Introduction

In forensic temperature based death time estimation (TDE) casework rectal temperature measurement is the method of choice, the geometry and material parameters of the measurement locus’s actual surrounding are important for modelling the cooling process. Its low conductivity (see e.g.1 Table 6.23) makes body fat quantity and localization a potentially important issue in modelling post mortem temperature decline. The Finite Element Method (FEM) for TDE was developed to compute physics-based body core cooling. Using a CT scan to generate an FE mesh2,3 seems promising. One step is segmentation: matching a ‘material label’ to each of the CT scans pixels. The CT grey-scale values of body fat from muscle- and organ tissue are similar (see e.g.2,3), making segregating challenging. In4 we described an algorithm “CT histogram-based Fat Estimation and quasi-Segmentation” (CFES) for fat—muscle segmentation which was applied to only one human corpse and validated against synthetic and CT phantom data. The actual paper now presents CFES results on a sample of R = 32 CT scans of human corpses.

All three quality measures used, the τ-value (signal theory based), the anatomical misclassification rate D, and the distance measure S applied to the grey-scale value histogram, depend on CFES’s weighting exponent α (see4) directly as S or indirectly, as τ and D do.

Methods

The study was reviewed and approved by the ethics committee of the University Hospital Jena. All of our experiments were performed in accordance with the relevant guidelines and regulations (see subsection ‘Ethics approval’ below).

Fat tissue (fat, abbreviated ‘F’) has an expectation value of the CT attenuation equivalent below the water value (0 HU), while the value of muscle-, organ-, and connective tissues (muscle or ‘M’) is slightly above 0 HU (see e.g.5).

We will use the assumption that to any pixel in every slice a material type Mi can be assigned unambiguously accepting a small amount of misclassification. The term FM frequency denotes the number of pixels of a fixed grey-scale value in a CT slice.

In statements, true for fat as well as for muscle, we write Ξ as a symbol to be substituted by the letter ‘F’ for fat or by an ‘M’ for muscle. For any finite set D the symbol #D means the number of elements in D. For any function f and any set B in the preimage of f, the term f(B) stands for the set of images f(b) of elements b in B. The asterisk ‘*’ attached to a quantity-symbol means the estimated value of the quantity, whereas an attached cross ‘+’ means the true value of the indexed quantity. Generally symbols without ‘*’ or ‘+’ index are random variables. If dependence of a random variable on the real number parameter α is emphasized, the variable is indexed by ‘α’. For any quantity U—which is interpreted as a random variable—and any set V in the image of U the symbol μV(U) stands for the mean of the quantity U, where the variation of U is bounded to the set V of possible U-values. For our list of symbols we assume for conciseness, that the slice contains fat and muscle pixel only. This is not the case in reality which is considered by constraining to the image parts where F and M are the only components. We use the abbreviations LS for Least Squares and WLS for Weighted Least Squares to specify estimation procedures.

For conciseness we will omit arguments or indices of quantity symbols if the omitted parts are unambiguously clear from the context.

We suppose the random variables yq for q = 1, … , Q in a slice Y to be stochastically independent.

The problem of fat-muscle quasi-segmentation, called FM quasi-segmentation in the following, makes it necessary to distinguish the partial histograms XF and XM of the grey-scale value histogram X(Y) of a slice Y. As XF and XM are connected to the grey-scale value distributions fF and fM which are amalgamated to f = z∙fF + (1 − z)∙fM we chose the approach to firstly estimate z, fF and fM from the data. This enables us to directly compute the pixel numbers QF and QM solving the additional problem of FM quantification. Moreover the FM quasi-segmentation problem can be solved via definition of a threshold tFM in the grey-scale value histogram segregating XF as effective as possible from XM.

For all computations we used the 8-bit CT image version resulting in N = 256 possible grey-scale values. Though this is a reduced grey-scale value resolution compared to the original 12-bit version, it yielded a better filling of the histograms compartments. Since the images were of format 512 × 512 pixels, we had Q = 262,144 pixels bearing grey-scale values for our histograms X(Y).

The signal-to-signal separation measure τ

The classical signal-to-noise ratio (SNR) (see e.g.6) in the field of digital image processing is usually defined as the quotient of the mean signal amplitude μ(signal) and the standard deviation of the noise σ(noise):

$$SNR{ := }\frac{{\mu \left( {signal} \right)}}{{\sigma \left( {noise} \right)}}$$
(1)

