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61

Measuring Blood Glucose Concentrations inPhotometric Glucometers Requiring Very Small

Sample VolumesNevine Demitri,Student Member, IEEE,Abdelhak M. Zoubir,Fellow, IEEE,

Abstract—Glucometers present an important self-monitoringtool for diabetes patients and therefore must exhibit high accu-racy as well as good usability features. Based on an invasive,photometric measurement principle that drastically reduces thevolume of the blood sample needed from the patient, we presenta framework that is capable of dealing with small blood samples,while maintaining the required accuracy. The framework consistsof two major parts: 1) image segmentation; and 2) convergencedetection. Step 1) is based on iterative mode-seeking methodsto estimate the intensity value of the region of interest. Wepresent several variations of these methods and give theoreticalproofs of their convergence. Our approach is able to deal withchanges in the number and position of clusters without anyprior knowledge. Furthermore, we propose a method based onsparse approximation to decrease the computational load, whilemaintaining accuracy. Step 2) is achieved by employing temporaltracking and prediction, herewith decreasing the measurementtime, and, thus, improving usability. Our framework is testedon several real data sets with different characteristics. We showthat we are able to estimate the underlying glucose concentrationfrom much smaller blood samples than is currently state-of-the-art with sufficient accuracy according to the most recent ISOstandards and reduce measurement time significantly comparedto state-of-the-art methods.

Index Terms—Blood glucose measurement, clustering, imagesegmentation, kinetic modelling, mean-shift

I. I NTRODUCTION

D IABETES MELLITUS describes a group of metabolicdiseases, affecting 347 million people worldwide. It

occurs when the pancreas cannot produce enough insulinor when the body cannot use the insulin it produces [1].The healthy glucose range lies between70mg/dlto180mg/dl[2]. A condition termed hypoglycaemia occurs when theblood sugar level drops below70mg/dl. This condition isassociated with a high short-term risk. Hyperglycaemia, incontrast, occurs for blood sugar levels above200mg/dl andhas longer-term effects on the body functions. Several studieshave been able to statistically associate a significant delay ofthe onset, or slowing down the progression of complicationsthrough intensive treatment guided by frequent blood glucoseself-monitoring [3], [4]. For this purpose, hand-held invasivedevices containing glucose biosensors are used by patientsto regularly and reliably self-monitor their glucose levels.This typically requires to extract a blood sample from the

N. Demitri and A. M. Zoubir are with the Signal Processing Group, Instituteof Telecommunications, Technische Universitat Darmstadt, Merckstr. 25,64283 Darmstadt, Germany.

patient’s finger using a lancet up to4− 5 times daily and can,therefore, present a hurdle to regular self-control. To reducethe inhibition and pain threshold for the patient, we presenta framework based on a novel approach that uses a bloodsample in the nano litre-range (nl-range), which is up to amagnitude of 100 smaller than is common in current state-of-the-art devices [5], [6], hereby drastically reducing the painsensation for the patient.

Chemical Test Strip

Camera

Region of Interest (ROI)

LEDs

Segment

ROI

Estimate intensity

value

Map to glucose

concentration

Deliver result to

the user

Emitted

Light

Reflected

Light

Fig. 1: The photometric measurement principle used to mea-sure the glucose concentration in a blood sample.

The approach used in our work, illustrated in Fig. 1, is basedon a photometric measurement principle, where the bloodsample is placed on a chemical test strip that reacts with theblood glucose, resulting in a color change. By illuminatingthe test area and capturing the reflections, the color changecan be measured and associated with the underlying glucoselevel. To counter the common problem of ambient light noise[7] in photometry, the measurement area is placed completelyinside the device, and thus protected from ambient light. Inthis approach the blood sample, and thus the region where thereaction takes place, is very small compared to the chemicaltest strip. The whole test strip is observed by an image sensorresulting in frames at discrete time instantsn = n0, n1, . . .

where n = tfs, t ∈ R being the continuous time andfsis the sampling frequency. The frames show both the regionwhere the reaction takes place as well as surrounding areas.The former represents the region of interest (ROI) and hasto be extracted. The underlying intensity value representing

2

the color change is termed relative remissionr ∈ R, whichis, finally, mapped to the underlying glucose concentrationg ∈ R. This process is carried out for the whole durationof the chemical reaction, producing a set of frames. Typically,the chemical reaction exhibits three different stages:

1) Constant relative remission where the reaction betweenthe glucose and the chemical agent has not started.

2) The moistening period starts atn = nD and is charac-terised by a rapid drop of the relative remission valuerDfollowed by a slow decrease, which can be modelled byan exponential decay.

3) Convergence is reached when the chemical reaction sat-urates atn = nC at a converged remission valuerC .

Figure 2 depicts an idealised course of a typical chemicalreaction for a low and a high glucose case. Different

Time (s)nD n

rC

rD

1) 2) 3)

High glucose levelLow glucose level

C

r(%

)

0 5 10 1550

60

70

80

90

100

Fig. 2: The typical course of relative remissionr over time,termed kinetic curve, for a low (blue solid) and a high (greendashed) glucose level. The three distinct stages of the chemicalreaction are illustrated.

behaviours of the frames can be observed depending on thestage and the underlying glucose concentration. Figures 3 (a)and 3 (d) show that frames captured before the chemicalreaction starts are characterised by a constant intensity over thewhole region. With the onset of the chemical reaction, the ROIstarts to shift in the direction of lower intensity. Dependingon the underlying glucose concentration, the intensity willconverge to a different final relative remission valuerC . Forthe low glucose range (Figs. 3 (b) and 3 (e)), this can bevery close to the initial reflectance behaviour of the constantstage (n < nD), whereas for high glucose concentrations(Figs. 3 (c) and 3 (f)) at least three distinct areas can typicallybe identified. The ROI itself is best recognisable as the areabetween the dotted lines in Fig. 3 (c). It is characterised bya granular structure. Furthermore, we observe a thick edgebetween the ROI and the background that takes on valuesthat lie in between both areas. In Fig. 3 (c), it is the areabetween the solid and the dotted line. This occurs due tothe inhomogeneous distribution of the blood sample over theedges, such that a weaker reaction takes place in this area. Asthe images are often degraded by noise, it can be difficult todistinguish the ROI from the other image regions, particularlyin low-contrast cases. The position of the ROI is unknown,as it depends on the blood flow over the test strip, as wellas movements of the test strip in the camera field. Moreover,the ROI is often disturbed by artefacts such as air bubbles or

dust particles that can change their position, size, and shapeduring the reaction. The result of this is a change in the numberand position of clusters in the image over time as well as fordifferent glucose concentrations.

Inte

nsity

Va

lue(%)

100

80

60

40

20

30

50

110

90

70

(a)

Inte

nsity

Va

lue(%)

100

80

60

40

20

30

50

110

90

70

(b)

Inte

nsity

Va

lue(%)

100

80

60

40

20

30

50

110

90

70

(c)

10080604020Intensity Value(%)

Occ

urre

nce

(d)

10080604020Intensity Value(%)

Occ

urre

nce

(e)

10080604020Intensity Value(%)

Occ

urre

nce

(f)

Fig. 3: Examples of observed images and their respectivehistograms. (a) and (d) show an observation atn < nD, (b)and (e) show an observation of a low glucose measurement atn > nC , (c) and (f) show an observation of a high glucosemeasurement atn > nC .

