Estimation of intrinsic volumes from digital grey-scale images

Estimation of in trinsic v olumes from digital grey-scale images Anne Marie Sv ane 1 1 Departmen t of Mathematical Sciences, Aarh us Univ ersity , amsv ane@imf.au.dk Abstract Lo cal algorithms are common to ols for estimating in trinsic v olumes from blac k- and-white digital images. Ho wev er, these algorithms are t y pically biased in the design based setting, ev en when the resolution tends to infinity . Moreo v er, im- ages recorded in practice are most often blurred grey-scale images rather than blac k -and-white. In this paper, an extended definition of lo cal algorith ms, ap- plying directly to grey-scale images without thresholding, is suggested. W e in v estigate the asymptotics of these new algorithms when th e resolution tends to infinit y and apply this to construct estimators f or surface area and inte- grated mean curv ature that are asymptotically unb iased in certain natural settings. 1 In tro duction In this pap er, we shall in v estigate the class of so-called local algo r ithms [6, 15] used for estimation of surface area and in tegrated mean curv ature from digital images. These algorithms are commonly used in applied sciences for analysing digital output data from e.g. microscopes and scanners, see [6, 8 , 10]. The main reason for the popu- larit y of lo cal alg o rithms is that they allow simple linear time implemen tations [11]. Ho w ev er, this efficiency is us ually paid for by a lac k of accuracy [4, 15]. Lo cal algorithms ha v e so far only b een defined for blac k-and-white images, see e.g. [14]. In a black-and-white image of a geometric ob ject X ⊆ R d , eac h pixel is coloured blac k if the midp oint lies in X and white otherwis e. The use of lo cal algorithms th us requires that w e are able to measure precis ely whether or not a giv en p oin t b elongs to X . In practice, how ev er, suc h exact measuremen ts are t ypically not p ossible, since the ligh t coming fro m eac h p oint is spread out followin g a p oint spread function (PSF). The result is a grey-scale image where each pixel is assigned a grey- tone corresponding to a measured ligh t in tensit y betw een 0 and 1. The standard w ay of o v ercoming this problem is to con v ert the image to blac k-and-white by choosing a t hreshold limit β ∈ [0 , 1) and con v erting all pixels with grey-v alue greater than β to blac k and all others to white . As a natural w a y o f assess ing a local algorithm, w e test it in t he design based setting where the ob ject under study has b een ra ndomly translated with resp ect 1 to the o bserv er b efore the image is recorded. Ideally , the estimator should be unb i- ased, at leas t asy mptotically when the resolution tends to infinity . F o r blac k-and- white images, surface area and integrated mean curv ature can generally not b e estimated without an asymptotic bias unless it coincides with the Euler characteris- tic [4, 15, 1 6 ]. T o make the asymptotic beha viour of grey-scale images we ll-defined it is necess ary to mak e some ass umptions on ho w the PSF c hanges with increased resolution. In this pap er, w e will assume that measuremen ts b ecome more accurate with increased resolution, see the precise a ssumption and a discussion of this in Subsection 2.1 b elow . Most previous asymptotic studies for black -and-white images do not take the thresholding pro cess in to accoun t. F or volume estimators, some first studies for sim- ple PSF’s w ere p erfo r med in [2, 5] and in [13], estimators for the Euler c haracteristic are studied in 2D. The first purp o se of this pa p er is to study the effect of thresholding on lo cal alg o rithms. On the other hand, most of the information hidden in the grey-v alues is thro wn a w a y in the thresholdin g pro cess. The se cond purp ose of this pap er is to giv e an extended de finition of lo cal algorithms, see Subsection 2.3, that exp loits the a v ail- able information in the grey-scale image b etter but still leads to fast computations. Finally w e are going to study the asymptotic bias of these estimators. 1.1 Results and applicatio ns The asymptotic studies of lo cal algorithms will b e based on three theoretical formul as extending [5, Theorem 2]. In Section 3 this theorem is extende d t o larger classes of PSF’s and in Section 5 it is extended to a second order form ula. The tec hniques in v o lv ed in the pro ofs are similar to those used in [5] and [14]. F rom the theoretical fo rmulas we obtain some applications to surface area es- timation in Section 4, and in Section 6 w e apply the results to es timators for the in tegrated mean curv ature and the first o r der bias of surface area estimators in finite high resolution. W e summarize some of the main findings here. Assuming only mild cond itions on the PSF, w e first consider surface area esti- mators applied to the class of so-called gen t le sets, se e Definition 3.1 b elow. This class include s for instance a ll manifolds and all p olycon v ex sets . F or blac k-and-white lo cal algorithms applied to thresh olded images, w e find that the asymptotic bias is the same as for