As the SNR was originally designed to quantify the detectability of a narrow signal peak in a broad band of noise in frequency domain, its concept has to be adapted to our detection scenario, where the separability of two distinctly narrow signals peaks has to be measured. We will define something like an analogon τ to the SNR for our FM separation problem. The measure τ will be referred to as a ‘signal-to-signal ratio’. The grey-scale value pdfs fΞ(y,Θ) of fat (Ξ = F) and muscle (Ξ = M) in a CT slice are Normal distributions Φ(y,EΞ,SΞ) (see6) in good approximation, completely determined by their moments EΞ, SΞ. Therefore we look for a separability measure τ depending on EΞ, SΞ. The separability τ should become better with growing distance of the maxima EF and EM. The τ value should be rising with falling widths SF and SM making the peaks in the grey value histogram sharper. Both effects lead to smaller overlap of fF and fM. The problem shows obvious parallels to the problem addressed by Welch’s t-test7. Thus the FM separability τ, as applied to a CT slice Y, is defined as:

$$\tau \left( Y \right) = \frac{{E_{M} - E_{F} }}{{\sqrt {\frac{1}{{Q_{M} }} S_{M}^{2} + \frac{1}{{Q_{F} }} S_{F}^{2} } }}$$
(2)

Inserting the definition of the pixel numbers QF = z∙Q and QM = (1 − z)∙Q of the components F and M into (2 we yield:

$$\tau \left( Y \right) = \frac{{\sqrt Q \cdot \left( {E_{F} - E_{M} } \right)}}{{\sqrt {\frac{{S_{F}^{2} }}{z} + \frac{{S_{M}^{2} }}{1 - z}} }}$$
(3)

Since there is no direct access to the actual values of the variables EF, EM, SF, SM, z one has to approximate them in formula (3) by using estimators EF*, EM*, SF*, SM*, z* instead, yielding an estimator τ* for τ.

The anatomical FM misclassification measure D

The quality of the quasi-segmentation result W*(Y) is determined by an informed observer, comparing W*(Y) to the single CT image Y. This is formalized by the function d(Y, W*(Y)), called the (single slice FM) misclassification:

$$d\left( {Y,W^{*} \left( Y \right)} \right){ := }\left\{ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} if \begin{array}{*{20}c} {No\,anatomical\,error\,detected\,comparing\,Y\,to\,W^{*} \,\left( Y \right)} \\ {Anatomical\,error\,detected\,comparing\,Y\,to\,W^{*} \,\left( Y \right)} \\ \end{array} } \right.$$
(4)

Given a CT scan Y = {Y1, … , YL} of an human abdomen we define the (FM) misclassification rate D(Y, W*(Y)) as the mean of the single-slice FM misclassification:

$$D\left( {\underline {Y} ,W^{*} \left( {\underline {Y} } \right)} \right){ := }\mu_{{\underline {Y} }} \left( {d\left( {Y,W^{*} \left( Y \right)} \right)} \right) = \frac{1}{{\# \underline {Y} }}\mathop \sum \limits_{{Y \in \underline {Y} }} d\left( {Y,W^{*} \left( Y \right)} \right) = \frac{1}{L}\mathop \sum \limits_{{Y \in \underline {Y} }} d\left( {Y,W^{*} \left( Y \right)} \right)$$
(5)

The (sample FM) misclassification rate F(Ψ,W*(Ψ)) refers to a sample Ψ = {Y1, … , YR} of CT scans Yr of different human abdomina analogously:

$$F\left( {\Psi ,W^{*} \left( \Psi \right)} \right)\,{ := }\,\mu_{\Psi } \left( {D\left( {\underline {Y} ,W^{*} \left( {\underline {Y} } \right)} \right)} \right) = \frac{1}{\# \Psi }\mathop \sum \limits_{{\underline {Y} \in\Psi }} D\left( {\underline {Y} ,W^{*} \left( {\underline {Y} } \right)} \right) = \frac{1}{R}\mathop \sum \limits_{{\underline {Y} \in\Psi }} D\left( {\underline {Y} ,W^{*} \left( {\underline {Y} } \right)} \right)$$
(6)

The values d(Y, W*(Y)), D(Y, W*(Y)) and F(Ψ,W*(Ψ)) depend on the CFES’ parameter value α via W*(Y) of the scans Y in Ψ. If this fact will be emphasized in the following we will append an index α to the respective symbol, as in: W*α(Y), d(Y, W*α(Y)), D(Y, W*α(Y)), F(U, W*α(U)). If we talk generally about the influence of parameter α on FM misclassification, we use the symbols dα, Dα, Fα.

The weighted least squares distance Sα

The (WLS) distance Sα(X(Y), X(Θ)) of a slice Y with reference to a parameter vector Θ, which was used as target parameter for CFES’ nonlinear estimation process, can be thought of as another quality measure defined. It is computed from the empirical histogram X(Y) : = {x0(Y), … , xN-1(Y)} and the theoretical histogram X(Θ) : = {x0(Θ), … , xN-1(Θ)} as follows:

$$S_{\alpha } \left( {X\left( Y \right), X\left( \Theta \right)} \right){ := }\mathop \sum \limits_{{n \in G_{0} }} \left( {x_{n} \left( Y \right) - x_{n} \left( \Theta \right)} \right)^{2} \cdot x_{n} \left( Y \right)^{\alpha }$$
(7)

As a seemingly evident hypothesis we primarily assume:

(S):

For any CT slice Y: A small value of the distance Sα(X(Y), X(Θ)) between measured grey value histogram X(Y) and theoretical grey value histogram X(Θ) given an actual value of the parameter Θ, or a high value of the FM separability τα(Y), should lead to a good FM quantification in terms of a correctly estimated value of the FM proportion parameter z as well as to a good FM quasi-segmentation measured via the single-slice FM misclassification dα(Y,W*(Y)).

