EXPLOITING 3 D ULTRASOUND FOR FETAL DIAGNOSTIC PURPOSE THROUGH FACIAL LANDMARKING

In the last decade, three-dimensional landmarking has gained attention for different applications, such as face recognition for both identification of suspects and authentication, facial expression recognition, corrective and aesthetic surgery, syndrome study and diagnosis. This work focuses on the last one by proposing a geometrically-based landmark extraction algorithm aimed at diagnosing syndromes on babies before their birth. Pivotal role in this activity is the support provided by physicians and 3D ultrasound tools for working on real faces. In particular, the landmarking algorithm here proposed only relies on descriptors coming from Differential Geometry (Gaussian, mean, and principal curvatures, derivatives, coefficients of first and second fundamental forms, Shape and Curvedness indexes) and is tested on nine facial point clouds referred to nine babies taken by a three-dimensional ultrasound tool at different weeks' gestation. The results obtained, validated with the support of four practitioners, show that the localization is quite accurate. All errors lie in the range between 0 and 3.5 mm and the mean distance for each shell is in the range between 0.6 and 1.6 mm. The landmarks showing the highest errors are the ones belonging to the mouth region. Instead, the most precise landmark is the pronasal, on the nose tip, with a mean distance of 0.55 mm. Relying on current literature, this study is something missing in the state-of-the-art of the field, as present facial studies on 3D ultrasound do not work on automatic landmarking yet.