blac k-and-white images, se e Subsection 4.1. In particular, lo cal al- gorithms applied to thresholded images are not asymptotically un biased. In con trast to this, Subsection 4.2 sho ws that if the PSF is rota t io n inv ariant, asymptotically un biased estimators based directly on the grey-v alues are plen ty . As a simple example, the estimator coun ting the n um b er o f grey-v alues in some in t erv al I ⊆ (0 , 1) has the correct asymptotic mean up t o a constan t factor. Moreo ve r, if I is symmetric around 1 2 , the first order bias in high resolution v anishes. This algorithm is clearly bot h simple a nd fast and it can ev en be applied if the grey-v alues in the output data are only given discretely . F or mor e general classes of PSF’s we obtain a description of the w or st case asymp- totic error. Thi s could b e used to searc h for estim ators minimizing the asymptotic bias. 2 With stronger assum ptions on both the PSF and the smo othness of the boun- dary of the underlying s et X , w e find in Subsection 6.3 that if the PSF is rota tion in v arian t, also asymptotically un biased estimators for the integrated mean curv ature do ex ist. One example is giv en b y the estimator that coun ts the num b er of grey- v alues in the in terv al ( β , 1 2 ) and subtracts the n um b er of grey-v alues in ( 1 2 , 1 − β ) for a suitable β ∈ (0 , 1 2 ) . F or thresholded images, the asymptotic mean is a little more complicated than in the black -and-white case, no w dep ending on the PSF, see Subsection 6.1. All results of this pap er are the oretical. The practical usefulness of lo cal algo- rithms f o r g rey-scale images is discussed in the final Section 7. 2 Lo cal digital algorithms In this section w e in tro duce lo cal digital estimators f o r the surface area 2 V d − 1 ( X ) and in tegrated mean curv ature 2 π ( d − 1) − 1 V d − 2 ( X ) of a ‘sufficien tly nice’ set X ⊆ R d . Both quan tities are examples of the so-called intrinsic v olumes V q , q = 0 , . . . , d , hence w e shall us e this term when referring to both, see e.g. [12]. 2.1 Mo dels for digital images Let L b e the lattice in R d spanned by the ordered basis v 1 , . . . , v d ∈ R d and let C v = L d i =1 [0 , v i ] be the fundamen tal cell of the lattice. As w e shall later b e scaling the lattice, w e ma y a s w ell assume that the v olume det( v 1 , . . . , v d ) of C v is 1. F or c ∈ R d , we let L c = L + c denote the translated lattice. W e shall think of the pixels in a digital image as the translations of C v that ha v e midp oin ts in L c . Let X ⊆ R d b e a geometric ob ject. The information ab out X hidden in a black -and-white image corres p onds to the set o f black pixel midpoints X ∩ L c . F or grey-scale images, w e a ssume that the light coming from eac h p oin t is spread out follo wing a p oin t spread function whic h is indep enden t of the position of the p oin t. That is, the ligh t that reac hes the observ er is given b y the in tensit y function θ X,ρ : R d → [0 , 1] where the in tensit y measu red at x ∈ R d is giv en b y θ X,ρ ( x ) = Z X ρ ( z − x ) dz . Here ρ is the PSF, whic h is assu med to be a measurable function satis fying ρ ≥ 0 and R R d ρd H d = 1 where H d is the d - dimensional Hausdorff measure. A digital image in the grey-scale setting is the restriction o f θ X,ρ to the observ ation lattice L c . A simple example of a PSF is ρ B = H d ( B ) − 1 1 B where B ⊆ R d is a Borel set of non-zero finite v olume and 1 B denotes the indicator function. At ev ery p o in t x ∈ R d , θ X,ρ B ( x ) measures the v olume of ( x + B ) ∩ X . F or instance, if B equals C 0 = C v − 1 2 P d i =1 v i , the gr ey-v alue of a pixel measures the fraction of the pixel that is containe d in X . This PSF is studied thoroughly in 2D in [2]. Another intere sting 3 example is when B is the ball B ( R ) of radius R > 0 . It may a lso b e relev ant to consid er v arious con tin uous appro ximations to these caused b y imprecisions of measuremen ts near the bo undary o f B . In other applic ations, it is more relev an t to consider a PSF with non-compact suppo rt. The main example to hav e in mind is the Gaussian ρ Gauss ( x ) = 1 ( √ 2 π σ ) d exp  − | x | 2 2 σ 2  whic h yields a go o d a ppro ximation of most PSF’s o ccurring in practice [7]. W e sa y that a PSF is reflection inv a rian t if ρ ( x ) = ρ ( − x ) and rota tion in v arian t if ρ ( x ) dep ends only on | x | . F or instance, ρ B ( R ) and ρ Gauss are rotation in v arian t, while ρ C 0 is only reflection in v arian t. A c hange of resolution to a − 1 for some a > 0 corresp onds to a c hange of lattice from L to a L . W e assume that the precision of the measu remen ts c hanges in suc h a w a y that t he PSF in resolution a − 1 is ρ a ( x ) = a − d ρ ( a − 1 x ) . The corresp onding in tensit y function is denoted θ X,ρ a ( x ) = Z X ρ a ( z − x ) dz = a − d Z X ρ ( a − 1 ( z − x )) dz . The PSF is omitted f r o m the notation when ev er it is clear fro m the con text. In some applications, e.g. for ρ B or in some cases where the blurring is caused by the optical device, this transformation of ρ with the resolution is natural. F or ρ C 0 it simply means that pixels b ecome sm aller in higher resolution. In other situations, e.g. if the ligh t is spre ad out b efore it reac hes the lense, it ma y be imp o ssible for the observ er to affect the blurring or a differen t transformation is more realistic. Ho w ev er, in this paper w e restrict ourselv es to the ab o v e setting. 