The minimisation of (7) as well as the formulae for the sensitivity analysis of the WLS approach are described in more detail in the supplementary information. There we derive the equations (S1.3) for small deviations of the WLS parameter estimation, the components of the Jacobi matrix (S2.1)–(S2.13), the components of the estimators covariance matrix (S3.3) and explain our assumption (S4.3) for the covariance matrix of the input, the grey-scale value histogram.

The sample of human body scan data

CFES was applied to CT scans of human abdomens. The abdomen’s upper and lower bounds were defined by anatomical landmarks: The central diaphragm region characterizes the upper scan limit, while the visible conjunction of the upper part of os pubis (facies symphysialis) marks the lower limit.

After excluding three of the originally 35 corpses from the sample due to missing CT data or due to artefacts deteriorating the CT scan, the sample consisted of R = 32 human bodies of 11 female and 21 male subjects. The prosecution had levied on the bodies and ordered the CT scans as well as the autopsies. For every body age (mean = 57.41 y, stddev. = 20.22 y), body weight (mean = 71.37 kg, stddev. = 16.59 kg), body height (mean = 1.70 m, stddev. = 0.08 m) and body mass index (mean = 24.43 kg/m2, stddev. = 4.92 kg/m2) were registered.

For all of the scans Y a LightSpeedVCT (GE Medical Systems) was used with a filtered back projection reconstruction, a ‘body’ filter type and with a spiral pitch factor of 0.984. Table 1 shows an overview of our scan’s CT parameter values. For our pixel-volume calculation the parameter ‘Spacing between slices’ (column 5 of Table 1), was relevant rather than the parameter ‘Slice Thickness’ (column 6 of Table1).

Table 1 CT parameter of the corpse sample (three of the originally 35 corpses excluded).
Full size table

Former usage of study material

The CT-data and its CFES-evaluations were used in the thesis27 of Schenkl, who is the first author of the present article.

Ethics approval

The study was reviewed and approved by the ethics committee of the University Hospital Jena. According to the ethics committee’s statement written consents of the kinship for the abdominal CT slices shown in the article were not needed since the bodies were confiscated by the local prosecution who directed the CT scans for investigations. Moreover, the representations of the slices are totally anonymized.

Results

Quality parameters: single analysis

Figure 1A presents misclassification rate Fα(Ψ,W*(Ψ)) as a function of α. The optimum exponent α = 1 with Fα = 0.036 = 3.6% is marked. Figure 1b is the a misclassification rate Dα histogram for α = 1.

Figure 1
figure 1

(a) Mean FM misclassification rate Fα for whole body CT sample Ψ as a function of weight exponent α. (b) Histogram of the FM misclassification rate Dα for α = 1. (c) Histogram of μYr(τ*α) with α = 1. (d) Histogram of μYr(ln Sα) for α = 1.

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Figure 1c, d show the mean separability μYr(τ*α)- and the mean μYr(lnSα) histogram respectively, taken over all CT slices Yr,l for an abdominal scan Yr for each one of the R bodies.

Quality parameters: multiple parameters analysis

In Fig. 2a a scatterplot of mean separability μYr(τα*) (X-axis) versus misclassification rate Dα (Y-axis is shown while Fig. 2b presents a scatterplot of the mean log minimal distance μYr(ln Sα) and misclassification rate Dα (for both α = 1). Figure 2c is a scatter plot of mean separability μYr(τ*α) (X-axis) and mean log WLS distance μYr(lnSα) (Y-axis) with α = 1 showing a distinctive linear relation (Linear regression analysis: R2 = 0.772, p < 0.001, slope a = 0.007 and intercept b = 20.145).

Figure 2
figure 2

(a) Scatterplot: FM misclassification rate Dα and mean FM separability μYr(τα*) of all individual scans Yr in the sample Ψ for α = 1. Triangle inserted to show non-function type relation. (b) Scatterplot: FM misclassification rate Dα and mean log WLS distance μYr(ln Sα) of all individual scans Yr in sample Ψ for α = 1. (c) Scatterplot: mean log WLS distance μYr(ln Sα) and mean FM separability μYr(τ*α) of each individual scan Yr in the sample Ψ with α = 1. The drawn line represents a linear model fitted with by a least square approach. (d) Example slice from scan LFD_5: Original CT slice. (e) Example slice from scan LFD_5: Grey value histograms: measured: X (thin line), fitted: X(Θ) (fat line), partial fat: XF(Θ) (dotted line), partial muscle XM(Θ), fat vertical line: threshold tFM. (f) Example slice from scan LFD_5: Quasi segmented image (light grey: muscle, dark grey: fat, white boundary area: bones), the weight exponent was α = 1.