INTRODUCTION
Fetal diagnosis has recently viewed a growth in the medical field.The earliest and earliest possibility to see if the baby "in mommy's tummy" is sane and grows well offers both to future mothers and practitioners the opportunity to state, months before the childbirth, if the baby needs or will need special care.Many syndromes involve physical dysmorphisms that, if seen early, could bring the doctors to follow the mother with a tailored medical path.These kinds of syndromes may affect the functioning of internal organs or of the whole organism, bring cognitive and behavioural deficits, or growth retardation, and cause peculiar physical and facial features.Especially in the earliest months of pregnancy, these defects could not be immediately identified, both because the baby's lineaments are not formed yet and because of poor accuracy of echography tools.
To improve the quality of echographies, in the last decade 3D tools have been employed, although these have not "taken the field" yet.In this context, face or body landmarks were taken into consideration for prenatal diagnosis.Johnson et al. (2000) demonstrated that three-dimensional ultrasonography (US) could be used in identifying and localizing cleft lip, as well as normal facial anatomy, prenatally.The shown advantage of 3D US is that the rendered image provides landmarks for planar images.Rotten et al. worked on 3D sonography for diagnosing retrognathia and micrognathia (Rotten et al., 2002).Then, they identified some landmarks to be analyzed for studying facial features: forehead, nasal soft tissues and nasal bones, upper lip, hard palate, oral cavity, inferior lip, and chin.The objective was also to assess the use of 3D ultrasound imaging in fetal facial examination (Rotten and Levaillant, 2004).Pilu et al. (2006) dealt with detecting anomalies in midline structures of fetal brain VEZZETTI E ET AL: Fetal ultrasound: a new 3D diagnosis protocol using three-dimensional ultrasound.The diagnosis is made when the main cerebellum landmarks, the fastigium of the fourth ventricle and the main fissures, could not be identified.Paladini et al. (2006) used 3D ultrasound to show how abnormalities of the cerebellar vermis and posterior fossa could be differentiated sonographically thanks to the employment of a set of morphometric parameters.They focused on simple anatomical landmarks easily detectable and measurable on the midsagittal ultrasound image of the fetal head (tentorium, maximum craniocaudal vermian diameter, clivus).Rochelson et al. (2006) used 3D sonographic multiplanar display with the use of spatial rotation to identify landmarks in the coronal (midline between the orbits at the nasal bridge) and transverse (midline at the level of the anterior orbit) planes, for establishing a reproducible and consistent facial profile in the midsagittal plane.The final goal is to apply geometric morphometric statistical analysis to quantify shape differences in normal and abnormal fetal skulls.Faure et al. (2007) described a new three-dimensional ultrasound rendering technique to examine the normal fetal posterior palate, providing diagnostic information.The sonographic visualization rates of seven defined anatomical landmarks of the fetal palate were computed for each gestational age: maxilla with alveolar ridge and tooth buds, maxilla palatine process, interpalata suture, palate bone horizontal plate, transverse palatal suture, posterior nasal spine and pterygoid process.Viñals et al. (2007) used 3D multiplanar and Volume Contrast Imaging (VCI) in the coronal plane to assess the integrity of the median fetal cerebral structures.Landmarks of structures such as the cerebellar vermis and the fourth ventricle were accurately defined through the use of the VCI filter.Bromley et al. (2007) examined whether the third-trimester fetus could be assessed sonographically using three-dimensional volume data sets, with the objective of determining fetal presentation, amniotic fluid volume, placental location, and estimating fetal weight.This identification was done through gross anatomical landmarks.Using as well 3D volumes, but of fetal face, Plasencia et al. (2007) investigated the effect of deviations from the exact midsagittal view on measurements of the frontomaxillary facial angle at 11+0 to 13+6 weeks' gestation.Landmarks identifying the midsagittal plane of fetal face have been manually detected by sonographers.Pugash et al. (2008) discussed the relative value of both prenatal US and fetal Magnetic Resonance Imaging (MRI) for prenatal diagnosis, with the support of landmarks to identify face and body parts.Turan et al. (2009) used 12 heart landmarks for second-trimester examination of fetal heart using three-dimensional US techniques.Final aim was to accurately discriminate between normal and abnormal cardiac anatomy.Individual anatomic landmarks were identified in 89.7-99.1%.Wong et al. (2009) examined the secondary palate at various gestational ages by the aid of oblique planes from stored 3D ultrasound volumes of the fetal face.The uvula served as a reliable landmark for viewing the soft palate.
The research group of Sepulveda, Martinez-Ten, and Wong also used three-dimensional volumes and 3D ultrasound to prenatally diagnose physical malformations.Firstly, they introduced the retronasal triangle as a new first-trimester sonographic landmark, useful in the early screening for cleft palate (Sepulveda et al., 2010) and for the presence or absence of the nasal bones (Martinez-Ten et al., 2010).Then, they dealt with micrognathia diagnosis in the first trimester of pregnancy.Facial landmarks were here cited as identifiers for various face parts (Sepulveda et al., 2012a).Finally, they determined whether systematic examination of primary and secondary palates aided in the identification of orofacial clefts in the first trimester (Martinez-Ten et al., 2012).They also reviewed techniques, advantages, limitations, and clinical applications of 3D ultrasound and fetal MRI (Sepulveda et al., 2012b) and pointed out that anatomical landmarks had a key role in assessing fetus's normal structure (Sepulveda et al., 2012c).
Another work group, the one of Lituania and Tonni, dealt with prenatal diagnosis.Firstly, they reported on the application of OmniView, a new 3D sonographic software, and its applications in the prenatal sonographic study of the fetal hard and soft palate, recognizable by the two main anatomic landmarks, the uvula and velum.As positive outcome, OmniView allows visualization of all anatomic landmarks of this specific targeted area: labia, primary palate, alveolar ridge, posterior palate, uvula, velum, and tongue (Tonni and Lituania, 2012).As a sequel, the soft palate and especially the uvula were used to diagnose clinical features of Stickler syndrome, a rare connective tissue disorder (Lituania and Tonni, 2013).Persico et al. (2010) used 3D ultrasound to investigate the effect of deviations from the exact midsagittal plane on the measurement of nasal-bone length at 16-24 weeks' gestation.For each parasagittal and oblique view obtained, they looked for the presence or absence of all the sonographic landmarks used for identification of the midsagittal plane.The sonographic landmarks commonly used to examine the fetal profile and to measure nasal-bone length, including the nose, upper and lower lips, the maxilla and the chin, were also visible in parasagittal and oblique sections of the profile.Borrell et al. (2011) assessed fetal anatomy using three-dimensional volumes acquired at 11 to 13 weeks in 223 pregnancies to identify the appropriate sections for evaluation of ten fetal anatomy landmarks.A limitation of the study is that these body landmarks retrieved from the 3D volumes were not assessed prospectively on ultrasound, and thus comparison was not possible.Manganaro et al. (2011) evaluated cleft lip and palate through MRI and US, although not 3D.The main facial landmarks were measured and analyzed for each fetus (forehead, occiput, orbits, nose, lips, chin, mandible).
As can be seen from the previous literature, in many cases landmarks are manually extracted or even represent a marginal or a landing point of the method.In other words, "landmarking" is not undertaken in this field yet.As a reply, this work stands for a preliminary study on facial soft-tissue landmark positioning on fetal 3D face shells, i.e., point clouds representing faces, extracted from three-dimensional echographies.Landmark extraction is a common prestarting point for many applications involving faces.We focused on this phase to provide a tool for the subsequent steps of diagnosis.In particular, we have developed an algorithm for automatically extracting soft-tissue landmarks on babies' faces.The method employed to build up the algorithm totally relies on geometrical criteria.More in detail, descriptors taken from the Differential Geometry, such as mean and Gaussian curvatures, coefficients of the first and second fundamental forms, principal curvatures, and other geometrical descriptors such as the Shape and Curvedness Indexes, are used to identify landmark positions on faces.
The reason for this research is that, although echographies are going towards 3D, procedures and techniques for analysing the achieved three-dimensional information and supporting the diagnosis are not inparallel developed.As a matter of fact, 3D tools are mainly aimed at providing lifelike three-dimensional images of the babies, without concretely exploiting the real potentialities that a 3D tool could offer for supporting fetal diagnosis.