The main contribution of this work is the development ofa framework to measure the concentration of an analyte ina fluid from an image-based measurement procedure; in ourcase the glucose concentration in blood. It is noteworthy, thatthis framework is not confined to this sole application but issuitable for other medical applications that rely on photometry.We show that our framework exhibits high accuracy, whilereducing the blood sample volume from the micro litre-range(µl-range) to the nl-range. The proposed methodology entails:(i) the detection of the onset of the chemical reaction, using aNeyman-Pearson hypothesis test; (ii) the segmentation of theregion of interest and accurate estimation of the underlying rel-ative remission value for each incoming frame at time instantn = nD, . . . , nC without prior knowledge on the number orshape of the regions. We build up on our previous work in [8],[9], by incorporating the median instead of the mean whichexhibits better performance, as well as, providing convergenceproofs for all the derived methods. Furthermore, a data-drivenheuristic is introduced to choose the subset size individuallyfor the scalable version, which further reduces the subset sizein most cases; (iii) the derivation of a model for the chemicalkinetics that is incorporated in a temporal tracking and pre-diction setup to enhance accuracy and reduce measurementtime drastically; (iv) the mapping of the estimated remissionto the underlying glucose concentration. The validation ofourproposed framework is performed on an extensive collectionof real data sets taken in six different scenarios to ensurerobustness of the framework in different settings. Using thesesets, we are able to identify a minimum range for the bloodsample volume to maintain the required accuracy. We note that

3

some aspects of this work appeared in [8], [9] and [10], whilethe substantial part of this manuscript is novel.Some notation: In this work, scalars are represented by lowercase, non-bold lettersa, while lower case bold letters denotevectorsa and al indicates thel-th element of the vector.Capital bold letters are matricesA and constants are givenby capital non-bold lettersA. We denote estimates bya andsets byA. The use ofn in the superscript denotes time indicesa(n) andj in the superscript denotes iteration stepsa(j).The paper is organised as follows: In Section II we discuss thestate-of-the-art. Section III presents the proposed framework,detailing the different challenges and stages of the procedure.It includes proofs of convergence of the proposed weightedmean-shift and medoid-shift algorithms. Section IV introducesthe used data sets as well as the validation criteria. Themain part of this section comprises the experiments and thediscussion thereof. We conclude with a summary and anoverview on future work in Section V.

II. STATE-OF-THE-ART

Traditionally, glucose self-monitoring devices are basedonan invasive procedure that uses a photometric or an electro-chemical approach to infer the glucose concentration fromthe blood sample [6], [11], [12]. This requires the extractionof a blood volume of1µlto25µl. While much research isbeing performed to replace the current devices by non-invasivealternatives [13]–[15], it seems that the traditional technologieswill continue to maintain their position [6], at least in thenear future. Reducing the blood volume needed, and herebythe induced pain remains a necessary field of research [6].Traditional photometric measurement principles use a bloodsample in theµl-range that completely covers up the test field.A photodiode is applied to capture the resulting reflections[16]. The use of much smaller blood samples renders thisapproach inaccurate, as the ROI becomes much smaller and thesignal-to-noise-ratio rises. This motivates the use of a camerato observe the chemical reaction. To the best of our knowledge,the problem of measuring blood glucose using the setup inFig.1 and an image processing-based approach has only beentackled in a limited number of studies. In [5], the authorspropose to use a histogram-based approach and the setup inFig. 1 to estimate the intensity of the region of interest. Tothis end, two clusters are assumed, one corresponding to theROI and the other to the background. The authors proposeto measure the displacement of the ROI cluster w.r.t. thebackground cluster. The underlying assumptions are that justtwo clusters exist in the image and that the background clusterexhibits a constant relative remission value over time. Theseare rather strong assumptions, which are hardly fulfilled inpractice. Our observations give evidence to a variable numberof clusters in the images, as seen in Fig. 3. The variability isglucose level-specific, temporally dependent, and caused bythe variability of the chemical reaction for different bloodsamples of different patients. Furthermore, the backgroundcluster does not experience a constant intensity but changesover time due to temperature and humidity issues, as wellas leakage of small amounts of blood to the background

cluster. We assume that the number of clusters in the imagesis unknown and propose an approach that is able to dealsystematically with this.

III. PROPOSEDALGORITHM

Our framework is summarised in Fig. 4. The input isgiven by the frames obtained by the camera, representing thereflectivity behaviour of the observed area. Each incomingframe is directly processed. A hypothesis test is performedonthe pre-processed, gray-scale reflectance imageX

(n) of sizeMx×My, obtained at framen, to detect whether the chemicalreaction has started, i.e., ifn ≥ nD. In this case, we proceedby segmenting the region of interest. After segmenting theimage, the relative remission values of the different regionsare estimated and the converged relative remission valuerCcorresponding to the ROI is identified. The current estimatealong with the history of estimates is used to test for con-vergence of the chemical reaction. If convergence is detected,the underlying intensity value is mapped to its correspondingglucose levelg. If not, the next frame is processed in thesame manner. The different blocks as depicted in Fig. 4 willbe described in further detail in the sequel.

No

Yes

A.

Pre-processing

C. Segmentation

frame

B. Detection

of Drop Time

E. Cluster

AssignmentF. Convergence

Test

G. Mapping to

Glucose

Concentration

n

X(n)

g rC

n = n+ 1

Fig. 4: The proposed approach.

A. Pre-processing

The raw framesX(n)raw obtained by the camera are normalised

w.r.t. the initial reflectivity prior to the start of the chemicalreaction. The normalisation is performed using the calibrationframes as described in [8] and results in a gray-level imageX

(n)norm with pixels in the range between0− 100.

In our previous work [8] we asserted that windowing the im-ages results in superior segmentation and remission estimationresults. We termed this process binning and usedB to denotethe size of the window. Binning is used as the last step inthe pre-processing stage unless stated otherwise. Finally, thepre-processed image is denoted byX

(n).

