

(7)

where W and GT are the binarized low-rank matrix approx-imation of thermal sequence and GT, respectively. Figure 3presents the results of the subsurface defect detection usingsemi-, convex-, and sparse-NMF compared with state-of-the-art approaches, such as PCT [1], candid covariance-freeincremental principal component thermography (CCIPCT) [2],NMF-gd, NMF-nnls [28], and sparse-PCT [3]. The qualitativeresults indicated the accuracy of semi-, convex-, and sparse-NMF relative to the state-of-the-art factorization methods. Theresults of semi-, convex-, and sparse- NMF relatively indicatemore discriminative patterns between defect and backgroundfor all three specimens, where sparse- NMF, similar to Sparse-PCT, showed lesser noisy approximation.

C. Depth and size of the defects

A thresholding should be applied to the approximatedmatrix to calculate a binary low-rank matrix approximationusing the presented approaches. Figure 4 and Table 2 showthe effect of different defect sizes and depths on the overallaccuracy of detecting different defects. Moreover, Figure 4represents changes in the threshold by using different pixelintensities (5% variation in each step) and its effect on theJaccard accuracy index. Table 2 indicates the highest accuracyfor each case by changing the pixel intensity.

In the case of aluminum, Tabel 2 presents the quantitativeassessment of semi-, convex-, and sparse-NMF in comparisonwith the state-of-the-art thermographic approaches and showed70.8%, 42.8%, and 85.1% accuracy with 27.5s, 1s, and 39.52scomputational load, respectively. The CCIPCT and PCT hadJaccard index range of 90.8% (< 1s) and 48.1% (0.25s), re-spectively. NMF-gd and NMF-nnls were accurate, with valuesof 84.8% (14.81s) and 81.2% (45.18s), respectively. Overall,big defects and middle-depth defects (1mm < d ≤ 3mm)showed better accuracy than the other defects (Table 2).

For the CFRP specimen, the overall range of the Jaccardaccuracy was lower due to bigger defects, thereby affectingthe detection of spatio-thermal defect patterns (Figure 3.c).PCT, CCIPCT, and sparse- PCT exhibited the highest accuracy,that is, > 95% (1s, 7.2s, and 63.7s), whereas convex-NMFand semi-NMF had 81.7% and 97.4% accuracy within 17.53s

6

CCIPCT PCT NMF-gd NMF-nnls SparsePCT SparseNMF

AL

PLE

XI

CFR

P

a.

b

.

c

.

SemiNMF CvxNMF

Fig. 3. (a-c) Low-rank matrix factorization for aluminum, Plexiglas, and CFRP plates for active thermography.

AL

Defects Depth ≤ 1mm 1mm <Defects Depth ≤ 3 mm

PLE

XI

Defects Depth ≤ 1mm 1mm < Defects Depth ≤ 3 mm Defects Depth > 3mm

CFR

P

Defects size ≤ 10 mm Defects size > 10 mm CFRP-overall

PLEXI-overall

a. b.

Diffe

ren

t De

pth

s

c.

Diffe

ren

t Size

s

AL-overall

d.

Fig. 4. Different depths and sizes of the defects for three specimens. To estimate such accuracy, we changed the pixel intensity by 5% in each step (X-coordinate), applied different binarized thresholds, and calculated Jaccard index accuracy. Columns a-c represent the accuracy of defect detection for depthsand sizes with the respect to the type of specimens.

and 286.57s, respectively (for s > 10mm). Sparse-NMFshowed low accuracy due to hard tuning of the regularizationparameters to obtain basis elements in the decomposition; assuch, the thermal defective pattern were poorly detected (¡10%) (Table 2). PLEXI showed higher accuracy for less deep

defects than the other depths, and Convex-NMF and NMF-nnls showed the two highest accuracies (86.8%- 9.37s and81.3%- 96.33s, respectively). Overall, the average of defectdetection for PLEXI reached 36.45% through Jaccard indexmeasurement (Table 2 and Figure 8).

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D. Results of thermal breast cancer screening

Seven low-rank matrices were extracted from every par-ticipant, where each case had 23 thermal sequences. Outof 7 low-rank matrices, one matrix manually selected basedon having relatively the best contrast property. The selectedrepresentative images are shown in Figure 5 for each partic-ipant. Heterogeneous thermal patterns were detected for 31participants in breast cancer screening indicating sickness orabnormality (healthy participants but with symptoms) in thebreast area (see Figure 5.a-c). More homogeneous thermalpatterns were detected among the healthy participants (healthywith no symptoms, Table 1, Figure 5.d-f).