METHODS
A facial landmark is a point which all faces share and has a particular biological meaning.In this study we have considered soft-tissue landmarks, which lie on the skin and can be identified once the face of the subject is digitally reconstructed, using proper reverse 3D techniques.In human face up to fifty-nine softtissue landmarks could be collected, but in our study we have used only 13 landmarks, as shown in Fig. 1.We have chosen to work only with landmarks that lie in the central part of the face because the borders of the face are too much affected by noise and moreover it is difficult to have an echography of the whole face, with also the borders.

Fig. 1. Landmarks detected by our proposed method: PN-pronasal, SN-subnasal, ALA-alae, EN-endocanthions, N-nasion, IE-inner eyebrow, CH-chelion, LSlabrum superior, LI-labrum inferior. These landmarks form a pivotal set of core points for describing facial shapes and features.
Considering the morphological features of the face, in order to extract the landmarks it is necessary to employ a refining procedure that firstly identifies the region, collecting the significant points, then extracts the specific landmarks.Relying on different peculiarities of different facial regions where the landmarks are located, different combinations of the first, second and mixed derivatives, the Coefficients of the Fundamental Forms E, F, G, e, f and g, the curvatures K, H, k 1 and k 2 , and Shape and Curvedness Indexes S and C as descriptors have been employed.A short description of their meanings is reported in the Appendix.
The localization and extraction processes used in the algorithm to detect the landmarks are explained and graphically represented.

PRONASAL
The pronasal (PN) is the point on the nose tip.It is surely the point most easily identifiable by human eye, especially because it is the most salient, when the face is well oriented.These are its most noticeable geometrical features: 1. it has high values of the Shape Index (S > 0.55); 2. in our reference is an absolute maximum, so it is a critical point (its first derivatives with respect to x and y are approximately equal to zero); 3. the principal curvature k 2 has a local (absolute, most of the times) maximum in it; 4. the Gaussian curvature K has a local maximum in it.
Actually our algorithm indentifies an area of interest through conditions 1 and 2 (in particular, the condition that the first derivative with respect to x is approximately equal to zero), then it extracts the landmark using condition 3, namely maximizing k 2 .The steps of the process are explained in the scheme below.