B. Detection of Drop Time

As illustrated in Fig. 2, the chemical reaction exhibits aconstant stage forn < nD, where the glucose in the bloodsample has not yet been detected. The timenD when the dropoccurs, i.e., when the chemical reaction starts, is dependenton the particular chemical reagent, the blood sample at hand,

4

and the underlying temperature and humidity conditions in thedevice. We suggest using the Neyman-Pearson hypothesis testof variance to detectnD. The advantage of using the Neyman-Pearson test is that it provides a meaningful way to set athreshold by incorporating the image statistics and controllingthe false-alarm rate. Figure 5 illustrates the probabilitydensityfunctionsfx of the intensities of two exemplary images takenat time n1 < nD and nC > n2 > nD, i.e., directly afterthe occurrence of the drop. Extensive analysis of real datashowed that it is reasonable to assume that the mean-adjustedframes forn < nD follow a zero-mean normal distributionwith varianceσ2

1 , whereas after the drop the mean-adjustedframes can be modelled by a zero-mean normal distributionwith varianceσ2

2 > σ21 . This increase in variance can be

explained by the onset of the chemical reaction and therebythe onset of the color change on the test strip. Hereby,σ2

1

can be estimated as the sample variance from available datasets. However,σ2

2 is unknown and depends on the underlyingglucose concentration.For the remainder of this work, we will deal with vectorisedforms of the images and denote these by

x(n) = vec

(X

(n)), (1)

wherex(n) is of size1×L,L =Mx ·My. Now, the hypothesistest can be formulated as

H0 :fx(n) ∼ N (0, σ2

1), n < nD (2)

H1 :fx(n) ∼ N (0, σ2

2), n ≥ nD

where x(n) is the mean-adjusted version ofx(n) and f

x(n)

denotes the pdf ofx(n). Calculating the likelihood-ratio leadsto the following test statistic

T(x(n)

)=

L∑

l=1

(x(n)l

)2 H1

≷H0

L

2L ln(δ) + ln(

σ22

σ21)

1σ21− 1

σ22︸ ︷︷ ︸

δ′

, (3)

where the thresholdδ is set to ensure a nominal false alarmrate. Assumingx(n) to be spatially i.i.d., the distribution ofT(x(n))σ21

underH0 and T(x(n))σ22

underH1 can be shown to follow

a χ2-distribution withL degrees of freedom [17]. The spatiali.i.d. assumption is justifiable in our case as we are observingframes prior to the onset of the reaction where no structure ispresent in the image.L represents the number of pixels in animage and is, therefore, quite large. Thus, we can approximatethe χ2-distribution by a Gaussian distribution [17] and theprobability of false alarmPFA becomes

PFA = Q

(δ′

σ21

), (4)

where Q(·) is the complementary cumulative distributionfunction of the standard Gaussian distribution. Given a fixedprobability of false alarmPFA, the threshold can be calculatedasδ′ = Q−1(PFA) · σ2

1 .

x

fx

n1n2

-30 -20 -10 10 20 30 40 50

0.05

0.15

0.25

0.35

0.1

0.2

0.3

Fig. 5: Pdf estimate of mean-adjusted intensities of a frameatn1 < nD (blue solid) andn2 ≥ nD (green dashed).

C. Segmentation Using the Mean-Shift Algorithm

1) The Standard Mean-Shift Algorithm:Our underlyingassumption is that the glucose images will contain a ROI,where the blood sample is distributed and, thus, the colorchange of the ROI represents the reflectivity caused by the un-derlying glucose concentration. This area will not necessarilybe completely spatially connected as it may contain artefactswhere the reaction has not taken place, or granularities in thechemical that do not contribute to a proper reaction. Otherareas in the image will not correspond to the ROI, either,because the blood sample is unevenly distributed in theseareas, or because the areas are dry, i.e., not covered withblood. We interpret the different areas contained in the imageas clusters with specific cluster centres overlaid by noise.We assume that data points converging to a certain clustercenter will belong to the corresponding area. The noise hereis assumed to be Gaussian. This assumption is based on thecalibration frames where no reaction has started.The mean-shift algorithm [18] has become a popular approachfor clustering and mode location estimation. It has been widelyapplied for medical image segmentation as it does not requirethe knowledge of the actual number of clusters [8], [19]–[22]. We use the mean-shift algorithm (MS), as well as twoextensions of it: the robust mean-shift (R-MS) [9] and thescalable, sparse mean-shift (SS-MS) [10]. Furthermore, weextend these variants to the medoid-shift [23].Assume a set ofL pixels in a gray-scale image1 at framen thatcan be expressed asx(n)l ∈ R, l = 1, . . . , L as in Eq. (1). Wewill omit the frame notation, henceforth, for better readability.A consistent estimator of the density is given by the kerneldensity estimator (KDE) [24] with bandwidth parameterh

fK(x) =1

Lh

L∑

l=1

K

(x− xl

h

), (5)

whereK(x) is a radially symmetric kernel function with astrictly decreasing profile forx ≥ 0. The bandwidthh of thekernel function is the only parameter that has to be tuned andis crucial to the performance of the mean-shift algorithm, as itaffects the number of clusters. Extensive work has dealt withthe choice of the bandwidth parameter, e.g. [24], [25].Generally, the mean-shift algorithm is derived by taking the

1Including spatial information for gray-scale images will increasethe dimensions by two, leading to a three-dimensional data vectorx = [xintensity

, xposition,Mx , x

position,My ].

5

zeros of the gradient of the KDE

∇fK(x) = 0 (6)

and reformulating to get the mean-shift vector ofx as

m(MS)h,K′ (x) =

[∑Ll=1 xlK

′(x−xl

h

)∑Ll=1K

′(x−xl

h

) − x

], (7)

whereK ′ = dKdx . Each data point is shifted according to

Eq. (7) in the direction of steepest ascent until convergence,such that the sequence of successive locations of a data vector

x(j+1) =

∑L

l=1 xlK′(

x(j)−xl

h

)

∑Ll=1K

′(

x(j)−xl

h

)

, j = 1, 2, ... (8)

wherex(j) denotes thej-th iteration step. Typically, the mean-shift algorithm tends to overestimate the number of modes. Itis usual to post-process the mean-shift results with a simpleclustering method to group nearby modes together, e.g. usinga mode pruning step with bandwidthh [8], [19].Another way to express the KDE, which allows for extensionsof the mean-shift, is to consider the kernel to be an innerproduct in the Hilbert spaceH, such that [26]

K

(x− xl

h

)= 〈Φ(x),Φ(xl)〉, (9)

whereΦ : R → H is a mapping function and〈·〉 denotes theinner product.K(x−xl

h ) is a positive definite kernel function.Hence, the KDE can be formulated as

f(x) = 〈Φ(x),L∑

l=1

wlΦ(xl)〉. (10)

In this formulation, the uniform weights1L have been substi-tuted bywl > 0 to attain a more general form. It has beenshown [19] that the mean-shift algorithm converges and thatthe mode estimator is asymptotically consistent and unbiased.

2) The Robust Mean-Shift Algorithm:In [9], the robustmean-shift (R-MS) is derived as an alternative to accountfor heavy-tailed noise in the data. Here, the sample mean ofvectors in Eq. (10) is substituted by a robust M-estimate

µΦ = argminµΦ

L∑

l=1

ρ

(Φ(xl)− µΦ

σ

), (11)

whereρ(·) is a monotone, differentiable loss function, such asHuber’s loss function [27] and the scaleσ is initialized witha robust estimate based on the mean absolute deviation [9].This leads to the robust KDE

f(x) = 〈Φ(x), µΦ〉 =⟨Φ(x),

L∑

l=1

wR-MSl Φ(xl)

⟩

=1

h

L∑

l=1

wR-MSl K

(x− xl

h

), (12)

where the robust weightswR-MSl can be determined using Itera-

tively ReWeighted Least Squares (IRWLS) [9], [28], resulting

in

wl =

σ·ψ(

Φ(xl)−µΦσ

)

Φ(xl)−µΦif Φ(xl)−µΦ

σ 6= 0

ψ′(0) if Φ(xl)−µΦ

σ = 0, (13)

whereψ = ρ′. The robust mean-shift vector reads

m(R-MS)h,K′ (x) =

[∑Ll=1 w

R-MSl xlK

′(x−xl

h

)∑L

l=1 wR-MSl K ′

(x−xl

h

) − x

]. (14)

For a more detailed description of the R-MS the reader isreferred to [9].