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Figure 2d–f present an example slice with μYr(τα*) = 170.24/Dα = 4% with single-slice misclassification dα = 0 but low mean separability τα* = 128.35 and middle-to-high log distance ln Sα = 23.38 for α = 1.

As slope a > 0 contradicts the naive intuition (S) (‘high separability τα causes low distance Sα’), we further studied the effect using simulated images.

Quality parameters: Monte Carlo analysis of mean log WLS distance μYr(ln Sα) and mean FM separability μYr(τα*)

We used two series YA,1, … ,YA,R and YB,1, … ,YB,R (R = 20) of simulated images (Fig. 3a,b) with EF = −80 HU, EM = 50 HU and S = SF = SM = 10 HU, 12 HU, 14 HU, 15 HU, 17 HU, 20 HU, 25 HU, 30 HU, 35 HU, 40 HU, 45 HU, 55 HU, 60 HU, 70 HU, 80 HU, 90 HU, 100 HU, 150 HU, 200 HU, 250 HU. For series A and B respectively z was set z = 0.3 and z = 0.5. All images YA,r and YB,r were evaluated by CFES algorithm with α = 1. The true values and the estimated values of EF, EM, SF, SM, z, and the quality parameters ln Sα, the true and the estimated value of τα were recorded.

Figure 3
figure 3

(a) Monte Carlo simulated CT image YA,r with components fat (dark grey), muscle (light grey), air (black). The true image parameters were: EF+ = −80 HU, EM+ = 50 HU, SF+ = SM+  = 70 HU, z+ = 0.3. (b) Grey-scale value histograms X (empirical), X(Θ) (fitted via CFES with α = 1), XF(Θ), XM(Θ) with: Thin line: measured, fat line: estimated, dotted: fat, dashed: muscle. (c and d) Scatterplots of mean true value μYA,r (τα+) vs. mean estimated value μYA,r (τα*) with z = 0.3. (c) The diagrams containing all of the synthetic images YA,r generated. (d) Diagram as in the upper right part only diagrams YA,r were considered with standard deviations S = SF+  = SM+  ≥ 70 HU. The parameter S = SF+  = SM+ labels the data-points. The weight exponent α = 1 was used in Sα.

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The τα+ vs. τα*—scatterplots in Fig. 3c, d respectively represent the full sample YA,r (r = 1, … ,R) and the YA,r with τα+-range of [0, 400] only.

Figure 4a is the scatterplot of ln Sα vs. τα over the YA,r—and the YB,r sample (A: z = 0.3: fat rings, B: z = 0.5: fat rectangles). The standard deviation value S labels every data point. A dashed rectangle marks the area × [0, 700] × [20.5, 24.5] of Fig. 2c.

Figure 4
figure 4

(a) Scatterplot for the variables FM separability τα* and logarithm of weighted least squares sum ln Sα of the Monte Carlo images YA,r and YB,r with r = 1, … , 20 (where A: z = 0.3 and B: z = 0.5). The data-points are labelled with the number of their grey-scale value histograms true standard deviations S+  = SF+  = SM+. The dashed rectangle on the left side marks the τα* − ln Sα—area of the diagram in Fig. 2d. (b) In the middle: Scatterplot for the variables FM separability τα* and logarithm of weighted least squares sum ln Sα of the Monte Carlo images YA,r with r = 1, … , 20. The data-dots are labelled with the value of their grey-scale value histograms true standard deviations S+  = SF+  = SM+. Encircling the central scatterplot five empirical and estimated grey-scale value histograms X (thin, drawn), X(Θ) (fat, drawn), XF(Θ) (fat, dashed), XM(Θ) (fat, dotted) are presented, each is connected to its corresponding data-dot in the central scatterplot by a fat drawn pointer-line. (c) 3-D scatterplot of the variables FM separability estimator τα*, the logarithm of the weighted least squares distance ln Sα and relative error REz of the z-estimator for the image-series YA,r and YB,r with r = 1, … , R. Black bubbles : A: z = 0.3 void bubbles: B: z = 0.5. Images YA,r and YB,r were excluded if S ≤ 5 or S ≥ 100 as well as if REZ ≥ 0.2. The weighting exponent was set to α = 1. (d) Computed diagrams of the z-estimator’s standard deviation S(z*) as a function of the standard deviations S = SF = SM of the partial fat and muscle distributions in the scenario YA and YB with EF = −80 HU, EM = 50 HU, z = 0 and ranges 20 HU ≤ S ≤ 90 HU (A: left) and 20 HU ≤ S ≤ 130 HU (B: right). Note the different ranges on the ordinate. The graphs were generated with a true fat ratio value z+  = 0.3. However the choice of the true fat ratio value z+ had virtually no influence on the result when it was changed from z+  = 0.3 to z+  = 0.5. The weight exponent was chosen α = 0 which means z* is an LS-estimator rather than an WLS estimator as it is in the case α = 1.