SUBNASAL
The subnasal (SN) is the point which lies exactly below the nose, in that little dimple above the mouth.These are its geometrical features: 1. it is a critical point (its first derivatives with respect to x and y are approximately equal to zero); 2. the coefficient f is close to zero and f(x,y+Δy) < 0, f(x,y-Δy) < 0; 3. the coefficient e has a local maximum in it; 4. the coefficient g has a local minimum in it.
In order to extract this landmark, the algorithm first identifies a region in the neighborhood of the pronasal, then it uses condition 2 to narrow the area, then it extracts the landmark with condition 4, namely minimizing g.The steps of the process are explained in the scheme below.

ALAE
The alae (AL) are the two points which lie on the left and the right of the widest part of the nose.Their geometrical features are: 1. they belong to the points whose Shape Index lies in the range corresponding to the surface of ridge (S Î [0.325,0.625]); 2. the derivative of z with respect of D x is positive on the right ala (from an external point of view) and negative on the left; 3. the coefficient e is positive; 4. the coefficient E has two local maximums in them.
The elaborated algorithm identifies two areas of interest using conditions 1 and 2, then it extracts the two landmarks using condition 4. The steps of the process are explained in the scheme below.

ENDOCANTHION
The endocanthions (EN) are the two points at which the inner ends of the upper and lower eyelid meet.These are their geometrical features: 1. they belong to the points whose Shape Index lies in the range corresponding to the surface of cup or rut; 2. in our reference system, they are local minimum, so they are critical points; Our algorithm identifies two areas of interest using conditions 1, 3, 5, 6, then it extracts the landmarks with condition 8.The steps of the process are explained in the scheme below.

NASION
The nasion (N) is a point at the top of the nose, nearly between the eyes.In the horizontal direction, it lies on the high part of the nose bone; in the vertical direction it is in the hollow under the forehead.These are its geometrical features: 1. it belongs to the points whose Shape Index lies in the range corresponding to the surface of ridge, saddle ridge, saddle point or saddle rut; 2. it is a critical point (its first derivatives with respect to x and y are approximately equal to zero); 3. the mean curvature H is close to 0 (H Î (-0.5, 0.5)); 4. the coefficient f is nearly 0 (f Î (-0.1,0.1)); 5. the coefficient g has a local minimum; 6. the Gaussian curvature K is negative in it; 7. the principal curvature k 1 has a local maximum.
In order to localize the landmark, the algorithm starts selecting a region of interest starting from the two endocanthions and the pronasal.Then it uses conditions 1 and 2 to narrow the area and finally it uses condition 7 to extract the landmark.The steps of the process are explained in the scheme below.

INNER EYEBROWS
The inner eyebrows (IE) are that points which lie at the connection between the nose bone and the eyebrows themselves.These are their geometrical features: 1. it belongs to the points whose Shape Index lies in the range corresponding to the surface of ridge or dome (S > 0.5); 2. the coefficient g is strictly negative in them; 3. the coefficient f is a local maximum in right IE and a local minimum in the left IE.
The elaborated algorithm firstly indentifies two regions of interest starting from the endocanthions, then it narrows the areas using condition 1; finally it extracts the landmarks using condition 3. The steps of the process are explained in the scheme below.

LABRUM SUPERIOR
The labrum superior (LS) is the point which lies on the upper lip of the mouth approximately in the middle, on that little bump under the subnasal.The labrum superior is the most evident landmark of the mouth, so it is the first localized.These are its geometrical features: 1. it belongs to the points whose Shape Index lies in the range corresponding to the surface of the dome; 2. the coefficient g has a local minimum in it; 3. the curvedness index C has a local maximum in it; 4. the Gaussian curvature K has a local maximum in it.
In order to extract the landmark, the process selects a region of interest starting from the subnasal, then it uses condition 1 to narrow the area and finally, with condition 3, it extracts the landmarks.The steps of the process are explained in the scheme below.