2.2 Lo cal al gorithms for blac k-and-white ima ges W e first recall the definition o f lo cal algorithms in the case of blac k-a nd-white images, see e.g. [15, Section 2] for more details and justifications o f suc h algorithms. An n × · · · × n latt ice cell is a set of the form C n z = ( z + L d i =1 [0 , nv i )) where z ∈ L . The set of lattice p oints lying in suc h a cell is denoted b y C n z , 0 = C n z ∩ L . An n × · · · × n configuration is a pair ( B , W ) where B , W ⊆ C n 0 , 0 are disjoin t with B ∪ W = C n 0 , 0 . W e index these b y ( B l , W l ) for l = 0 , . . . , 2 n d − 1 where B 0 = W 2 n d − 1 = ∅ . A lo cal digital algorithm in the sense of [15] estimates V q b y a w eigh ted sum of configuration counts : Definition 2.1. A lo c al digital estimator ˆ V q for V q b ase d on the image X ∩ a L c is an estimator of the form ˆ V a L c q ( X ) = a q 2 n d − 1 X l =1 w l N a L c l ( X ) 4 wher e N a L c l ( X ) = X z ∈ a L c 1 { z + aB l ⊆ X } 1 { z + aW l ⊆ R d \ X } is the total numb e r of o c curr enc es of the c onfigur ation ( B l , W l ) in the imag e and the c onstants w l ∈ R ar e c al le d the we i g hts. Note that w e ass ume the weigh ts to be homogeneous in the sense of [15]. In all the algorithms used in pra ctice, the w eigh ts are homogeneous. Supp ose a grey-scale digital image is thres holded at lev el β ∈ [0 , 1) . This means that the set of blac k lattice p oints is no w { z ∈ a L c | θ X a ( z ) > β } . Replacing X ∩ a L c b y this set in Definition 2.1, the resulting estimator b ecomes ˆ V ( β ) a L c q ( X ) = a q 2 n d − 1 X l =1 w l N ( β ) a L c l ( X ) (2.1) where N ( β ) a L c l ( X ) = X z ∈ a L c 1 { θ X a ( z + aB l ) ⊆ ( β , 1] } 1 { θ X a ( z + aW l ) ⊆ [0 ,β ] } . 2.3 Lo cal al gorithms in the grey-scale setting W e no w suggest a more general definition of lo cal algorithms based directly on grey-scale images. An n × · · · × n configuratio n in the grey-scale setting is a p oint θ X a ( aC n z , 0 ) ∈ [0 , 1] n d . E ach configuration m ust th us be w eigh ted not b y a finite col- lection of we ights but b y a function f : [0 , 1] n d → R . This leads to the follow ing definition: Definition 2.2. A lo c al estimator ˆ V q for V q is an estimator of the form ˆ V ( f ) a L c q ( X ) = a q X z ∈ L c f ( θ X a ( aC n z , 0 )) wher e f : [0 , 1 ] n d → R is a Bor e l function. T o ensur e fi n iteness of estimators o n c omp act sets, we assume that f (0) = 0 and, if ρ has non-c om p act supp ort, w e also assume supp f ⊆ (0 , 1] . T o ensur e inte gr ability of z 7→ f ◦ θ X a ( aC n z , 0 ) when X i s c omp act, w e mor e over assume that f i s b ounde d. In the definition, one could of course let f dep end on b oth lattice distance a > 0 and p osition z ∈ R d , but due to the homogeneit y and translation inv a riance of in trinsic v olumes, w e restrict ourselv es to the es timators in Definition 2.2. Algorithms of this ty p e ha v e already b een considered in [2] and [9]. Also (2 .1) is a special case of this, but Definition 2.2 allo ws a m uc h more refined use of the grey-v alues. Apart from (2.1), w e shall mainly consider estimators with n = 1 , correspo nding to estimators of the form ˆ V ( f ) a L c q ( X ) = a q X z ∈ L c f ( θ X a ( az )) (2.2) where f : [0 , 1] → R is as in Definition 2.2. 5 2.4 Con v ergence i n the design based setting W e shall in v estigate lo cal algorithms for digital grey-scale images in the design based setting. This may b e mo deled as the situation where X ⊆ R d is held fixed while the observ ation lattice L c is the translation of L by a uniform random translation v ector c ∈ C v . The mean of a lo cal estimator applied to a grey-scale imag e is then E ˆ V ( f ) a L c q ( X ) = a q E X z ∈ L c f ( θ X a ( aC n z , 0 )) = a q − d Z R d f ◦ θ X a ( z + aC n 0 , 0 ) H d ( dz ) . (2.3) As a natural w ay of asse ssing a local algorithm in the des ign based setting, w e study the mean estimator when the resolution go es to infinit y . Ideally , the algorithm w ould be asymptotically un biased, i.e. lim a → 0 E ˆ V ( f ) a L c q ( X ) = V q ( X ) for all sets X in some family S of subsets of R d . In the case of surface area estimation, w e more generally consider the w orst case asymptotic relative mean error as a measure for how we ll an algorithm work s: Err ( ˆ V ( f ) d − 1 ) = sup X ∈S     lim a → 0 E ˆ V ( f ) a L c d − 1 ( X ) − V d − 1 ( X ) V d − 1 ( X )     . 