3) The Scalable Sparse Mean-Shift Algorithm:The compu-tational complexity of the mean-shift algorithm is proportionalto the square of the total number of data pointsL. To overcomethe problem of high computational demand, a scalable sparseversion has been derived in [10]. A more detailed discussionof the computational complexity of both versions is providedin [10]. Using only a subset of the data points, the SS-MS isable to achieve high accuracy while reducing computationalpower. The gist lies in substituting the mean in Eq. (10) by asparse approximation [29], [30]. A short outline will be givenin the sequel, for more details the reader is referred to [10].The sparse approximation of the mean can be formulated as

minI|I|=N

minαi,i∈I

∣∣∣∣∣

∣∣∣∣∣

L∑

l=1

wl〈Φ(x),Φ(xl)〉 −∑

i∈I

αi〈Φ(x),Φ(xi)〉

∣∣∣∣∣

∣∣∣∣∣

2

.

(15)where the index setI ⊆ {1, . . . , L} of cardinality |I| = N isdefined to be a subset of the full set of indices, the weightsfollow α ∈ R

N and1 ≤ N ≪ L. The scalable, sparse KDE

fK(x) =1

h

∑

i∈I∗

αI∗,iK

(x− xi

h

), (16)

whereαI∗,i denotes thei-th weight obtained using the optimalsetI∗. The resulting mean-shift vector has the form

m(SS-MS)h,K′ (x) =

[∑i∈I∗ αI∗,ixiK

′(x−xi

h

)∑

i∈I∗ αI∗,iK ′(x−xi

h

) − x

]. (17)

A robust sparse formulation (RSS-MS) can be derived [10].To solve Eq. (15), we need to: 1) find a solution for the inneroptimisation problem; i.e., solve forα: 2) find an optimal setI∗. For a fixedI, αI = Ξ

−1I ξI . Here

ΞI = (〈Φ(xi),Φ(xj)〉)i,j∈I (18)

ξI =

L∑

j=1

wj〈Φ(xm),Φ(xj)〉,m ∈ I, (19)

where ΞI is the Gram matrix. To ensure convergence,αi < 0, ∀i = 1, ..., N . Generally,αi will not always fulfill theproperty. We, therefore, normalise the weightsαi by settingto zero all weightsαi < 0 and re-normalising such that∑i αi = 1. The optimal index subset is found by assuming a

fixedN and maximising an incoherence functionνI

I∗ = maxI⊆{1,...,L}

νI , (20)

6

where

νI = minj /∈I

maxi∈I

〈Φ(xi),Φ(xj)〉. (21)

This is intuitive in the sense that to find the most representativesubset of data vectors, we choose the ones that are mostincoherent to each other. In [10], we outline an algorithm tomaximize νI and determineI∗. The question remaining ishow to choose the cardinalityN of the subset of indicesI.Clearly, in some cases this can be given by the applicationat hand knowing a certainNmax, or by using test data forcross-validation, as in [10], whereN is set to be sufficientlyhigh to incorporate the worst-case scenario. We introduce adata-driven selection ofN , which results in a smallerN formost data sets than cross-validation. Figure 6 shows typicalexamples of the progression of a normalised version ofνI

νI =νI,|I|=N

max νI(22)

for the images given in Fig. 3, using different values ofN = 1, . . . , L. We observe that the value ofνI drops quicklyafter a certain value ofN and that the behaviour of the curveis similar for different images. This signifies that the firstNν samples chosen by the algorithm contribute highly to theincoherence in the image and it is sufficient to use a subsetI of cardinalityNν for the sparse representation. We propose

νI

N

n < nD

High Glucose

Low Glucose

0

1

0.2

0.4

0.6

0.8

100 101 102 L = 103

0.1

0.3

0.7

0.9

Fig. 6: An example of the progression of normalisedν withincreasing cardinalityN .

to chooseNν based on the gradient ofνI in Fig. 6 and set athresholdTν

∆ν(Nν) =νI,|I|=Nν

νmax−νI,|I|=Nν−1

νmax≤ Tν . (23)

The choice ofTν is based on heuristics and chosen specificallyfor the data at hand by observing the trade-off betweenaccuracy and computation.

4) Convergence Properties of the Derived Algorithms:It remains to prove that the mean-shift algorithm in all itsvariations will converge. In [19], the convergence proof ofthestandard MS is given. We extend it for the cases of the R-MSand the SS-MS.

Theorem 1. If wl > 0 holds for the weights from Eq. (10)and the kernel functionK(·) has a convex and monotonicallydecreasing profile, the sequence of trajectory points{x(j)},j = 1, 2, . . . will converge and the sequence{fK

(x(j)

)},

j = 1, 2, . . . is monotonically increasing.

The proof given in Appendix A is a generalised version ofTheorem 1 in [19]. We need to ensure that the weights willfulfill wR-MS

l > 0 for R-MS andαI∗,i > 0 for SS-MS.wR-MSl

are derived using the IRWLS and are, thus, certain to followthe required property for monotone loss functionsρ(·) . Thecalculation ofαI∗,i ensured that it follows this property.

D. Segmentation Using the Medoid-Shift Algorithm

The medoid-shift was derived by Sheikhet al. in [23] as analternative to the mean-shift that uses the medoid instead ofthe mean. The medoid is defined as the point in the set thathas the smallest distance to all other points. We derive herearobust medoid-shift and a sparse scalable medoid-shift. Thisresults in the following formulation for successive locationsof x:

x(j+1) = argmin

x∈{xl}

L∑

l=1

∣∣∣∣∣∣x− xl

∣∣∣∣∣∣2

wlK′

(x(j) − xl

h

), (24)

wherewl represents the weights associated with each kernel.For the standard medoid shift (MedS)wl = 1

L and for therobust medoid-shift (R-MedS)wl = wR-MS

l . For the sparsescalable medoid-shift

x(j+1),SS-MedS= argmin

x∈{xi}

∑

i∈I∗

∣∣∣∣∣∣x− xi

∣∣∣∣∣∣2

× (25)

αI∗,iK′

(x(j) − xi

h

)

Unlike the mean-shift, the medoid-shift always converges topoints contained in the data set. The advantage of this isthat the medoid-shift vector needs only to be computed oncefor every data sample and hence, needs less iterations thanthe mean-shift. Moreover, it does not need a further heuristiclike the mean-shift to group together neighbouring clusters.The convergence of the sequence of trajectory points in themedoid-shift is guaranteed and the proof is analogous to thatof the mean-shift. The only difference is that for the medoid-shift we additionally need to ensure that there are no cyclesin the sequence of trajectory points, i.e.,x

(j) 6= x(j+c), for all

c > 0. The proof thereof is given in Appendix B.