LABRUM INFERIOR
The labrum inferior (LI) is the point which lies on the lower lip of the mouth, approximately in the middle, closer the hollow located above the chin.Its geometrical features are the same of the labrum superior: 1. it belongs to the points whose Shape Index lies in the range corresponding to the surface of the dome; 2. the curvedness index C has a local maximum in it; 3. the coefficient g has a local minimum in it; 4. the Gaussian curvature K has a local maximum in it.
In order to localize the LI, the algorithm identifies a region of interest using the coordinates of the labrum superior and of the subnasal.Then, the pro-

EXPERIMENTAL VALIDATION
cess to extract the landmark is equal to the one of the labrum superior.The steps of the process are explained in the scheme below.
From September 2012 to January 2013, 30 threedimensional volumes of 30 fetuses at 22-32 weeks' gestation were acquired.Written informed consent was obtained from the parents for publication of clinical details, clinical images, and videos.Principles outlined in the Declaration of Helsinki have been followed.

CHELIONS
The cheilions are the two points on the outer corners of the mouth, where the outer ends of the upper and lower lips meet.The geometrical features that we have indentified are: 1. it belongs to the points whose Shape Index lies in the range corresponding to the surface of rut or cup; Among these acquisitions, 9 were selected and processed for the purposes of the study.The remaining ones were excluded because of high noise or practical "inconveniences" such as baby's hands on face, or simply too much inaccuracy in the scanning process.
2. the coefficient f is close to 0 in them (f Î (-0.1, 0.1)); 3. the absolute value of the coefficient e is maximum in them; 4. the second derivative D xy has a local maximum in the right chelion and a local minimum in the left one.
The process uses the coordinates of the labrum superior and inferior and of the two alae to identify a region of interest.Then it uses condition 1 to narrow the area and it extracts the landmark using condition 4.
The ultrasound equipment was a Voluson system (GE Healthcare, Wauwatosa, WI, USA), with a RAB 4-8 (real time 4D convex transducer probe).The GE RAB 4-8 has a frequency range of 4 to 8 MHz and is used for OB applications (Footprint 63.6×37.8mm, FOV 70°, V 85°×70°).In Table 1 the used scan settings for each 3D static acquisition and the fetal age of each baby are shown.
Using 4D VIEW software, it is possible to see the images acquired with the Voluson System on three orthogonal planes, i.e., axial, sagittal, and coronal (Fig. 2).The plane chosen for the facial shell modeling is the midsagittal (Fig. 3).
The distance between two successive slices is 0.4 mm.For each slice composing the whole volume, the relative DICOM format file is created and stored.The final store is exported into Simpleware ScanIP software for the 3D model reconstruction.Face data were collected in points clouds, then these shells were imported in Matlab® and triangulated.The triangular mesh was then converted into a square grid, onto which a Matlab® algorithm for facial landmarking based on the theoretical foundations of the previous section was elaborated, implemented, and run.
In the following paragraphs the results of the landmark extraction procedures will be presented using one face as sample.

PRONASAL
The area of extraction of the pronasal is firstly reduced by selecting all the points whose Shape Index is greater than the threshold value 0.5 (Fig. 4).
Then, among these points, the points with D x Î [-u, u] were selected, where u is the instrument uncertainty (Fig. 5).Finally, the algorithm extracts the landmark maximizing the k 2 (Fig. 6).Fig. 4. On the left.Graphical representation of the Shape Index.On the right.The narrowing of the area of interest (in red or yellow) by choosing only the points whose Shape Index is greater than 0.5.[-u, u], where u is the instrument uncertainty.

SUBNASAL
The first step of the process of extraction of the subnasal is defining a region of interest using the pronasal already extracted (Fig. 7).Then, this region is reduced using the conditions on f (Fig. 8).At the end, the landmark is extracted minimizing g (Fig. 9).

ALAE
The two alae were localized using different geometrical descriptors: two conditions are firstly applied using the index S and the first derivative D x , then the landmarks are extracted maximizing E. The condition on the derivative is only used to distinguish between the left and right part of the face, so the algorithm can distinguish between right ala and left ala.In Figs.10-12 are shown the different steps of the process.