3 First order form ulas W e first deriv e some abstract form ulas, from whic h the first order asymptotic b e- ha viour of the in tegral in (2.3) can b e determined. These extend the formula b y Kiderlen and Rata j give n in [5, Theorem 2] for PSF’s of the form ρ B to some larger classes of PSF’s. Though only considered in their paper as a cor rection term to v ol- ume estimators, the formu las ha v e applications to surface a rea estimation as w ell. W e first in tro duce some notation and state their results in the language of the presen t pap er. F or a closed set X ⊆ R d , w e let exo ( X ) denote the p oints in R d not having a unique nearest p oin t in X . Let ξ X : R d \ exo ( X ) → X b e the natural pro jection taking a p oint in R d \ exo ( X ) to its unique neares t p oint in X . W e define the normal bundle of X to be the set N ( X ) =  x, z − x | z − x |  ∈ X × S d − 1   z ∈ R d \ ( X ∪ exo ( X ) ) , ξ X ( z ) = x  . F or ( x, n ) ∈ N ( X ) w e define the reac h δ ( X ; x, n ) = inf { t ≥ 0 | x + tn ∈ exo ( X ) } ∈ (0 , ∞ ] . F ollowin g [5], w e in tro duce the class of gen tle sets : Definition 3.1. A close d set X ⊆ R d is c al le d gen tle if (i) H d − 1 ( N ( ∂ X ) ∩ ( B × S d − 1 )) < ∞ for any b ounde d Bor el set B ⊆ R d . 6 (ii) F or H d − 1 –almost al l x ∈ ∂ X ther e exist two b al ls B in , B out ⊆ R d b oth c ontain- ing x and such that B in ⊆ X , int( B out ) ⊆ R d \ X . The condition (ii) in the definition means that for almost all x ∈ ∂ X there is a unique pair ( x, n ( x )) ∈ N ( X ) with ( x, n ( x )) , ( x, − n ( x )) ∈ N ( ∂ X ) . F or n ∈ S d − 1 , w e let H − t,n denote the halfspace { x ∈ R d | h x, n i ≤ t } . F or short w e sometimes write H n = H − 0 ,n . F or a set S ⊆ R d , h ( S, n ) = sup {h s, n i | s ∈ S } denotes the suppor t function and ˇ S = {− s | s ∈ S } . Finally , ⊕ denotes Mink o wski addition and for t ∈ R , w e use the notation t + = t ∨ 0 = max { t, 0 } . W e are now ready to state the first order for mula [5, Theorem 2]: Theorem 3.2 (Kiderlen, Rata j) . L et X ⊆ R d b e a clo s e d gentle set and A ⊆ R d b e b ounde d me asur able. L et β , ω ∈ (0 , 1 ] and B , W, P , Q ⊆ R d non-empty c omp a ct with H d ( P ) , H d ( Q ) > 0 . Then lim a → 0 a − 1 Z ξ − 1 ∂ X ( A ) 1  θ X,ρ P a ( x + aB ) ⊆ [ β , 1]  1  θ X,ρ Q a ( x + aW ) ⊆ [0 ,ω )  dx = Z ∂ X ∩ A ( ˜ ϕ ρ P ( β , n ) − ˜ ϕ ρ Q ( ω , n ) − h ( B ⊕ ˇ W , n )) + d H d − 1 wher e ˜ ϕ ρ ( β , n ) = sup { t ∈ R | θ H n ,ρ ( tn ) ≥ β } for n ∈ S d − 1 and β ∈ (0 , 1] . Observ e in the definition o f ˜ ϕ ρ that the contin uous function t 7→ θ H n ,ρ ( tn ) = θ H − − t,n ,ρ (0) = θ H n ,ρ a ( atn ) is dec reasing so that ˜ ϕ ρ ( β , n ) is finite decreasing for β ∈ (0 , 1] . 3.1 The case of compact supp ort W e first g eneralize Theorem 3.2 to PSF’s that are almost ev erywhere b ounded and compactly suppo r ted. Note how the op en and closed ends of the inte rv als hav e b een switc hed in the statemen t of the theorem. F or this reason, the functions ˜ ϕ are replaced by ϕ ρ ( β , n ) = inf { t ∈ R | θ H n ,ρ ( tn ) ≤ β } = sup { t ∈ R | θ H n ,ρ ( tn ) > β } . Theorem 3.3. L et X ⊆ R d b e a close d gentle set and A ⊆ R d b ounde d me asur able. L et I and J b e non-empty finite ind ex sets. F or i ∈ I a n d j ∈ J , let β i , ω j ∈ [0 , 1) , B i , W j ⊆ R d b e non-empty c o m p act, and ρ i , ρ j b e almos t everywher e b ounde d PSF’ s with c omp act supp ort. Then lim a → 0 a − 1 Z ξ − 1 ∂ X ( A ) Y i ∈ I 1  θ X,ρ i a ( x + aB i ) ⊆ ( β i , 1]  Y j ∈ J 1  θ X,ρ j a ( x + aW j ) ⊆ [0 ,ω j ]  dx = Z ∂ X ∩ A (min i ∈ I { ϕ ρ i ( β i , n ) − h ( B i , n ) } − max j ∈ J { ϕ ρ j ( ω j , n ) + h ( ˇ W j , n ) } ) + d H d − 1 . 7 In the proo f w e shall use the following notation when a fixed x ∈ ∂ X is under- sto o d. W e write H := H − h x,n ( x ) i ,n ( x ) and after p o ssibly shrin king B in and B out , w e assume that they bo t h hav e the common radius r > 0 . F or t ∈ R and s ∈ R d w e write θ X a ( t ; s ) := θ X a ( x + a ( tn + s )) . The map ( t, s ) 7→ θ X a ( t ; s ) is con tin uous when X is gen tle since ∂ X ⊕ B ( ε ) ↓ ∂ X as ε ↓ 0 and H d ( ∂ X ) = 0 so that b y monotone con v ergence, | θ X a ( t ′ ; s ′ ) − θ X a ( t ; s ) | =     Z X − a ( t ′ n + s ′ ) ρ a ( z ) d z − Z X − a ( tn + s ) ρ a ( z ) d z     ≤     Z ∂ X ⊕ B ( a ( | t − t ′ | + | s − s ′ | )) ρ a ( z − ( x + a ( tn + s ))) d z     go es to 0 for ( t ′ , s ′ ) → ( t, s ) . F or x ∈ ∂ X and s ∈ S ⊆ R d , let t X + ( a, β ; s ) = inf { t ∈ ( − δ ( ∂ X , x, − n ) , δ ( ∂ X , x, n )) | θ X a ( t ; s ) ≤ β } t X − ( a, β ; s ) = sup { t ∈ ( − δ ( ∂ X , x, − n ) , δ ( ∂ X, x, n )) | θ X a ( t ; s ) > β } t X + ( a, β ; S ) = sup { t X + ( a, β ; s ) | s ∈ S } t X − ( a, β ; S ) = inf { t X − ( a, β ; s ) | s ∈ S } . When t 7→ θ X a ( t ; s ) is decreasing, w e write t X ( a, β ; s ) = t X + ( a, β ; s ) = t X − ( a, β ; s ) . Since θ H a ( t ; s ) is independen t of a , w e sometime s just write θ H 0 ( t ; s ) , t H (0 , β ; s ) , etc. Moreo v er, θ H 0 ( t ; s ) = θ H 0 ( t + h s, n i ; 0) , so t H + (0 , β ; S ) = ϕ ( β , n ) + h ( ˇ S , n ) t H − (0 , β ; S ) = ϕ ( β , n ) − h ( S, n ) . Pr o of of The or em 3.3. F or an y gen tle set Y ⊆ R d , let f Y a ( x ) = Y j ∈ J 1  θ Y ,ρ j a ( x + aW j ) ⊆ [0 ,ω j ]  g Y a ( x ) = Y i ∈ I 1  θ Y ,ρ i a ( x + aB i ) ⊆ ( β i , 1]  . Let D b e suc h that B i , W j , supp ρ i , supp ρ j ⊆ B ( D ) for all i, j . This ensures that supp f X a g X a ⊆ ∂ X ⊕ aB (2 D ) and hence b y [3, The orem 2.1] that Z ξ − 1 ∂ X ( A ) f X a ( x ) g X a ( x ) dx = d X m =1 mκ m Z N ( ∂ X ) 1 A ( x ) (3.1) × Z δ ( ∂ X ; x,n ) 0 t m − 1 f X a ( x + tn ) g X a ( x + tn ) dtµ d − m ( ∂ X ; d ( x, n )) 8 where κ m is t he v olume of the unit ball in R m and µ m ( ∂ X, · ) are certain signed measures o f lo cally finite total v ariation. First observ e that Z δ ( ∂ X ; x,n ) 0 t m − 1 f X a ( x + tn ) g X a ( x + tn ) dt ≤ Z 2 aD 0 t m − 1 f X a ( x + tn ) g X a ( x + tn ) dt ≤ m − 1 a m (2 D ) m . Since eac h µ d − k has lo cally finite total v ariation a nd A is bounded, Lebesgue’s theo- rem of dominated con v ergence and the iden tification of µ d − 1 giv en in [5, Equation (8)] yields lim a → 0 a − 1 d X m =1 mκ m Z N ( ∂ X ) 1 A ( x ) × Z 2 aD 0 t m − 1 f X a ( x + tn ) g X a ( x + tn ) dtµ d − m ( ∂ X ; d ( x, n )) = lim a → 0 a − 1 κ 1 Z N ( ∂ X ) 1 A ( x ) Z 2 aD 0 f X a ( x + tn ) g X a ( x + tn ) dtµ d − 1 ( ∂ X ; d ( x, n )) = Z ∂ X ∩ A  lim a → 0 Z 2 D − 2 D f X a ( x + atn ) g X a ( x + atn ) dt  H d − 1 ( dx ) if the limit exists . Th us w e consider the inner in tegral for x ∈ ∂ X fixed. Observ e that θ B in a ( t ; s ) ≤ θ X a ( t ; s ) , θ H a ( t ; s ) ≤ θ R d \ B out a ( t ; s ) . Th us f B in a ( x + atn ) ≥ f X a ( x + atn ) ≥ f R d \ B out a ( x + atn ) g B in a ( x + atn ) ≤ g X a ( x + atn ) ≤ g R d \ B out a ( x + atn ) (3.2) and hence Z 2 D − 2 D f R d \ B out a ( x + atn ) g B in a ( x + atn ) dt ≤ Z 2 D − 2 D f X a ( x + atn ) g X a ( x + atn ) dt (3.3) ≤ Z 2 D − 2 D f B in a ( x + atn ) g R d \ B out a ( x + atn ) dt. If Y i , i = 1 , 2 , are gen tle sets for whic h t 7→ θ Y i a ( t ; s ) are decre asing on ( − 2 D , 2 D ) for all s with | s | ≤ D , then Z 2 D − 2 D f Y 1 a ( x + atn ) g Y 2 a ( x + atn ) dt = Z min i ∈ I { t Y 2 − ( a,β i ; B i ) } max j ∈ J { t Y 1 + ( a,ω j ; W j ) } 1 { min i ∈ I { t Y 2 − ( a,β i ; B i ) } > max j ∈ J { t Y 1 + ( a,ω j ; W j ) } dt (3.4) = (min i ∈ I { t Y 2 − ( a, β i ; B i ) − max j ∈ J { t Y 1 + ( a, ω j ; W j ) } ) + . 9 F rom no w on, we assume 2 aD ≤ r . This guaran tees that x + a ( B (2 D ) ⊕ [ − 2 D n, 2 D n ]) ⊆ conv( B in ∪ B out ) and thus (3.4) holds fo r Y i equal to B in , H , or R d \ B out . Here [ x, y ] denotes the line segmen t b etw een x and y . Moreo ver, ( H ∩ ( x + a ( tn + s + B ( D ))) − aν n ) ⊆ B in ∩ ( x + a ( tn + s + B ( D ))) R d \ B out ∩ ( x + a ( tn + s + B ( D ))) ⊆ ( H ∩ ( x + a ( tn + s + B ( D ))) + aν n ) whenev er aν ≥ r − p r 2 − a 2 (2 D ) 2 . It follows that θ H a ( t + ν ; s ) ≤ θ B in a ( t ; s ) ≤ θ H a ( t ; s ) θ H a ( t ; s ) ≤ θ R d \ B out a ( t ; s ) ≤ θ H a ( t − ν ; s ) for suc h ν . Th us | t H ( a, β ; s ) − t B in ( a, β ; s ) | , | t H ( a, β ; s ) − t R d \ B out ( a, β ; s ) | ≤ a − 1 ( r − p r 2 − (2 aD ) 2 ) ≤ M a (3.5) for some M > 0 dep ending only on r and D . Therefore,     Z 2 D − 2 D f B in a ( x + atn ) g R d \ B out a ( x + atn ) dt − Z 2 D − 2 D f H a ( x + atn ) g H a ( x + atn ) dt     ∈ O ( a ) . But for all a , Z 2 D − 2 D f H a ( x + atn ) g H a ( x + atn ) dt = (min i ∈ I t H − (0 , β i ; B i ) − max j ∈ J t H + (0 , ω j ; W j )) + = (min i ∈ I { ϕ ρ i ( β i , n ) − h ( B i , n ) } − max j ∈ J { ϕ ρ j ( ω j , n ) + h ( ˇ W j , n ) } ) + . Th us, the right hand side of (3.3) is forced to conv erge with this limit. A simi lar argumen t applie d to t he left hand side finally forces the middle term to con ve rge with this limit as w ell, pro ving the theorem. Remark 3.4. If the one or more of the interv als [0 , β i ] or ( ω j , 1 ] are replaced by [0 , β i ) o r [ ω j , 1 ] , resp ectiv ely , with β i , ω j ∈ (0 , 1] , the theorem clearly holds with the correspo nding ϕ replace d b y ˜ ϕ by a similar argumen t, as long as the in tersection of all the in terv als do es not contain either of 0 or 1. 3.2 Generalization to PSF’s with non-compact supp ort The pro of of Theorem 3.3 generalizes to the case where supp ρ is non-compact and satisfies: Prop erty 3.5. There is a C > 0 suc h that ρ ≤ C almost ev erywhere and the function t 7→ − R ∂ H n ρ ( z − tn ) dz is con tin uous for ev ery n ∈ S d − 1 . 10 The condition is satisfie d for most PSF’s o ccurring in practice, e.g. if ρ is b ounded and ρ ( z ) ∈ O ( | z | k ) f o r some k < − d . Prop ert y 3.5 clearly ensures: Lemma 3.6. L et ρ have Pr op erty 3.5 . Then t 7→ θ H n ( tn ) is de cr e asing C 1 with d dt θ H n ( tn ) = − Z ∂ H n ρ ( z − tn ) dz . W e first need some tec hnical lemmas. Let µ ( R ) = Z | z |≥ R ρ ( z ) dz . By in tegrabilit y of ρ , lim R →∞ µ ( R ) = 0 . Lemma 3.7. L et x ∈ ∂ X and a > 0 b e fixe d. L et K > 0 an d 0 < R < r − 2 aK , and supp ose ρ is a PSF with ρ ≤ C alm ost sur ely for some C > 0 . Then ther e is a c onstant M > 0 dep ending on l y on r , C , and K such that for al l t ∈ R and s ∈ R d with | s | , | t | ≤ K , 0 ≤ θ H a ( t ; s ) − θ B in a ( t ; s ) ≤ M a − d ( R + a | s | ) d +1 + µ ( a − 1 R ) 0 ≤ θ R d \ B out a ( t ; s ) − θ H a ( t ; s ) ≤ M a − d ( R + a | s | ) d +1 + µ ( a − 1 R ) . Pr o of. Observ e that R is c hosen so small that x + a ( tn + s ) + B ( R ) ⊆ con v ( B in ∪ B out ) whenev er | s | , | t | ≤ K . Since B in ⊆ H ⊆ B in ∪ A ∪ R d \ ( x + a ( tn + s ) + B ( R )) where A = ( H \ B in ) ∩ ( x + a ( tn + s ) + B ( R )) , this ens ures that θ H a ( t ; s ) ≤ Z B in ρ a ( z − ( x + a ( tn + s )) dz + a − d Z A ρ ( a − 1 ( z − ( x + a ( tn + s )))) dz + a − d Z R d \ ( x + a ( tn + s )+ B ( R )) ρ ( a − 1 ( z − ( x + a ( tn + s )))) d z ≤ θ B in a ( t ; s )) + a − d C Z B d − 1 ( R + a | s | ) ( r − p r 2 − | z | 2 ) dz + µ ( a − 1 R ) ≤ θ B in a ( t ; s ) + M a − d ( R + a | s | ) d +1 + µ ( a − 1 R ) where B d − 1 ( D ) denotes the ball in R d − 1 of radius D . The second inequalit y is similar, using R d \ B out ⊆ H ∪ ( R d \ ( x + a ( tn + s ) + B ( R ))) ∪ B with B = ( R d \ ( B out ∪ H )) ∩ ( x + a ( tn + s ) + B ( R )) . 