E. Cluster Assignment

The segmentation leaves us with a finite number of clustercentres corresponding to the different image regions. Typically,we attain2 − 4 clusters; 1) one corresponding to the ROI,2) another to the dry test strip area and optionally 3) oneto the border between the dry area and the ROI, and 4) onecorresponding to artefacts in the image. We need to identifywhich cluster center corresponds to the ROI. This can be doneusing a data-driven approach, exploiting information fromthestructure of the images. Using the size of the clusters, we canidentify the background region and the ROI, as they are thelargest. The ROI will correspond to the one exhibiting thelower intensity of the two remaining regions according to thenature of the chemical reaction.

7

F. Convergence Test

Convergence of the chemical reaction is a very crucialissue for the estimation accuracy. Ideally, the chemical kineticcurve evolves as in Fig. 2. Final convergence is typicallyreached after10 sto15 s. This leads to a long measurementtime for the patient and, hereby, reduced usability, which canbe a cause of irregular self-control. So far, the convergenceof kinetic curves in hand-held devices has been defined asthe point at which the slope of the kinetic curve reaches athresholdTslope over a predefined time period [5]. This pointis reached after around3 sto15 s, depending on the underlyingmeasurement. The assumption made, hereby, is that in the idealcase, measurements of equal glucose concentration will followa similar progression at all times and, therefore, the intensityvalue estimate at the predefined slope threshold will be relatedto the underlying glucose concentration. The mapping functionbetween intensity and glucose can be adjusted to accountfor the inaccuracy. While this can lead to satisfactory resultswhen no information is given on the kinetic curve, we willshow that it can also result in erroneous, premature estimates.We propose to improve the performance of this approachdrastically by incorporating a model of the kinetic behaviourof the chemical reaction [7], along with state estimationtechniques to predict the actual convergence values ahead oftime. Hereby, stages 2) and 3) of the kinetic curve depicted inFig. 2 are modelled.The glucose oxidase taking place on the chemical test stripcan be modelled by the differential equation [7]

r(n) = (rD − rC) · e−nτ + rC + v(n), n > nD, (26)

where r(n) is the remission measurement value at timen,r0 is the initial remission value after the drop, andrC is theconvergence value of the relative remission after the chemicalreaction has converged. We definerC to be the state that wewant to predict. Furthermore,τ is the reaction rate, andv(n)is a zero-mean, white, Gaussian noise process with unknownvarianceσ2

v describing the measurement noise.Due to the nature of the chemical reaction, it is realistic toassume thatτ andrC are correlated. Building up on [7], weperform a regression analysis using a real data set of estimatedconvergence valuesrC = [rC,1, ..., rC,NM

] and correspondingrates τ = [τ1, ..., τNM

], NM being the number of availablemeasurements. Hereby, we establish a linear relation betweenτ andrC to be the most suitable least-squares (LS) fit, as canbe seen in Fig. 7

τ = ∆τ · rC + τ0, (27)

where∆τ > 0 andτ0 < 0. Using a least-squares (LS) fit, weestimate the regression parameters∆τ andτ0, to obtain fromEq. (26) the following nonlinear relation for the kinetic curve

r(n) = (rD − rC) · e−n·(∆τ ·rC+τ0) + rC + v(n) (28)

= f(rC , v(n)) , n > nD.

We use an Extended Kalman Filter (EKF) to process themeasured kinetic curve and perform an online prediction ofthe remission convergence valuerC . For that, we formulate

τ

rC

LS fit

Data points

70 80 90 10050 60

0

-0.1

-0.2

-0.3

-0.4

Fig. 7: Regression analysis for the relation betweenτ andrC

the prediction and measurement equations as follows

rC(n) = a(n)rC(n− 1) + w(n) (29)

r(n) = h(rC(n), v(n)) , (30)

wherea(n) is the transition matrix describing the transitionof the state estimaterC from time n − 1 to n. In our case,a(n) = 1, as rC is a static state of the chemical reaction.The process noise is modelled by the random variablew(n)and can be used to account for uncertainty in the model.h(rC(n), v(n)) describes the observation model, relating theestimated staterC(n) to the measurementr(n). It is given bythe partial derivative of the process model w.r.t. the staterC

h(rC(n), v(n)) =∂f(rC(n), V (n))

∂rC(n)(31)

= 1− e−n·(∆τ ·rC(n)+τ0)

× (1 + rD ·∆τ · n− rC(n) ·∆τ · n) .

We alternate between the prediction and the correction stepofthe EKF, including a new measurementr(n) in each iteration.

G. Mapping to Glucose Concentration

Finally, we map the relative remission estimaterC to theunderlying glucose concentration, which we will deliver tothe user. To this end, a calibration functionfCalib : rC → g

is coded into the glucometer to perform the mapping duringmeasurement. The calibration function has to be generateda priori by means of photometric measurements performedunder lab conditions, using blood samples with known under-lying glucose concentrations.

IV. EXPERIMENTAL RESULTS AND DISCUSSION

A. Data Sets

TABLE I: The real data sets used for validation.

Set NM Ng Υ Volume

A 48 5 6.45µm/ Pixel 10 nl to 100 nlB 78 16 6.45µm/ Pixel 10 nl to 100 nlC 78 16 6.45µm/ Pixel 10 nl to 100 nlD 48 4 30 µm/ Pixel 10 nl to 100 nlE 200 10 30 µm/ Pixel around1 nlF 200 10 30 µm/ Pixel Standard

8

To evaluate our proposed framework, we use real data sets,obtained from a setup as in Fig. 1 using blood samples fromblood donations injected with a glucose solution correspondingto the amount of glucose needed. Altogether, we have sixdifferent data sets, comprising 452 different measurementswhich corresponds to a total of 263.064 processed images.The sets differ in terms of the chemical used as well asthe resolution of the camera and the volume of the bloodsample. This results in a different blood flow over the teststrip and thereby different positions and shapes of the regionof interest, as well as prominence of the reaction. Informationon the different data sets is given in Table I. The number ofmeasurements in the data sets is denoted byNM , the numberof different glucose concentrations tested in each data setby Ng, and the resolution of the images byΥ. The volumeof the blood drop ”Standard” indicates state-of-the-art rangesaround1µlto25µl [6]. Each measurement containsNf = 580frames obtained at a frame rate offs = 30 fps. All datasets are pre-processed applying a binning size ofB = 5for Υ = 6.45µm /Pixel andB = 1 for Υ = 30µm/Pixel.After binning, all images containL = 1210 pixels. For theKDE, a Gaussian kernel and a fixed bandwidth parameterh

as in [8] are employed.Tslope is chosen to be10−2. For therobust versions, as in [9], Huber’s loss function is used anditsparameter tuned to achieve 95% asymptotic efficiency in theGaussian case; IRWLS is initialised with uniform weights.