ENDOCANTHION
The localization of endocathions was performed firstly selecting the points whose Shape Index lies in the range corresponding to the surfaces of cup and rut (Fig. 13), then the algorithm uses the conditions on the second derivatives (Figs. 14 and 15) and on the coefficient F (Fig. 16) to narrow the areas, as previously explained.At the end, the landmarks are extracted maximizing the coefficient e (Fig. 17).The condition on the coefficient F is only used to distinguish between right and left endocanthion.

NASION
In order to localize the nasion, the algorithm firstly identifies a region of interest using the pronasal and the endocanthions, already extracted.Then, it select the points whose Shape Index SÎ [-0.375,0.625]and that are critical points; finally it maximize the principal curvature k 1 to extract the landmark.The behavior of the principal curvature is not so evident because the local maximum is not very high in comparison with the other maximums.In Figs.18-21 are shown the steps of the process.

INNER EYEBROW
Firstly, the process identifies two regions of interest using the coordinates of the endocanthions (Fig. 22), then the areas are narrowed with the condition on the Shape Index (Fig. 23).Finally, the right IE is extracted maximizing the coefficient f, instead the left IE is extracted minimizing f (Fig. 24).

LABRUM SUPERIOR
The process of extraction of the labrum superior is very simple, using the subnasal it identifies a region of interest (Fig. 25), then it narrows the area using the Shape Index (Fig. 26) and it extracts the landmark maximizing the curvedness index C (Fig. 27).

LABRUM INFERIOR
The first step of the process of extraction of the labrum inferior consists in identifying a region of interest using the subnasal and the labrum superior (Fig. 28).Then, the process is equal to the one of the labrum superior (Figs. 29 and 30).

CHELION
The extraction of the chelions is not easy, because most of the geometric descriptors does not have a significant behavior in these points and the discriminating features that we have identified were not so evident in all the scans.So, the algorithm firstly identifies two regions of interest using the coordinates of the labrum superior and inferior and of the alae (Fig. 31).Then, it narrows the area using the condition on the Shape Index (Fig. 32) and it extracts the landmarks maximizing the second derivative D xy for the right chelion and minimizing D xy for the left one (Fig. 33).

RESULTS
The 13 resulting landmarks of the nine shells studied are shown in Fig. 34.
In parallel with our extraction, it was asked to four practitioners to take the shells into exam and determine where the true landmarks were located.A set of coordinates of "real landmarks" has been collected for each physician and, although the indicated coor-dinates were most of the times the same, a mean has been computed to state the final real ones.
The coordinates of landmarks extracted with our algorithm were compared with the real points through a brief statistical study.Euclidian distances between the correct landmarks and the respective points given by our algorithm were computed.The values of these distances are shown in Table 2, where the unit of measurement is the millimetre.
where N is the number of landmarks used.The results are shown in Table 3.
As can be seen from Tables 2 and 3, all errors lie in the range between 0 and 3.5 mm and the mean distance for each shell is in the range between 0.6 and 1.6 mm.It is important to underline that the quality of the results does not depend on the number of weeks of the fetus.For instance, Gio and Gian are two fetuses of 22 weeks, but Gio is the one with the best results while, on the contrary, Gian has one of the highest mean error values.Instead, results are highly depending on echography quality.Lisa, which shows the worst result, has a bad quality because the baby had a hand next to the face, so it was difficult to extract a clear image of the whole face.
In order to check whether some landmarks were more subject to errors, mean and variance were computed also for every landmark.The values are given in Table 4, while the trends of mean and variance were graphically represented in Fig. 35.The localization seems to be quite accurate.The values show that the position of the pronasal is the most accurate with a mean distance of 0.55 mm.Instead, the least precise landmarks belong to the mouth region, in particular the two chelions have the highest values, with a mean distance of 1.82 mm for the right one and 1.48 mm for the left one.As explained in the section relating to chelions in section 3, the extraction of the two chelions is not trivial, moreover often in these areas the mouth is not well defined.