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The center of Fig. 4b repeats Fig. 4a with associated histograms X (thin line), XΞ*(Θ) (dotted: Ξ = F, dashed: Ξ = M), X*(Θ) shown for the marks S = 12 HU, 25 HU, 40 HU, 60 HU, 100 HU.

Figure 4c investigates the ‘umbrella’ in Fig. 4a, b. It shows a 3-D scatterplot (YA,r or YB,r excluded if S ≤ 5 or S ≥ 100 or if REZ ≥ 0.2.) of separability τα*, logarithmic distance ln Sα and the relative error REZ =|z+  − z*|/z+ of z-estimation (Black filled bubbles = YA,r (z = 0.3), void bubbles = YB,r (z = 0.5)). The view is leading through the left and right front sides as well as through the top side of the cube.

Figure 4d presents the Gaussian estimation of our WLS estimation z*’s standard deviation S(z*) with α = 0, where WLS = LS and EF = −80 HU; EM = 50 HU, z = 0.3 (graph for z = 0.5 is nearly identical), S = SF = SM varied between S = 20 HU and S = 90 HU (Fig. 4d left) and with ΔE : = EM − EF between S = 20 HU and S = ΔE/2 = 130 HU respectively (Fig. 4d right). The curves in Fig. 4d were generated for α = 0, since this guarantees the weighting to be of no influence. The latter was preferable as for each of the values r = 1–3 the weight-matrix else would have been different for every two slices else, thus introducing noise in the evaluation.

General results

As examples of the CFES performance Fig. 5a,d,g,j show four slices Y, their grey-scale value histograms X(Y), X(Θ*), XF(Θ*), XM(Θ*) (Fig. 5b,e,h,k) and the respective quasi-segmentation results W*(Y) (Fig. 5c,f,i,l). Horizontal image-line 1, 2, 3, 4 corresponds to LFD 06, 22, 24, 16 (see Table 1) respectively. While the first three slices (a), (d), (g) lead to acceptable results, slice (j) was added as a poor result caused by weak contrast and artefacts. LFD 16 was excluded from our statistical evaluations.

Figure 5
figure 5

Four slices Y in (a), (d), (g), (j) from four different persons at different abdominal positions. Respective histograms with thin line = measured data X, thick line = fitted histogram X(Θ*) , partial histograms XF(Θ*), XM(Θ*) in (b), (e), (h), (k) and quasi-segmentation results W*(Y) in (c), (f), (i), (l). The parameter α was α = 1.

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Discussion

General discussion

The exponent α’s calibration in the paper4 yielded optimum values α = −1 for synthetic CT images and α = −1 for the real-world phantom. Application of CFES to our body CT scan sample Ψ and evaluating the misclassification rate Fα(Ψ, W*(Ψ)) leads to Fig. 1a. This indicates α = 1 to be the optimal parameter value with Fα = 3.6% for our body sample though the values for α = −1, α = 2 and α = 3 lie only slightly above: The Fα-values for α = −3, − 2, − 1 are Fα = 54%, 30%, 6% which is distinctively higher.

The three quality measures τα, Dα and Sα are a posteriori quantities. This is evident for Dα, and for Sα. Only for synthetic CT images the separability τα+ could be determined in advance. τα can only be estimated from a priori parameter estimators as described in4. The parameter τα obtains its justification from analogies in signal processing and in statistics in contrast to the WLS difference Sα, whose minimality is necessary for estimating the parameter vector Θ. The misclassification rate Dα is relying on a priori expert knowledge.

Quality measures should quantify firstly the suitability of the input data (the separability of fat from muscle in a CT-image), secondly the efficacy of the segmentation (WLS fitting of the model X(Θ) to the histogram X and computation of the threshold tFM) and at last the correctness of the result (the anatomical correctness of FM segmentation). Our measures τα, Sα and Dα cover all three aspects: τα measures the shape of the X(Y) to quantify the CT-image Y’s suitability for CFES application, Sα judges the final ‘distance’ between X and X(Θ) only, while Dα evaluates W*(Y) without distinction of genesis.

In4 we gave a synopsis of tissue segmentation studies which we review in Table 2 for their quality measures.

Table 2 Studies in literature about CT fat segmentation: quality measures.
Full size table

Our choice of τα, Sα and Dα may be partly explained since we were not able to determine a gold standard as e.g. manual dissection. Sα as the minimized quantity of quasi-segmentation seems to be a natural choice. As we knew the single material grey-scale value pdf to be Gaußian, we did not want to use non-parametric fitting methods as e.g.8. Usage of τα was inspired by the t-test statistic, used for deciding whether two Gaußians have different expectation values. Moreover there are similarities of τα to the signal theoretic quantity of signal to noise ratio (SNR), quantifying the separability of two signals in frequency domain. Finally the misclassification rate Dα seemed to be a canonical choice for a measure of the pure CFES-output quality.

In the slice of Fig. 2d–f the dα implies good quality, whereas Sα indicates low quality, therefore we will investigated in more detail the interrelations between the three quality measures. Yet for α-calibration there is no better parameter than Dα or Fα since the usage of Sα as a quality measure for fat quantification runs into problems we will discuss below.