E. Outcomes of thermal breast cancer screening

Statistical analyses were performed to measure the thermalheterogeneity of the breast area by using texture analyses.The thermal measurements were encoded in the gray levelintensity. The thermal level co-occurrence matrices (TLCMs)of the breast area were calculated for the ROI [40]. For eachpatch, TLCM with a horizontal offset of 4 (two distances (0, 1)and two angles [0, π/2]) were computed to capture thermalpatterns in the ROI.

Four properties of the TLCM matrices were determined tomeasure the level of contrast, dissimilarity, correlation, energy,and homogeneity among the pixels in the ROI by using thefollowing equations:Contrast:

levels−1∑i,j=0

Pi,j(i− j)2 (8)

Dissimilarity:levels−1∑i,j=0

Pi,j |i− j|2 (9)

Homogeneity:levels−1∑i,j=0

Pi,j1 + (i− j)2

(10)

Energy: √√√√levels−1∑i,j=0

P 2i,j (11)

Correlation:levels−1∑i,j=0

Pi,j

(i− µi)(j − µj)√(σ2i )(σ2

j )

(12)

We stratified the participants based on these properties andcompared with the binary GT data obtained from mam-mography information (symptomatic or non-symptomatic).Kurskal Wallis test was performed to determine the statisticaldifference between two groups (symptomatic versus non-symptomatic) of participants. Semi-, Convex-, and sparse-NMF showed statistically significant separation of the twogroups of participants when split on contrast-based TLCM(p = 0.03, Figure 6.a). Sparse- PCT, PCT, CCIPCT, andNMF-nnls did not show strong stratification ability because

they were not significantly discriminated (p > 0.1). NMF-gdshowed slight separation strength, which was not significant(p = 0.09, Figure 6.a). The plot of the TLCM properties forthe participants also showed potential separability between thegroups (Figure 6.b-d), which confirmed the results of boxplotsfor separating participants (Figure 6.a). A logistic regressionmodel fitted for multivariate thermal covariates (contrast, dis-similarity, correlation, homogeneity, and energy) was used toexamine the hypothesis that the thermal heterogeneity canbe used as a biomarker to stratify among participants (todetermine symptomaticity). The accuracy levels were 74.1%,75.9%, and 77.8% for Semi-, Convex-, and Sparse- NMF,respectively. The two highest accuracies were found to belongto NMF-gd and sparse- PCT, with values of 81.5% and 79.6%,respectively. CCIPCT and PCT were commonly showed 75.9%accuracy (Figure 7).

F. Robustness in thermal defect patterns

Noise in the sequence of thermal images is inevitable dueto the nature of thermal acquisition. The sensitivity of acquisi-tions also depends on the acquisition’s conditions, which mightintensify or mitigate the noise level in different systems. Here,the robustness of matrix factorization methods was evaluatedwhile noise was added to the input signal. The results of semi-, convex-, and sparse- NMF were estimated by Jaccard indexmeasure for additive Gaussian noise input. Table 2 representsthe accuracy for semi-, convex-, and sparse- NMF comparedwith the state-of-the-art approaches. SNR for each method wascalculated by the following equation (from [41]):

SNR3 = 10 log10|µS − µN |2

σ2N

(13)

where µS and µN are the average levels of signal in the ROIand noise, respectively. σN is the standard deviation of noisein the reference or sound region of the image [41]. Many otherdefinitions have been established for the SNR. In this study,we follow one definition for the entire comparative analysis(following [41]). In general, low-rank matrix factorizationprevents additive noise in the spatio-thermal defect patterndetection due to low-rank noise reduction procedure in suchalgorithms. Here, the robustness results of each method weredetermined by the SNR metric, and Gaussian noise was addedto the input image set. Low-rank matrix factorization analyseswere performed to each case study while the noise levelwas increased from 3 % to 20 %. The most robust methodsin the presence of noise were sparse-PCT and sparse-NMFbecause of the additive regularization parameters and relativelynonlinear (and sparse) behavior of these methods. This findingmight compensate for the low accuracy in the non-noisyenvironments and presents an advantage over the other state-of-the-art approaches. CCIPCT and Semi-NMF methods weremore stable than SNR of input with higher additive noise;however, these methods had comparatively lower overall SNRthan sparse-based methods. The computational process waspartially performed with a PC (Intel Core 2Quad CPU, Q6600,2.40 GHz, RAM 8.00 GB, 64-bit Operating System) by usingMATLAB and Python programming language [42].