CONCLUSION
In this work we have implemented a landmarking algorithm for extracting facial landmarks from nine fetuses at 22-32 weeks' gestation obtained through 3D ultrasound.The attempt was to give a further utility to 3D ultrasound, that nowadays mainly has the role of providing future mothers with a threedimensional rendered image of their babies.Although some work has been undertaken on this tool, no automatic landmarking algorithms for diagnostic purposes have been done yet.This study tries to answer to this lack.
The method designed to elaborate the algorithm relies on the behaviour that some geometrical descriptors have on faces, when they are computed pointby-point on facial shells.The results obtained, validated with the support of four practitioners, show that the localization is quite accurate.All errors lie in the range between 0 and 3.5 mm and the mean distance for each shell is in the range between 0.6 and 1.6 mm.The landmarks showing the highest errors are the ones belonging to the mouth region; in particular the two chelions are the least precise with a mean distance of 1.82 mm for the right one and 1.48 mm for the left one.Instead, the most precise landmark is the pronasal with a mean distance of 0.55 mm.

APPENDIX
The First and Second Fundamental Forms are used to measure distance on surfaces and are defined by , , 2 2  Some definitions of these descriptors are given.These are the forms implemented in the algorithm: For the role they play in the work, a little digression about their significance is needed.Their meaning is shown in Figs.36-38 and in Table 5.
3. the second derivative of z with respect to x D xx is negative in them; 4. the second derivative of z with respect to y D yy is negative in them; 5. the second derivative of z D xy is nearly 0 in them (D xy Î [-0.1,0.1]); 6. the coefficient F is negative on the right endocanthion and positive on the left; 7. the coefficient f is negative on the right endocanthion and positive on the left; 8. the coefficient e has a local minimum in them.

Fig. 6 .
Fig. 6.On the left.Graphical representation of the principal curvature k 2 .On the right.The extraction of the landmark maximizing k 2 .

Fig. 7 .
Fig. 7. Region of interest for the extraction of the subnasal.

Fig. 8 .
Fig. 8. On the left.Graphical representation of the coefficient f.On the right.The narrowing of the area of interest by choosing only the points that satisfy the conditions on f.

Fig. 9 .
Fig. 9. On the left.Graphical representation of the coefficient g.On the right.The extraction of the landmark minimizing g.

Fig. 10 .
Fig. 10.On the left.Graphical representation of the Shape Index.On the right.The narrowing of the area of interest (in red or yellow) by choosing only the points whose Shape Index lies in the range corresponding to the surface of rut.

Fig. 11 .
Fig. 11.On the left.Graphical representation of the derivative of z with respect to x.On the right.The narrowing of the area of interest (in red and yellow) by choosing, among the points selected in the previous step, only the ones whose derivative of z with respect to x is positive on the right side of the face and negative on the left side.

Fig. 12 .
Fig. 12.On the left.Graphical representation of the coefficient E. On the right.The extraction of the landmarks maximizing E.

Fig. 13 .
Fig. 13.On the left.Graphical representation of the Shape Index.On the right.The narrowing of the area of interest (in red) by choosing only the points whose Shape Index lies in the range corresponding to the surface of cup.

Fig. 14 .
Fig. 14.On the left.Graphical representation of the second derivative of z with respect to x.On the right.The narrowing of the area of interest (in red) by choosing, among the points selected in the previous step, only the ones whose second derivative of z with respect to x is positive.

Fig. 15 .
Fig. 15.On the left.Graphical representation of the second derivative Dxy.On the right.The narrowing of the area of interest (in red) by choosing, among the points selected in the previous step, only the ones that satisfy the condition on the second derivative Dxy.

Fig. 16 .
Fig. 16.On the left.Graphical representation of the coefficient F. On the right.The narrowing of the area of interest (in red) by choosing, among the points selected in the previous step, only the ones whose F is positive on the right side of the face and negative on the left side.

Fig. 17 .
Fig. 17.On the left.Graphical representation of the coefficient e.On the right.The extraction of the landmarks minimizing e.

Fig. 18 .
Fig. 18.Region of interest for the extraction of the nasion.