B. Validation Methods

1) Coefficient of Variation: The coefficient of variation(CV) is a very popular measure to assess the accuracy ofpharmacokinetic measurements [31]. In our work, we use theremission coefficient of variation CVr

CVr =1

Ng

Ng∑

γ=1

CVrg(γ), CVrg(γ)

=σRg(γ)

µRg(γ)

(32)

where γ = 1, . . . , Ng, σRg(γ)and µRg(γ)

are the samplestandard deviation, and respectively the sample mean of theelements of the setRg(γ). Rg(γ) denotes the test set of relativeremission estimates obtained from several measurements ofthesame underlying glucose concentrationg(γ).

2) The Clarke Error Grid: The Clarke Error Grid (CEG)[32] is a standard method for evaluating glucose measurementperformance. It plots the estimated glucose concentrationsagainst the actual glucose concentrations and classifies theerror according to its medical severity. To this end, differentregions are defined in the CEG as depicted in Fig. 13. Acommon specification is to have at least95% of all pointsin the region A, maximally5% of all points in the region B[33] and no points in the other zones.

3) The Glucose-Specific Mean Absolute Deviation:Toquantify the validation, we employ a glucose specific meanabsolute deviation (MAD) inspired by [32]. Instead of usingthe standard MAD between the true and the estimated glucoselevel, a glucose-specific MAD (gMAD) is defined. The gMADweighs the errors using a penalty function according to theirmedical severity on the basis of the CEG. Hereby, errors madefor hypoglycaemic cases below70mg/dl are weighed with the

highest factorwgMAD as they present critical short-term risks;hyperglycaemic cases which present more long-term risks areweighed with a slightly smaller factorwgMAD , and normalcases are given the factorwgMAD = 1. The gMAD reads

gMAD =1

N

NM∑

γ=1

∣∣∣g(γ)− ˆg(γ)∣∣∣ · wgMAD(g(γ)− ˆg(γ)), (33)

where the penalty function is given by

wgMAD(g − g) =

1.5σ(g)σ(g) if g ≤ 85 and g ≥ g;1σ(g)σ(g) if g ≥ 155 and g ≤ g;1 otherwise,

(34)with σ(·) and σ(·) being sigmoid functions that ensure asmooth transition as given in [32]. According to the mostrecent ISO standards [34], the maximal permissible error is±15mg/dl within a reference range of0mg/dlto75mg/dl and20% for a reference range above75mg/dl.

C. Results

First, we analyze the quality of the remission results. Herein,a comparison of the segmentation methods is given, as wellas an analysis of the different data sets. Next, we evaluatethe data-driven choice ofNν for the SS-MS. We, then, turnour attention to the effect of using the EKF to predict theconvergence estimates. Finally, we study the precision of theglucose estimates after the mapping operation.

1) Kinetic Curves & Remission Accuracy:In Fig. 8, wepresent a selection of the results of the kinetic curves, i.e.,the progression of the relative remission over time for thedifferent data sets, using MS (mean-shift), RSS-MS (robustsparse MS), MedS (medoid-shift), and R-MedS (robust MedS).For the time being, we will present the results using the

200

300

400

500

40 5 10 15

70

80

90

100

100

50

60

110

r(%

)

Glu

cose

Lev

el[

mg/

dl]

Time [s]

(a) Set A using MS

200

300

400

500

40 5 10 15

70

80

90

100

100

50

60

110

r(%

)

Glu

cose

Lev

el[

mg/

dl]

Time [s]

(b) Set C using RSS-MS

200

300

400

500

40 5 10 15

80

100

100

60

r(%

)

Time [s]

Glu

cose

Lev

el[

mg/

dl]

(c) Set E using MedS

200

300

400

500

40 5 10 15

80

100

100

60

r(%

)

Time [s]

Glu

cose

Lev

el

[mg/

dl]

(d) Set F using R-MedS

Fig. 8: Kinetic curves for different Sets using the differentvariants of MS and MedS. The red circles signify convergencepoints r(Stand. Conv)

C .

standard convergence criterion, i.e., when the slope reaches a

9

TABLE II: CV r values for the different data sets using the standard convergence criterion.

.

Set MS R-MS MedS R-MedS SS-MS RSS-MS SS-MedS RSS-MedS

A 1.59 1.29 1.17 1.22 1.87 1.81 1.52 1.52B 1.95 1.83 1.82 1.71 2.4 2.1 1.95 1.95C 1.15 1.51 0.90 0.99 1.50 1.41 1.39 1.39D 1.87 1.7 1.58 1.39 1.87 2.90 1.10 1.09E 3.98 4.17 3.39 3.25 5.9 5.6 4.3 4.2F 1.48 1.37 1.10 1.04 2.44 2.25 1.99 1.78

threshold ofTslope for a series of consecutive frames. Overall,we establish the expected progression of the kinetic curve withits three typical stages as in Fig. 2. We see that measurementsof the same underlying glucose concentration are, mostly, bun-dled together. Convergence is reached faster for low glucoseconcentrations than for high glucose concentrations. Moreover,our assumption that the decay rateτ of the reaction is corre-lated with the underlying glucose concentration is confirmed.The higher the underlying glucose concentration, the steeperthe decay.We observe a significant decrease in performance when com-paring Set E with Set F, which were taken using the exact samesetup and blood samples, however using different volumesof the blood samples. These results are underlined when

70 80 90 10050 60 11012030 40

Occ

urre

nce

Intensity Value(%)

(a)

70 80 90 10050 60 11012030 40

Occ

urre

nce

Intensity Value(%)

(b)

Fig. 9: Histograms of converged images of high glucose rangemeasurements from (a) Set B, (b) Set C.

analyzing the resulting CVr values for these two sets in theleft part of Table II. This indicates that blood sample volumesin the range of1 nl could be problematic. The volume range10nlto100nl, in contrast, seems to perform well.Table II shows that the medoid-shift versions are significantlybetter than the mean-shift versions for all sets. ObservingSetB, we notice that the difference in performance between thetwo methods is not as high as for other sets. The reason can beunderstood from Fig. 9. The more distinct the modes are, themore separable the clusters, the better mean-shift performs.By choosing the medoid instead of the mean and convergingto actual points in the data set, the medoid-shift doesn’t showthis bias in estimators.The results for the sparse scalable mean-shift are given inthe right part of Table II. We assert that the resulting CVr

values are slightly worse than their non-scalable counterparts.This degradation, however, is not too severe and the loss inaccuracy can be accepted to ensure lower computation. Onlyfor Set E do we notice a severe degradation in performance,

when using the SS-MS. This can be traced back to the factthat due to the very small blood volume, the ROI is muchsmaller than for the other measurements.

2) Data-driven Choice ofNν for the Sparse MS:Asportrayed in Section III-C3, we propose a data-driven approachto selectNν uniquely for every frame. Figure. 10 shows thechoice ofNν for Set C, withNν averaged over all frames foreach measurement. The tendency matches our expectations:very low glucose concentrations are characterized by lowcontrast and therefore higher coherence. The choice ofNν ishigher than for high glucose concentrations, i.e., more datapoints are needed to reliably represent the image. As theconcentration increases and the images become more distinctin the different regions, the variance in the choice ofNν forthe same glucose concentrations decreases. Taking the worst-case scenario would have resulted inN ≈ 370 data points,which is much higher than the data-driven choice ofN .