8

CC

IPC

T P

CT

NM

F-gd

NM

F-nn

ls Sp

arsePC

T Sp

arseNM

F

Healthy participants Sick participants

SemiN

MF

Co

vex-NM

F

a. b. c. d. e. f.

Fig. 5. Low-rank approximation of thermal sequence determined using different matrix factorization techniques. Columns (a-c) show symptomatic patients(diagnosed by mammography as cancer patients or healthy with symptoms), whereas columns (d-f) show the result of methods for healthy participants.

V. DISCUSSION

In this study, we used semi-, convex-, and sparse- NMFto extract thermal patterns for infrared diagnostic systems foractive and passive thermography imaging. This study was de-signed based on the general trend of dimensionality reductionand defect detection methods (such as [1], [23], [2], [19], [3],[4], [30]) and involved in-depth comparative analyses of spec-imens (for IRNDT) to identify potential patients with breastcancer (for DMR). In IRNDT, Convex- NMF showed higheraccuracy than semi- NMF and Sparse- NMF for detectingdefects (Table 2) possibly because the convex-NMF solutionsare sparse and significantly more orthogonal unlike semi-NMF.In general both semi- and convex- NMF can give identicalresults similar to K-means clustering, where sharper clusteringindicators can be given to convex-NMF. The restriction onbasis can have significant effect on subspace factorization

(creating more loss Equ.(5)), and more constrained resultsmore degeneration of the accuracy [33].AL showed higher defect visibility in less deep defects incontrast to PLEXI. This finding might be due to the differentthermal properties of the specimens with different depths. Theoverall performance of defect detection in the CFRP specimenwas lower than that in aluminum and PLEXI due to its smallerdefects (1mm >, Figure 2). CFRP showed lower overall defectvisibility due to less deep and bigger defects with the respect todifferent thermal properties. However, the results indicated thatbigger defects in CFRP led to higher defect distinguishability.The application of semi-, convex-, and sparse- NMF in low-rank matrix approximation is new in the field of thermography,but limited data are available for these approaches ([1], [23],[2], [19], [3], [4], [30]). This study is the first to conductsuch analyses for active and passive thermography. Moreover,

9

a. b. P-value: 0.1

P-value: 0.1

P-value: 0.1

P-value: 0.4 P-value: 0.09

P-value: 0.03

P-value: 0.03

P-value: 0.05

c.

d.

Non-Symptomatic participants Symptomatic participants

Fig. 6. Stratification of participants into symptomatic (abnormal-“a” suffix) and non-symptomatic (healthy-h suffix) groups for each matrix factorizationalgorithm as presented by boxplots and Kurskal Wallis test (a.). b-c represent the 3D views of grouping using TCLM’s properties.

Fig. 7. ROC graph for different matrix factorization approaches for multi-variate covariate logistic regression binary classification (abnormal or healthyparticipants).

comparative analyzing depths and sizes of defects for all thecurrent state-of-the-art approaches increases the contributionof this study.In DMR, Semi-, convex-, Sparse- NMF showed significantimprovements in stratifying symptomatic patients from healthyparticipants (Figures 7, and 6.a) unlike other approaches(Figure 6.a). Moreover, convex- NMF showed higher accuracythan the results of IRNDT in finding heterogeneous thermalpatterns, which might be due to the nature of passive thermog-raphy or better grouping thermal patterns because of betterclustering property of convex-NMF than in IRNDT. NMF-gd showed the maximum accuracy with slightly insignificantseparability (Figures 6.a and 5), indicating the better perfor-mance of gradient descent algorithm over non-negative leastsquare algorithm to optimize the minB,Af(B,A) . Thermaland infrared imagery has been used to determine the breast