As CT numbers generally depend on temperature (see26), the question arises whether the results presented are robust against sample temperature changes. The results in26, using conventional polychromatic 120kVp CT (CPI) (as well as virtual monoenergetic images by dual energy measurements (VMI) with 40kVp, 50 kVp, 60 kVp, 70 kVp, 100 kVp, 120 kVp, 140 kVp), seem to give evidence to the hypothesis of robustness: The temperature-changes in an interval of [10 °C, 60 °C] led to changes of [− 40 HU, − 80 HU] in the grey-level values for fat-substitute and of [50 HU, 60 HU] for muscle-substitute in CPI (see Fig. 5 in26) which lie in the range of inter-personal grey-level value variations of our study. Furthermore26 indicates that there is no influence of the convolution kernel chosen on the temperature dependence of the grey-level values (see Fig. 5 in26) for CPI. The paper26 implies possibilities for further research on quasi-segmentation e.g. by using VMI and/or temperature control for improving CFES-results.

Relations between the three quality measures

In Fig. 2a the sample’s location in the triangle tells us that high values of the μYr(τα*) seem to force low values on Dα (see (8A)), whereas high values of Dα constrain μYr(τα*) to low values (see (8B)). Nevertheless, there are abdominae with low values of Dα and of μYr(τα*) (see (8C)):

$$\mu_{{\underline {Y} r}} \left( {\tau_{\alpha }^{*} } \right) \uparrow = > D_{\alpha } \downarrow$$
(8A)
$$D_{\alpha } \uparrow = > \mu_{{\underline {Y} r}} \left( {\tau_{\alpha }^{*} } \right) \downarrow$$
(8B)
$$\exists : D_{\alpha } \downarrow \wedge \mu_{{\underline {Y} r}} \left( {\tau_{\alpha }^{*} } \right) \downarrow$$
(8C)

While (8A) and (8B) match our intuition (S) about relations of the quality measures, the last vertex of the triangle [(8C): low μYr(τα*) and low Dα] seems to contradict them grossly at first sight. Note however, that the minimum value of μYr(τα*) for Dα = 0% is μYr(τα*) ≈ 200. A ‘linear approximation’ to the upper bound of the relation is for α = 1:

$$0 \le D_{\alpha } \le A \cdot \mu_{{\underline {Y} r}} \left( {\tau_{\alpha }^{*} } \right) + B$$
(9)

where the line A μYr(τα*) + B is estimated by connecting the two data-points (x0, y0) = (103.0 , 21.35%) for scan LFD_19 and (x1, y1) = (654.0, 0.0%) for scan LFD_23 by:

$$A^{*} = \frac{{y_{1} - y_{0} }}{{x_{1} - x_{0} }} = \frac{21.35\% - 0.0\% }{{103.0 - 654.0}} = - 0.03875\%$$
(10)
$$B^{*} = y_{1} - A^{*} \cdot x_{1} = 0.0 + 0.03875 \cdot 654.0 = 25.3425\%$$
(11)

Figure 2d–f displays slice 119 of scan LFD_5 as an example of a slice Yr,l with low dα and with low τα* as well. This suggests the hypothesis of the threshold tFM to be robust against fluctuations of τα.

Figure 2b is a scatterplot of μYr(ln Sα) vs. Dα for α = 1. As in Fig. 2a we see a relation, not a functional graph but a ‘formed cloud’ filling one corner of the first quadrant. In analogy to Fig. 2a we see on Fig. 2b high values of μYr(ln Sα) seem to force low values of Dα [see (12A)], contradicting intuition (S), whereas high values of Dα constrain low values of μYr(ln Sα) [see (12B)] contradicting (S) as well. The third corner of the triangle meets intuition (S): low μYr(ln Sα) coincides with low Dα [see (12C)].

$$\mu_{{\underline {Y} r}} \left( {\ln S_{\alpha } } \right) \uparrow = > D_{\alpha } \downarrow$$
(12A)
$$D_{\alpha } \uparrow = > \mu_{{\underline {Y} r}} \left( {\ln S_{\alpha } } \right) \downarrow$$
(12B)
$$\exists :\mu_{{\underline {Y} r}} \left( {\ln S_{\alpha } } \right) \downarrow \wedge D_{\alpha } \downarrow$$
(12C)

The upper boundary approximation line of the cloud is:

$$0 \le D_{\alpha } \le C \cdot \mu_{{\underline {Y} r}} \left( {ln S_{\alpha } } \right) + G$$
(13)

Estimating the line C μYr(ln Sα) + G by connecting the data-points (x0, y0) = (21.62 , 21.35%) (scan LFD_19) and (x1, y1) = (23.88, 5.6%) (scan LFD_27) gives:

$$C^{*} = \frac{{y_{1} - y_{0} }}{{x_{1} - x_{0} }} = \frac{5.6\% - 21.35\% }{{23.88 - 21.62}} = - 6.97\%$$
(14)
$$G^{*} = y_{1} - C^{*} \cdot x_{1} = 5.6\% + 6.97\% \cdot 23.88 = 172.0\%$$
(15)