30 60 75 90 105120135 150

150

180 220260300

300

350

350

400450 550

200

250

100

Glucose Concentration [mg/dl]

Nν

Fig. 10: The choice ofNν for different glucose concentrationsfor Set C.

Let us now analyze the quality of the SS-MS when usinga data-driven choice ofNν versus a fixedNν = 0.37L. Theresults are given for the RSS-MedS in Table. III. Evidently,theaccuracy of both approaches is comparable. This underlinesthat a data-drivenNν maintains the accuracy needed.

TABLE III: Comparison of the CVr values for a data-drivenchoice ofNν vs. a fixedNν = 0.37L.

Choice ofNν A B C D E F

Data-driven 1.03 1.29 0.81 1.90 4.60 1.59Fixed 0.95 1.38 1.27 1.97 4.35 1.64

3) State Estimation using the EKF:The kinetic curves inFig. 8 show that convergence for same-glucose-level measure-

10

ments is not always reached at the same time which resultsin strongly varying remission estimates, although the courseof the kinetic curve is very similar. Let us take the550mg/dl

70

80

90

100

100

50

60200

300

400

500110

0 5 10 15

rC[%

]

Time [s]

MS kinetic curves

Glu

cose

Lev

el

[mg/

dl]

(a)

70

80

90

100

50

60

110

0 5 10 15

Kinetic curve

r(Stand. Conv)C

r(EKF Conv)C

Time [s]

r[%

]

(b)

Fig. 11: (a) State estimation ofrC using the EKF. The redcircles (◦) indicate the estimated convergence values. (b) Acomparison of the convergence estimates using the standardconvergence criterion and the EKF.

measurements in Fig. 8 (b) as an example. Convergence isfound at times betweentC ≈ 4 s and tC ≈ 9 s, leadingto remission estimates fromrC ≈ 50% to rC ≈ 59%.Figure 11 (a) shows the EKF state estimatesr

(EKF Conv)C for

this example. Evidently, the EKF converges to reliable stateestimates quickly and therefore, the resultingrC estimates notonly lie in the same range, but match the final convergencevalue of the kinetic behavior more accurately. This can be seenclearly in Fig. 11 (b). The convergence value reached by thestandard methodr(Stand. Conv)

C occurs after around11 s and doesnot match the actual saturation value. The convergence valuereached by EKFr(EKF Conv)

C , however, occurs after5 s and ismuch closer to the actual saturation value.

TABLE IV: Time gain and reduced error obtained using EKF.

Time Gain Error Benefit

Low Glucose Range (g ≤ 75mg/dl) 1.41 s 2.23 %High Glucose Range (g > 75mg/dl) 0.97 s 2.1 %

EKFStand.

CVr

1.5

2.5

3.5

0.5

1

2

A B C D E FSets

0

3

(a)

EKF - Low glcStand. - Low glc

EKF - High glcStand. - High glc

B C E FSets

gMA

D

5

15

10

25

0

20

35

30

(b)

Fig. 12: (a) Comparison of the CVr values for the standardconvergence criterion to EKF. (b) Comparison of the gMADvalues for the standard convergence criterion to EKF.

Table IV shows the time gain obtained using the EKF. Theseresults are the average of all time gain values obtained for allsets over all measurements. Comparing the remission estimatesr(Stand. Conv)C and r(EKF Conv)

C to the saturation values of thechemical reaction, we assert that the EKF produces more

accurate results, with a gain of more than2% in remission.Figure 12 presents a comparison of CVr values and gMADvalues for the standard convergence criterion and the EKFconvergence method. This analysis considers for each set themethod with the best results in Tables II and V, respectively,and compares it to its EKF-method-equivalent. The EKFconvergence always leads to improved CVr values. Similarly,for the high glucose range the gMAD results show the sameoutcome. For low glucose ranges the standard convergencecompares favourably. The reason for this is that the EKFmethod is quite sensitive to the model used. As can be seenin Fig. 8, the kinetic curves of low glucose measurementsexhibit a dip after the drop followed by a steady rise. This isnot embodied in the model in Eq. (28).

4) Mapping to the Underlying Glucose Concentrations:Finally, we present the results of the glucose estimatesg

obtained by mapping the remission estimates usingfCalib. The

(a) (b)

Fig. 13: Clarke’s Error Grid Analysis for Set E for (a) thestandard and (b) the EKF convergence criteria.

Clarke Error Grid plot of Set E in Fig. 13 (a) shows that 93%of the points lie in region A, 6% lie in region B and1% liein region D using the standard convergence criterion. Thereby,the results do not conform with the requirements. As indicatedby the results in Fig. 13 (b), using EKF state estimation topredict the convergence values leads to a significant improve-ment of the results. Now 100% of the estimates of Set E liein the A-region. Table V presents the mean gMAD values

Low Glucose (≤ 75mg/dl) High Glucose (> 75mg/dl)

Set MS R-MS

MedS R-MedS

MS R-MS

MedS R-MedS

B 14.4 13.0 6.6 6.6 13.4 11.8 17.0 17.2C 6.9 6.9 4.6 4.0 12.8 14.4 7.8 8.9E 22.6 29.4 24.4 21.0 30.8 37.4 25.5 23.6F 9.3 9.4 8.5 8.1 17.5 17.0 16.6 18.4

TABLE V: gMAD values using the standard convergencecriterion.

.

for g of each set separated according to the glucose range.This analysis is omitted for Sets A and D, as they do notcontain enough different glucose concentrations to constructthe mapping function. The results for most sets lie below the

11

ISO requirements, i.e., low glucose ranges show errors smallerthan gMAD = ±15mg/dl and high glucose ranges show anerror larger than 20%. However, we note that in Set E thelimit is often exceeded, confirming again the fact that bloodsample volumes around1nl are problematic.The previously made observation concerning medoid-shiftoutperforming mean-shift is confirmed again here. The robustversion seems to improve the performance in high glucoseranges more than in low glucose ranges.

V. CONCLUSION

Regular self-control using hand-held glucometers is anindispensable part of diabetes care and therefore, glucometersshould maintain high accuracy while improving usability. Wehave developed a full framework to measure the blood glu-cose concentration from glucose images using blood samplevolumes in the nl-range, which is much smaller than thestate-of-the-art, while complying with the most recent ISOstandards for accuracy [34]. Using the mean-shift principleand its variations, the robust and scalable sparse mean-shift,the intensity level of the region of interest is estimated. Wehave shown that the scalable version of the mean-shift withan individual selection of the number of data points givesgood results w.r.t. accuracy, while decreasing the computationtime. These variations are extended to the medoid-shift, whichoutperforms the mean-shift in our experiments. Furthermore,the convergence of all mean-shift and medoid-shift variationsis proven. We assert that the mean-shift and its variationsare suitable for segmentation applications, where the numberand size of the regions is unknown. The extended Kalmanfilter and a model for the chemical reaction are employedto enhance the accuracy of the estimate and reduce themeasurement time by around20%. A linear relation has beenfound between the convergence of the glucose reaction andits decay rate. We establish that inaccuracies in the derivedmodel can lead to degraded performance and, therefore, themodel has to be calibrated uniquely for each specific setupused. As future work, we aim to enhance this model, takinginto account not only the underlying glucose concentrationbutalso incorporating affecting parameters such as temperature,humidity, and the haematocrit level [12] and viscosity of thesample. Additionally, we see the need to further validate ourframework with clinical tests. Finally, we establish that verylow blood sample volumes of1nl seem to be at the limit ofwhat is acceptable in accuracy.