abnormality for past several years [43], [44]. Discussions aboutthe better positions for such imaging acquisitions [31] andabout the reliability of this modality for detecting the abnor-malities [45] have been reported. However, the association oflow-rank approximation of thermal heterogeneity with breastabnormality has not been discussed in literature and could havenovel contribution to the field.One limitation for applying the presented models is built basedon manually selection of the bases with better defect visibility.Moreover, the low-rank approximation does not guaranteethe best contrast in the predominant basis. In other word,the first rank approximation is not necessarily better thatthe second rank to show defects. Determining the internalparameters of each method (such regularization and number ofiteration) was not discussed here and was selected based on theoptimal performance of the algorithms because of comparativeanalyses among different methods and their validation. Thepresented techniques offer some advantages. First, the problemcan be more similar to clustering (particularly for active ther-mography) [33], [23], [46] due to the NMF’s properties andselecting non-negative low-rank approximation, which couldserve as unequivocal advantage on combining dimensionalityreduction and detecting defects. Second, these methods consid-erably alleviate the effect of motion artifacts and noise, whichcan be substantial improvements in infrared thermographyapplications in medicine. To our best of knowledge, this studyis the first to employ semi-, convex-, sparse- NMF methodsfor thermography in this extent.

VI. CONCLUSIONS

This study proposed comparative analyses and new ap-plications of semi-, convex-, and sparse- NMF algorithmsin infrared diagnostic systems. We applied these approachesto detect defects in NDT specimens (i.e. carbon fiber-reinforced polymer composites (CFRP), aluminum (AL),

10

Computational Time for different datasets [s]

0.04 0.25

14.81

45.18

18.31

39.52

27.5

0.87

Aluminum

CCIPCT

PCT

NMFgd

NMFnnls

Sparse-PCT

Sparse-NMF

SemiNMF

Convex-NMF

0.041 7.2

117.3

235.9

63.66 206.91

286.57

17.53

CFRFP

CCIPCT

PCT

NMFgd

NMFnnls

Sparse-PCT

Sparse-NMF

SemiNMF

Convex-NMF

0.12 1.85

46.83

96.33

17.89

263.06

122.68

9.37

PLEXI

CCIPCT

PCT

NMFgd

NMFnnls

Sparse-PCT

Sparse-NMF

SemiNMF

Convex-NMF

0.02 0.03

0.09

0.60

0.10

0.07

0.04

0.05

DMR

CCIPCT

PCT

NMFgd

NMFnnls

Sparse-PCT

Sparse-NMF

SemiNMF

Convex-NMF

Fig. 8. Computational time for semi-, convex-, and sparse- NMF with the respect to other common methods in thermography.

CCIPCT PCT NMFgd NMFnnls SparsePCT SparseNMF SemiNMF ConvexNMF

3% 8.55 5.75 22.1 17.74 22.65 20.76 9.41 5.65

5% 8.83 5.94 24.16 14.99 22.56 22.21 8.35 5.27

7% 9.04 6.06 23.34 14.85 22.66 18.65 9.99 5.12

10% 9.43 6.41 19.76 13.57 23.01 22.53 7.15 3.91

15% 9.93 2.23 15.48 12.62 23.47 18.64 8.97 3.1

20% 10.34 7.31 16.45 11.16 23.01 22.08 7.51 2.39

-5

0

5

10

15

20

25

30

SNR

[d

B]

SNR for additive Gaussian noise

3%

5%

7%

10%

15%

20%

Fig. 9. SNR analysis of the state-of-the-art approaches in thermography.

and poly(methyl methacrylate) (PMMA, known as Plexiglas-PLEXI) and measured thermal heterogeneity in breast cancerscreening test (DMR). We compared the appropriateness ofthese approaches to the similar state-of-the-art thermographicmethods, such as PCT, CCIPCT, Sparse- PCT, NMF-gd, andNMF-nnls, among dissimilar thermal datasets. The resultsindicated the significant performance of semi-, convex-, andsparse- NMF in detecting defects (average accuracy levelsof 68.6%, 91.8%, and 60.1%, respectively) and preservedthermal heterogeneity to discriminate between symptomaticand healthy participants (accuracy of 74.1%, 75.9%, and77.8%, respectively). Sparse- NMF and sparse- PCT showedsignificant robustness against noise, which was tested byadditive Gaussian noise from 0% to 20% than other thermo-graphic techniques. Future works should substitute the manualselection of the basis from the low-rank matrix approximationwith an automatic selection. In addition, an expansion of thevalidation set to a larger infrared imaging cohort can furtherconfirm the strength and limitations of these approaches.

ACKNOWLEDGMENT

This is the authors version of the article with doi:10.1109/TIM.2020.3031129 in IEEE Transactions on Instru-mentation and Measurement 2020.

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