Fig. 19 .
Fig. 19.On the left.Graphical representation of the Shape Index.On the right.The narrowing of the area of interest (in red) by choosing only the points whose Shape Index lies in the range corresponding to the surface of ridge, saddle ridge, saddle point or saddle rut.

Fig. 20 .
Fig. 20.On the left.Critical points, in red.On the right.The narrowing of the area of interest (in red or yellow) by choosing, among the points selected in the previous step, only the critical ones.

Fig. 21 .
Fig. 21.On the left.Graphical representation of k1.On the right.The extraction of the landmark maximizing k1.

Fig. 22 .
Fig. 22. Regions of interest for the extraction of the inner eyebrows.

Fig. 23 .
Fig. 23.On the left.Graphical representation of the Shape Index.On the right.The narrowing of the area of interest (in red) by choosing only the points whose Shape Index is greater than 0.5.

Fig. 25 .
Fig. 25.Region of interest for the extraction of the labrum superior.

Fig. 26 .
Fig. 26.On the left.Graphical representation of the Shape Index.On the right.The narrowing of the area of interest (in red) by choosing only the points whose Shape Index lies in the range corresponding to the surface of dome.

Fig. 27 .
Fig. 27.On the left.Graphical representation of the curvedness index C.On the right.The extraction of the landmark maximizing C.

Fig. 28 .
Fig. 28.Region of interest for the extraction of the labrum inferior.

Fig. 29 .
Fig. 29.On the left.Graphical representation of the Shape Index.On the right.The narrowing of the area of interest (in red) by choosing only the points whose Shape Index lies in the range corresponding to the surface of dome.

Fig. 30 .
Fig. 30.On the left.Graphical representation of the curvedness index C.On the right.The extraction of the landmark maximizing C.

Fig. 31 .
Fig. 31.Regions of interest for the extraction of the chelions.

Fig. 32 .
Fig. 32.On the left.Graphical representation of the Shape Index.On the right.The narrowing of the area of interest (in red) by choosing only the points whose Shape Index lies in the range corresponding to the surface of rut or cup.

Fig. 33 .
Fig. 33.On the left.Graphical representation of Dxy.On the right.The extraction of the landmarks maximizing Dxy for the right CH and minimizing Dxy for the left CH.

Fig. 35 .
Fig. 35.Graphical representation of values of sample mean and sample variance computed between the 9 shells for the 13 landmarks.
E, F, G, e, f and g are their Coefficients.Curvatures are used to measure how a regular surface x bends in .If D is the differential and N is the normal plane of a surface, then the determinant of DN is the product (-k 1 )(-k 2 ) = k 1 k 2 of the Principal Curvatures, and the trace of DN is the negative -(k 1 + k 2 ) of the sum of Principal Curvatures.In point P, the determinant of is the Gaussian Curvature K of x at P. The negative of half of the trace of DN is called the Mean Curvature H of x at P. In terms of the principal curvatures can be written 3  P DN where h is a differentiable function .It is, therefore, convenient to have at hand formulas for the relevant concepts in this case.To obtain such formulas let us parametrize the surface by u = x, v = y.The most used descriptors are surely the Shape and Curvedness Indexes S and C, introduced by Koenderink et al. (1992):

Fig. 38 .
Fig. 38.Indexes (S,C) are viewed as polar coordinates in the (k 1 , k 2 )-plane, with planar points mapped to the origin.The effects on surface structure from variations in the curvedness (radial coordinate) and Shape Index (angular coordinate) parameters of curvature, and the relation of these components to the principal curvatures (k 1 and k 2 ).The degree of curvature increases radially from the centre.

Table 1 .
Weeks' gestation and scan settings for each baby.

Table 2 .
Numerical values of the distances between the correct landmarks and the landmarks obtained with the algorithm for the 9 shells.Then, for each shell, sample mean E and sample variance σ of these distances d i were calculated:

Table 3 .
Values of sample mean and sample variance computed between the 13 landmarks of each shell.

Table 4 .
Values of sample mean and sample variance computed between the 9 shells for the 13 landmarks.