11 When n ∈ S d − 1 and s ∈ R d are giv en, w e shall sa y that β ∈ (0 , 1) is a regular v alue for θ H n 0 ( · ; s ) if t 7→ θ H n 0 ( t ; s ) = θ H n 0 ( t + h s, n i ; 0) has non-zero deriv ativ e on the set { t ∈ R | θ H n 0 ( t ; s ) = β } . Lemma 3.8. L et x ∈ ∂ X b e fixe d. L et ρ b e a PSF satisfying Pr op erty 3.5 and B ⊆ R d a c omp a ct se t. Supp ose β ∈ (0 , 1 ) is a r e gular value for θ H 0 ( · ; s ) for al l s ∈ B and let K > 0 b e given. Then ther e is a function γ ( a ) ∈ o (1) such that for al l s ∈ B , θ R d \ B out a ( t ; s ) < β for al l t ∈ ( t H (0 , β ; s ) + γ ( a ) , K ] θ B in a ( t ; s ) > β for al l t ∈ [ − K , t H (0 , β ; s ) − γ ( a )) whenever a is sufficiently smal l. In p articular, for a smal l enough | t B in ± ( a, β ; s ) − t H (0 , β ; s ) | , | t R d \ B out ± ( a, β ; s ) − t H (0 , β ; s ) | ≤ γ ( a ) whenever s ∈ B . Pr o of. Since g ( t, s ) = d dt θ H 0 ( t ; s ) is con tin uous and g ( t H (0 , β ; s ) , s ) < 0 for all s ∈ B b y assum ption, there is a δ > 0 such that M 1 = inf {− g ( t H (0 , β ; s ) + ξ , s ) | s ∈ B , | ξ | ≤ δ } > 0 . Let s ∈ B , write t 0 = t H (0 , β ; s ) , and let | ν | ≤ δ . By the mean v a lue theorem there is a | ξ | ≤ ν suc h that θ H 0 ( t 0 ; s ) − θ H 0 ( t 0 + ν ; s ) = − ν g ( t 0 + ξ , s ) ≥ M 1 ν. Put R ( a ) = a ε where d d +1 < ε < 1 . Lemma 3.7 sho ws that 0 ≤ θ R d \ B out a ( t ; s ) − θ H 0 ( t ; s ) ≤ M 2 a − d ( R ( a ) + a | s | ) d +1 + µ ( a − 1 R ( a )) for all t ∈ [ − K, K ] . Th us, whenev er δ ≥ ν > M − 1 1 ( M 2 a − d ( R ( a ) + a | s | ) d +1 + µ ( a − 1 R ( a ))) , w e get β > M 2 a − d ( R ( a ) + a | s | ) d +1 + µ ( a − 1 R ( a )) + θ H 0 ( t 0 + ν ; s ) ≥ θ R d \ B out a ( t 0 + ν ; s ) and since ν 7→ θ H 0 ( t + ν ; s ) is decreasing, this also holds when ν > δ as long a s t 0 + ν ≤ K . Th us we may tak e γ ( a ) to b e γ ( a ) = M − 1 1 ( M 2 a − d ( R ( a ) + a sup {| s | , s ∈ B } ) d +1 + µ ( a − 1 R ( a ))) . Then γ ( a ) ∈ o (1) since ε ( d + 1) − d > 0 and lim a → 0 µ ( a ε − 1 ) = 0 . The claim ab out θ B in a is sim ilar. F or the la st claim, c ho ose D > 0 su c h that µ ( D ) < β , 1 − β and B ⊆ B ( D ) . Then fo r a small enough, x + a ( tn + s + B ( D )) ⊆ B in ⊆ H for all t ∈ [ − a − 1 r , − 2 D ] , so for suc h t , θ H 0 ( t ; s ) ≥ θ B in a ( t ; s ) ≥ 1 − µ ( D ) > β . The claim for t B in ± ( a, β ; s ) no w follow s from the first part with K replaced b y 2 D . The claim ab out t R d \ B out ± ( a, β ; s ) is simil ar. 12 W e can no w state the main theorem for PSF’s with non-compact supp ort: Theorem 3.9. The or em 3.3 also holds for PSF’s ρ i , ρ j having non-c omp act supp ort and satisfying Pr op erty 3.5 if β i , ω j ∈ (0 , 1) ar e r e gular values for θ H n ,ρ i 0 ( · ; b ) an d θ H n ,ρ j 0 ( · ; w ) , r esp e ctively, for al l n ∈ S d − 1 , b ∈ B i , and w ∈ W j . Pr o of. The pro of go es as in the case of compact supp ort. W e no w choose D suc h that µ ( D ) < min { β i , 1 − ω j | i ∈ I , j ∈ J } and all B i , W j ⊆ B ( D ) to ensure that supp f X a g X a ⊆ ∂ X ⊕ aB (2 D ) . The s ame a rgumen t then reduces the pro of to a computation of the limit as a → ∞ of Z 2 D − 2 D f X a ( x + atn ) g X a ( x + atn ) dt (3.6) for eac h x ∈ ∂ X . W e still hav e the inequalities f R d \ B out a g B in a ≤ f X a g X a ≤ f B in a g R d \ B out a . Ho w ev er, θ B in a and θ R d \ B out a ma y not b e injectiv e. It is still true that Z 2 D − 2 D f B in a ( x + atn ) g R d \ B out a ( x + atn ) dt ≤  min i ∈ I { t R d \ B out − ( a, β i ; B i ) } − max j ∈ J { t B in + ( a, ω j ; W j ) }  + . (3.7) Moreo v er, Lemma 3 .8 yields f H a ( x + a ( t − γ ( a )) n ) g H a ( x + a ( t + γ ( a )) n ) ≤ f R d \ B out a ( x + atn ) g B in a ( x + atn ) and hence  min i ∈ I { t H − (0 , β i ; B i ) − γ ( a ) } − max j ∈ J { t H + (0 , ω j ; W j ) + γ ( a ) }  + ≤ Z 2 D − 2 D f R d \ B out a ( x + atn ) g B in a ( x + atn ) dt. (3.8) Both the righ t hand side of (3.7) and the left hand side of (3.8) con verge to  min i ∈ I { t H − (0 , β i ; B i ) } − max j ∈ J { t H + (0 , ω j ; W j ) }  + b y Lemma 3.8. Th us (3.6 ) is forced to con v erge with the same limit. Remark 3.10. The condition that β is a r egular v a lue is easily satisfied given Prop ert y 3.5, e.g. if ρ > 0 almost eve rywhere. 4 Applications to surface area estimation The form ulas of Sec tion 3 ha v e implications to surface area estim ation. W e deriv e some o f these b elow . 