Figure 2c is the scatterplot (α = 1) of μYr(ln Sα) vs. μYr(τα*). The distinct linear relationship with positive slope was significant in by linear regression analysis, which is a contradiction (16) to (S):

$$\mu_{{\underline {Y} r}} \left( {\tau_{\alpha }^{*} } \right) \uparrow = > \mu_{{\underline {Y} r}} \left( {\ln S_{\alpha } } \right) \uparrow$$
(16)

The deviations (8C), (12A), (12B), (16) from (S) were investigated, using the simulated CT images (YA,r)r=1,…,R and (YB,r)r=1,…,R (see Fig. 3a, b).

Figure 3c shows a scatterplot of τα+ vs. τα* with ranges [0, 6000] on the X- and on the Y-axis for sample YA,r. The matching of τα+ and τα* is almost perfect here, becoming slightly weaker in the region near (0,0). Zooming in on a range of [0, 400] for τα+ we notice strong deviations of the estimator values τα* in Fig. 3d. Note that the range of our sample Ψ’s mean separabilities μYr(τα*) is [0, 600]: The correlation of τα+ and τα* of our sample Ψ would have to be estimated from the zoomed in representation in Fig. 3d rather than from the diagram in Fig. 3c. The correlation of τα+ and τα* refutes the objection to the positive correlation μYr(τ*α) − μYr(lnSα) in Fig. 2c: “Maybe there is a negative correlation in the analogous μYr(τ+α) − μYr(lnSα) diagram, which cannot be detected as τ+α is not accessible for our real-world-sample Ψ, but the μYr(τ*α) − μYr(lnSα) diagram shows a positive correlation caused by a flaw of CFES generating a locally negative correlation between τα+ and τα*?” Despite the noise in Fig. 2c there is a distinctly positive relation, so the objection can be rejected.

The relations between τα* and ln Sα are demonstrated in Fig. 4a as scatterplot of τα* vs. ln Sα with α = 1 for the images YA,r (square marks). The partial graph over τα*-range [1150, 3000] corresponds with (S): Rising τα* is correlated with falling ln Sα and vice versa:

$$\forall \tau_{\alpha }^{*} \in \left[ {1150, 3000} \right]:\left( {\tau_{\alpha }^{*} \uparrow < = > \ln S_{\alpha } \downarrow } \right)$$
(17)

The curve in the range of [1150, 3000] is a function graph without random noise.

In the complementary partial graph over τα*-range [0, 1150] the curve is more noisy. Its tendency contradicts (S) since rising τα* coincides with rising ln Sα:

$$\forall \tau_{\alpha }^{*} \in \left[ {0, 1150} \right]:\left( {\tau_{\alpha }^{*} \uparrow < = > \ln S_{\alpha } \uparrow } \right)$$
(18)

The two τα*-ranges are associated to two ranges of S = SF = SM in the synthetic sample A: ‘τα* in [1150, 3000]’ corresponds to ‘S in [10 HU, 25 HU] whereas ‘τα* in [0, 1150]’ is associated to S ≥ 25 HU. Though ΔE = EM − EF = EF+  = 50 HU –(-80 HU) = 130 HU, even S = 25 HU means only a moderate grey-scale value distributions densities fM and fF overlap.

The area of the sample Ψ’s scatterplot μYr(τ*α) vs. μYr(lnSα) in Fig. 2c can be realized as the dashed rectangle in the τα* vs. ln Sα—scatterplot Fig. 4a: This area lies on the left side of [0, 1150] with trend (17) and comprises a vast variation of the ln Sα—values in the scatterplot of the synthetic samples A and B. The trend in Fig. 2c seems to be fairly linear despite varying fat-proportions z in the real-worlds CT scan sample Ψ. It does not depend on the fat-proportion z, since the umbrella-form with the fuzzy left end shows up in both plots of Fig. 4a. While the top of the umbrella at about τα* = 25 sank from ca. ln Sα = 25 to ca. ln Sα = 23.7 for z rising from z = 0.3 to 0.5, the maximum-location on the τα*-axis is nearly identical.

To see, how the overlap between z∙fF and (1 − z)∙fM of a slice is associated to its position on the umbrella, Fig. 4b shows the τα* vs. ln Sα scatterplot of (YA,r)r=0,…,R again. Additionally, we present five of the grey-scale value histograms X(Y), X(Θ), XF(Θ), XM(Θ) associated to the scatterplots data-points for the parameter values S = SF = SM = 12 HU, 25 HU, 60 HU, 100 HU, 150 HU. The umbrella’s top lies at S = 25 HU, and it is obvious, that the increasing overlap of the partial histograms XF(Θ) and XM(Θ) increasingly deteriorates z-estimation quality when moving from the right to the left side of the umbrella. The z-estimation quality can be recognized roughly from the overlap area under XF(Θ) and XM(Θ).