APPENDIX APROOF OF CONVERGENCE OF THE MEAN-SHIFT

ALGORITHM WITH WEIGHTS wl

Proof. Since L is finite, the sequence{fK(x(j)

)},

j = 1, 2, . . . is bounded. We will show that forx(j) 6= x(j+1):

fK(x(j)

)< fK

(x(j+1)

). Without loss of generality we

assumex(j) = 0, k being the profile of the kernelK(·) [19]

fK

(x(j+1)

)− fK

(x(j)

)=

1

h

L∑

l=1

wl (35)

×

[k

(x(j+1) − xl

h

)− k

(xlh

) ]

For convex profiles andx2 6= x1, x1, x2 ∈ [0,∞) it followsk(x2) ≥ k(x1) + k′(x1) (x2 − x1), such that

fK

(x(j+1)

)− fK

(x(j)

)≥ −

1

h

L∑

l=1

wl · k′(xlh

)· (36)

×

[||xl||

2 −∣∣∣∣∣∣x(j+1) − xl

∣∣∣∣∣∣2]

= −21

h· xT (j+1)

L∑

l=1

wlxl · k′(xlh

)

+1

h

L∑

l=1

wl

∣∣∣∣∣∣x(j+1)

∣∣∣∣∣∣2

· k′(xlh

)

= −L∑

l=1

wl

∣∣∣∣∣∣x(j+1)

∣∣∣∣∣∣2

· k′(xlh

)

Sincek(x) is monotonically decreasing,−k′(x) ≥ 0 for x ∈[0,∞).

∑Ll=1 −k

′(xl

h

)is strictly positive, as are the weights

wl. Hence fK(x(j+1)

)− fK

(x(j)

)> 0 and the sequence

{fK(x(j)

)} is convergent.

Without assumingx(j) = 0 and reformulating (35) we get

fK

(x(j+1)

)− fK

(x(j)

)≥ (37)

= −1

h

L∑

l=1

wl

∣∣∣∣∣∣x(j+1) − x

(j)∣∣∣∣∣∣2

· k′(x(j) − xl

h

).

Since fK(x(j+1)

)− fK

(x(j)

)converges to zero, then∣∣∣∣x(j+1) − x

(j)∣∣∣∣2 also converges to zero andx(j),j = 1, 2, . . .

is a Cauchy sequence.

APPENDIX BEXTENSION OF THE PROOF OF CONVERGENCE FOR THE

MEDOID-SHIFT WITH WEIGHTSwl

Proof. The choice of successive points in the medoid-shiftalgorithm, as given in (24), is carried out according to

L∑

l=1

1

h

∣∣∣∣∣∣x(j) − xl

∣∣∣∣∣∣2

wlk′

(x(j) − xl

h

)(38)

>

L∑

l=1

1

h

∣∣∣∣∣∣x(j+1) − xl

∣∣∣∣∣∣2

wlk′

(x(j) − xl

h

),

as equality of these two terms indicates convergence. This canbe reformulated as

L∑

l=1

1

hwlk

′

(x(j) − xl

h

)(39)

×

[∣∣∣∣∣∣x(j) − xl

∣∣∣∣∣∣2

−∣∣∣∣∣∣x(j+1) − xl

∣∣∣∣∣∣2]

= −L∑

l=1

1

hwl

∣∣∣∣∣∣x(j+1) − x

(j)∣∣∣∣∣∣2

k′(x(j) − xl

h

)> 0,

proving that fK(x(j+1)

)> fK

(x(j)

), hence that the

sequence{fK(x(j)

)}, j = 1, 2, · · · is strictly positive and

thusx(j) 6= x(j+c), for all c > 0. This proves that their are

no cycles and medoid-shift will converge.

12

ACKNOWLEDGEMENT

The authors would like to thank Dipl.-Phys. B. Limburgfrom Roche Diagnostics GmbH, Mannheim, for his supportand for providing the real glucose data.

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[5] J.M. Asfour et al., “Analyse Optischer Daten Mit Hilfe Von Histogram-men.,” Dec. 07 2011, European Patent Application, EP1843148A1.

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Nevine Demitri (S’11) was awarded a DAAD schol-arship to pursue her B.Sc. and M.Sc. degrees inTechnische Universitat Darmstadt, Germany. There,she received the B.Sc. in information and com-munication technology and M.Sc. in informationtechnology and electrical engineering in 2008 and2011, respectively. She is currently working towardsthe Ph.D. degree in the Institute of Telecommuni-cations, Signal Processing Group, Technische Uni-versitat Darmstadt. Her current research interestsinclude biomedical signal processing, image and

video processing, as well as machine learning.

Abdelhak M. Zoubir is a Fellow of the IEEE andIEEE Distinguished Lecturer (Class 2010- 2011).He received his Dr.-Ing. from Ruhr- UniversitatBochum, Germany, in 1992. He was with Queens-land University of Technology, Australia, from1992-1998 where he was Associate Professor. In1999, he joined Curtin University of Technology,Australia, as a Professor of Telecommunications andwas Interim Head of the School of Electrical &Computer Engineering from 2001 until 2003. In2003, he moved to Technische Universitat Darm-

stadt, Germany, as Professor of Signal Processing and Head of the SignalProcessing Group. His research interest lies in statistical methods for signalprocessing with emphasis on bootstrap techniques, robust detection andestimation and array processing applied to telecommunications, radar, sonar,automotive monitoring and safety, and biomedicine. He published over 300journal and conference papers on these areas. Professor Zoubir acted asGeneral or Technical Chair of numerous conferences and workshops. Recently,he was the Technical Co-Chair of ICASSP-14 held in May in Florence, Italy.Dr Zoubir has also held several positions in editorial boards; most notably, hewas the Editor-In-Chief of the IEEE Signal Processing Magazine (2012-2014).He was elected as Chair (2010-2011), Vice-Chair (2008-2009) and Member(2002-2008) of the IEEE SPS Technical Committee Signal Processing Theoryand Methods (SPTM), and a Member (2007-2012) of the IEEE SPS TechnicalCommittee Sensor Array and Multi-channel Signal Processing (SAM). Hecurrently serves on the Board of Governors of the IEEE SPS as an electedMember-at-Large (2015-2017), and has been a Member of the Board ofDirectors of the European Association of Signal Processing(EURASIP) since2008.