The importance of the umbrella—phenomenon can be demonstrated by relating it to the relative error REz : =|z* − z+|/z+ of our target variable z. In Fig. 4c, a 3-D scatterplot of τα* vs. ln Sα vs. REz for (YA,r)r=1,…,R (black bubbles) and (YA,r)r=1,…,R (white bubbles) with the additional constraints: 5 HU ≤ S ≤ 100 HU and REz < 0.2 is shown. It demonstrates the abrupt strong rising of the error REz in the target variable z* if the data points leave the right side of the ‘umbrella’ in Fig. 4a (which can be found projected on the bottom of the coordinate cube in Fig. 4c as well) and proceed to the left side.

As reason for the umbrella form of the τα* vs. ln Sα scatterplot (Fig. 4a), for the fuzziness and for the rise of the relative error REz on the left side of the umbrella (in Fig. 4c), a sort of pseudo-overfitting of the model f(y,Θ) can be hypothesized. While the well-known effect of overfitting is associated to using a model with too many parameters, our self-coined term ‘pseudo-overfitting’ means usage of a model with the correct number of parameters (in our case: 5 parameters: EF, EM, SF, SM, z) if the model is a mixture of distributions of the same type (in our case: the normal distribution mixture f(g,Θ) = z fF(g,Θ) + (1 − z) fM(g,Θ) ) and with an empirical distribution geometrically similar to a (mixture of less) distribution(s) of the same type (in our case: X(Y) seems to consist of only one normal distribution like in Fig. 4b the grey value distribution for S = 150 HU). Here we have to take into account similar effects as those caused by classical overfitting: The fitting process uses the seemingly redundant parameters to model some of the apparently random deviations, which leads to virtually getting ‘better’ results in terms of Sα though the separability τα decreases (see Figs. 2c and 4a). Associated with smaller values of Sα and τα one notes a dramatic loss in the estimators z*’s exactness which was measured by the relative error REz of the z-estimator z* (see Fig. 4d).

A principal constraint in fitting Gaußian mixtures

A large proportion of the problems reported in 4.1 and 4.2 seems to come from a principal difficulty in fitting Gaußian mixtures as we did for estimating the parameter vector Θ: The grey value histogram X(Y) of a CT slice Y containing fat- and muscle components only was assumed to be a mixture of the Gaußian components XF and XM for fat F and muscle M respectively. Figure 4d demonstrates an exponential growth of the z-estimator’s standard deviation S(z*) for linearly increasing standard deviation S = SF = SM of the F- and the of the M-component. This is made plausible by comparing the graph on the left side of Fig. 4 (lower right part) (z = 0.5, 20 HU ≤ S ≤ 90 HU) to the graph on the right side (z = 0.5, 20 HU ≤ S ≤ 130 HU): though the curve seems to be geometrically identical, the ordinates range has grown by a factor of order 20 while the domain was enlarged from [20 HU, 90 HU] to [20 HU, 130 HU] only. This effect does not depend on the actual value of z: The shown graphs with a true parameter value of z = 0.3 are virtually indistinguishable from analogous ones–which therefore are not shown here—with true parameter value z = 0.5. The computation results presented in Fig. 4d strikingly demonstrate the fact that the exactness S(z*) of z-estimation is rapidly deteriorating with increasing overlap of the single-material—normal distributions in the grey-scale value histogram. Given a distance of ΔE = EM − EF = 50 HU − (− 80 HU) = 130 HU between the maxima in the histogram, a value of S = SF = SM ≥ 60 HU leads to an approximate z-estimator standard deviation S(z*) ≥ 0.1 while S ≥ 90 HU causes S(z*) ≥ 1.0. Note that z is a ratio from the interval [0, 1] which makes an estimator z* with S(z*) ≥ 1 nearly useless.

The difficulty in fitting Gaussian mixtures discussed above and particularly the swiftly growing standard deviation S(z*) of the z-estimator with rising standard deviation of SF = SM seems to be no consequence of our special fitting algorithm WLS chosen: Due to the high number of pixels in a CT slice it is possible to apply asymptotic propositions to the probability distribution of the estimator. Jennrich28 shows, that if the data’s (in our case the grey-value histogram X) probability distribution lies in the exponential family (the multinomial distribution of our case belongs to this family) the Fisher Scoring algorithm for iteratively finding the Maximum Likelihood Estimator (MLE) value is identical to the Gauss–Newton algorithm which iteratively detects the Non-linear Least Squares estimator’s (NLLS) value, which is detected by CFES as well. Hence the probability distribution of z* computed here via CFES (NLLS estimation respectively) is nearly identical to the distribution of the MLE of z which asymptotically is the most effective estimator (see29,30), at least in case α = 0.

Note further that the standard deviation S(z*)'s rapid rising with growing overlap between HF and HM are not reported in the fat segmentation literature (see8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25) except maybe hinted in8.

A practical consequence of this issue will be to choose CT parameter values minimizing the single-material distribution standard deviations SF and SM.