13 4.1 Thresholded image s Theorem 3.3 and 3.9 apply directly to local algorithms for thresholded images: Corollary 4.1. L et X ⊆ R d b e c omp act gentle, β ∈ [0 , 1) , ρ a PSF and ( B l , W l ) a c onfigur ation. Supp ose B i = B l , W j = W l , β i = ω j = β , and ρ i = ρ j = ρ satisfy the c onditions of either The or em 3. 3 or 3.9. Then lim a → 0 a d − 1 E N ( β ) a L c l ( X ) = lim a → 0 a − 1 Z R d 1  θ X a ( z + aB l ) ⊆ ( β , 1]  1  θ X a ( z + aW l ) ⊆ [0 ,β ]  dz = Z ∂ X ( − h ( B l ⊕ ˇ W l , n )) + d H d − 1 (4.1) = lim a → 0 a d − 1 E N a L c l ( X ) . In p articular, i f ˆ V d − 1 is a lo c al estimator fo r black-and-white images, lim a → 0 E ˆ V ( β ) a L c d − 1 ( X ) = lim a → 0 E ˆ V a L c d − 1 ( X ) . The last equalit y in (4.1) is [5, Theorem 5]. The asymptotic mean of surface area estimators applied to thresholded images is th us the same a s for black-and-white images, so the asymptotic results in [4, 1 5, 16] carry o ve r: Corollary 4.2. L et d > 1 and let ˆ V d − 1 a lo c a l algorithm. L et ρ and β b e as in Cor ol lary 4.1. Then ˆ V ( β ) d − 1 is asymptotic al ly biase d on b oth the class of r -r e gular sets (se e Definition 5 .1 b elow) and on the class of c o m p act c o nvex p olytop e s with non-empty interior. 4.2 Surface area estimators with n = 1 Consider the nu m b er of pixels with threshold v alue in some in terv al I ⊆ (0 , 1 ) N a L c I ( X ) = |{ z ∈ a L c | θ X a ( z ) ∈ I }| where | · | denotes cardinalit y . This corresponds to the case n = 1 in Definition 2.2 with the function f = 1 I . Theorem 3.3 and 3 .9 yield: Corollary 4.3. L et X ⊆ R d b e a c omp act gentle set, ( β , ω ] ⊆ (0 , 1) , and ρ a PS F . Supp ose β i = β , ω i = ω , B i = W j = { 0 } , and ρ i = ρ j = ρ satisfy the c onditions of either The or em 3.3 or 3.9. Then lim a → 0 a d − 1 E N a L c ( β ,ω ] ( X ) = Z ∂ X ( ϕ ρ ( β , n ) − ϕ ρ ( ω , n )) d H d − 1 . (4.2) In p articular, if ρ is r otation invariant, ϕ ρ ( β ) := ϕ ρ ( β , n ) is indep enden t of n ∈ S d − 1 , so 1 2 ( ϕ ρ ( β ) − ϕ ρ ( ω ) ) − 1 N a L c ( β ,ω ] (4.3) is an asymptotic al ly unbia s e d estima tor for V d − 1 on the class of c omp act gen tle sets. 14 Remark 4.4. If one or more of the op en and closed ends of ( β , ω ] are c hanged, the correspo nding ϕ should b e replaced b y ˜ ϕ in (4.2), y ielding a s imilar statemen t for all I ⊆ (0 , 1) . F or I = (0 , 1) and ρ = ρ B where B is the closure of its in terior, this implies lim a → 0 a d − 1 E N a L c (0 , 1) ( X ) = Z ∂ X h ( B ⊕ ˇ B , n ) d H d − 1 . In particular, if X and B are con v ex, this is the mixed v olume 2 V ( X [ d − 1 ] , B ⊕ ˇ B ) , see [12, Section 5]. Remark 4.5. Ev en if the grey-v alues in the output data are group ed in to finitely man y (at least three) in terv als, an estimator o f the form N I can still b e applied. Supp ose ρ is as in Theorem 3.3. The limit in (4.2) can also be written as Z ∂ X ( ϕ ρ ( β , n ) − ϕ ρ ( ω , n )) d H d − 1 = Z ∂ X µ n (( β , ω ]) d H d − 1 = Z ∂ X Z (0 , 1) 1 ( β ,ω ] dµ n d H d − 1 where µ n for n ∈ S d − 1 is the Leb esgue–Stieltjes measure defined by the incre asing righ t con tin uous function β 7→ − ϕ ρ ( β , n ) . Lemma 4.6. F or any B or el set A ⊆ (0 , 1) , the function S d − 1 → R that is given by n 7→ µ n ( A ) is B or el me asur able. In p articular, for any c omp act gentle set X ⊆ R d , ther e is a Bor el me asur e µ X on (0 , 1) defi ne d by µ X ( A ) = Z ∂ X Z (0 , 1) 1 A dµ n d H d − 1 . Pr o of. Since ϕ ρ : S d − 1 × (0 , 1) → R is measurable, n 7→ µ n ( A ) is clearly measurable for A b elonging to the in tersection stable collection of ( β , ω ] for β , ω ∈ [0 , 1 ) . T he claim no w follo ws from Dynkin’s lemma. In tro duce the image measure µ X a = a − 1 H d − 1 ◦ ( θ X a ) − 1 on (0 , 1 ) . If f : (0 , 1 ) → R is b ounded measurable, E ˆ V ( f ) a L c d − 1 ( X ) = Z (0 , 1) f dµ X a . Similarly , b y standard argumen ts, Z (0 , 1) f dµ X = Z ∂ X Z (0 , 1) f dµ n d H d − 1 . Theorem 3.3 and 3.9 yield : 15 Corollary 4.7. If ρ is as in The or em 3 .3, µ X a c onver g e s we akly to µ X . In p artic- ular, for any f : (0 , 1) → R that is b ounde d me asur able and µ X -almost everywher e c ontinuous, lim a → 0 E ˆ V ( f ) a L c d − 1 ( X ) = Z (0 , 1) f dµ X . (4.4) If ρ satisfies Pr op erty 3.5 and [ β , ω ] ⊆ (0 , 1) c on tains on l y r e gular values for θ H n 0 ( · ; 0) for al l n ∈ S d − 1 , the r estriction of µ X a to ( β , ω ) c o nver ges we a k ly to µ X r estricte d to the same interval. In p articular, (4.4) h o lds in this situation a s wel l if supp f ⊆ ( β , ω ) . Pr o of. Supp ose first that ρ is as in Theorem 3.3. T aking f = 1 (0 ,ω ] sho ws that lim a → 0 µ X a ((0 , ω ]) = µ X ((0 , ω ]) for all ω ∈ (0 , 1) . Moreo v er, Remark 3.4 sho ws that lim a → 0 µ X a ((0 , 1)) = Z ∂ X ( ϕ ρ (0 , n ) − ˜ ϕ ρ (1 , n )) d H d − 1 . By monotone con v ergence, this equals µ X ((0 , 1)) = sup ω < 1 µ X ((0 , ω ]) = sup ω < 1 Z ∂ X ( ϕ ρ (0 , n ) − ϕ ρ ( ω , n )) d H d − 1 = Z ∂ X ( ϕ ρ (0 , n ) − inf ω < 1 { ϕ ρ ( ω , n ) } ) d H d − 1 = Z ∂ X ( ϕ ρ (0 , n ) − ˜ ϕ ρ (1 , n )) d H d − 1 . The weak conv ergence follow s. The non-compact case is similar. If ρ is rotation in v ariant, ϕ ρ ( β ) = ϕ ρ ( β , n ) , and hence µ := µ n , is independen t of n ∈ S d − 1 . Th us (4.4) reduces to: Corollary 4.8. Supp ose ρ is r otation invariant. Under the as s umptions of Cor ol- lary 4.7, lim a → 0 E ˆ V ( f ) a L c d − 1 ( X ) = 2 V d − 1 ( X ) Z (0 , 1) f dµ. That is, ˆ V ( f ) d − 1 is asymp totic al ly unbiase d if and only if 2 R (0 , 1) f dµ = 1 . If ρ is not rotation in v ariant, w e can get b ounds on the w orst case asymptotic relativ e mean error inste ad: Corollary 4.9. Under the assumptions of Cor ol lary 4.7, Err( ˆ V ( f ) d − 1 ) ≤ sup n ∈ S d − 1     2 Z 1 0 f dµ n − 1     (4.5) with e quality i f ρ is r efle ction invaria nt or f satisfie s f ( x ) = f ( 1 − x ) . F or any f , the function ˜ f ( x ) = 1 2 ( f ( x ) + f (1 − x )) satisfies ˜ f ( x ) = ˜ f (1 − x ) and Err( ˆ V ( ˜ f ) d − 1 ) ≤ Err( ˆ V ( f ) d − 1 ) . 16 Pr o of. Err( ˆ V ( f ) d − 1 ) is giv en b y sup X ∈S     lim a → 0 E ˆ V ( f ) a L c d − 1 ( X ) V d − 1 ( X ) − 1     = sup X ∈S     V d − 1 ( X ) − 1 Z ∂ X Z 1 0 f dµ n d H d − 1 − 1     ≤ sup X ∈S     V d − 1 ( X ) − 1 Z ∂ X     Z 1 0 f dµ n − 1 2     d H d − 1     ≤ sup n ∈ S d − 1     2 Z 1 0 f dµ n − 1     . Let n k ∈ S d − 1 b e a sequence with   2 R 1 0 f dµ n k − 1   con v erging to the latter suprem um and c ho ose an orthonormal basis u 1 k , . . . , u d − 1 k for n ⊥ k and a sequence t k > 0 suc h that lim k →∞ t k = 0 . Observ e t hat µ n ( A ) = µ − n (1 − A ) . Ass uming ρ reflection in v arian t or f ( x ) = f (1 − x ) , the asymptotic relative bias on the sets [0 , t k n k ] ⊕ L d − 1 i =1 [0 , u i k ] th us con v erges to the right hand side o f the inequalit y . The last claim follo ws from     lim a → 0 E ˆ V ( ˜ f ) a L c d − 1 ( X ) V d − 1 ( X ) − 1     ≤ 1 2      lim a → 0 E ˆ V ( f ) a L c d − 1 ( X ) V d − 1 ( X ) − 1     +     lim a → 0 E ˆ V ( f ) a L c d − 1 ( − X ) V d − 1 ( − X ) − 1      . 5 Second order form ulas T o obtain a second order v ersion of Theorem 3.3, w e need to b e able to control the second order b eha viour of the b oundary of underlying set X ⊆ R d . W e assume throughout the section that d > 1 . Thus we s hall restrict atten t io n to the class o f r -regular sets: Definition 5.1. A close d subset X ⊆ R d is c al le d r -r e gular fo r some r > 0 if for al l x ∈ ∂ X ther e exist two b al ls B in and B out of r adius r b oth c ontaining x such that B in ⊆ X and in t( B out ) ⊆ R d \ X . T he unique outwar d p ointing normal ve ctor at x is denote d by n ( x ) . It can b e pro v ed [1] that if X is r -regular, then ∂ X is a C 1 manifold with H d − 1 -almost ev erywhere differentiable normal v ector field. In particular, its principal curv atures k 1 , . . . , k d − 1 ≤ r − 1 , corresp onding to the orthog o nal principal directions e 1 , . . . , e d − 1 ∈ T ∂ X , can b e defined almost ev erywhere as the eigen v alues of the differen tial dn . Th us the second fundamen tal form I I x on the tangent space T x ∂ X is defined for H d − 1 -almost all x ∈ ∂ X . F or P d − 1 i =1 α i e i ∈ T x ∂ X , II x is the quadratic form g iv en by II x d − 1 X i =1 α i e i ! = d − 1 X i =1 k i ( x ) α 2 i whenev er d x n is defined. In particular, the trace is T r II = k 1 + · · · + k d − 1 . 17 The in tegrated mean curv ature 2 π ( d − 1) − 1 V d − 2 can t hus b e defined [1] for r - regular sets b y V d − 2 ( X ) = 1 2 π Z ∂ X T r( II ) d H d − 1 . Let T ε ∂ X = { ( x, α ) | α ∈ T x ∂ X, | α | < ε } . W e need the follow ing lemma. A pro of can b e found e.g. in [14]. Lemma 5.2. L et X b e an r -r e g ula r set. Ther e is a unique function q : T r ∂ X → R such that for α ∈ T x ∂ X , q ( x, α ) is the unique q ∈ [ − r , r ] with x + α + q n ( x ) ∈ ∂ X . Ther e is a c onstant C > 0 such that q ( x, α ) ≤ C | α | 2 for al l ( x, α ) ∈ T r ∂ X . Mor e o v er, lim a → 0 a − 2 q ( x, aα ) = − 1 2 II x ( α ) . W e also mak e the follo wing observ ation: Lemma 5.3. If X is r -r e gular, K > 0 , and ρ has c omp act supp ort, then for al l a sufficiently sma l l, the m a p t 7→ θ X a ( t ; s ) is de cr e asing on the interval [ − a − 1 r , a − 1 r ] for al l s ∈ B ( K ) . Pr o of. Supp ose supp ρ ⊆ B ( D ) . By Lemma 5.2, ( x + a ( tn + s + B ( D ))) ∩ X − aν n ⊆ ( x + a (( t − ν ) n + s + B ( D ))) ∩ X for all s ∈ B ( K ) , ν > 0 , a nd t, t − ν ∈ [ − a − 1 r , a − 1 r ] whenev er a is sufficie n tly small. Hence, θ X a ( t ; s ) = Z ( x + a ( tn + s + B ( D ))) ∩ X − ν n ρ a ( z − ( x + a (( t − ν ) n + s ))) dz ≤ Z ( x + a (( t − ν ) n + s + B ( D ))) ∩ X ρ a ( z − ( x + a (( t − ν ) n + s ))) dz = θ X a ( t − ν ; s ) . F or z ∈ R d and n ∈ S d − 1 , w e write z = ( z n ⊥ , z n ) where z n = h z , n i ∈ R and z n ⊥ ∈ n ⊥ is the pro jection of z o nto n ⊥ . Let x ∈ ∂ X . Define the quadratic appro ximation Q ( x ) to X at x b y Q ( x ) = { z ∈ R d | | ( z − x ) n | ≤ − 1 2 II x (( z − x ) n ⊥ ) } . If x ∈ ∂ X is understoo d, w e simply write Q := Q ( x ) . Definition 5.4. S upp ose ρ is c ontinuous with c omp act supp ort. F or x ∈ ∂ X and β 0 a r e gular value for θ H 0 ( · ; s ) , define ψ Q ( x ) ( β 0 ; s ) = ψ 1  t H (0 , β 0 ; s ); s  ψ 2  t H (0 , β 0 ; s ); s  = − ψ 1  t H (0 , β 0 ; s ); s  d dβ t H (0 , β ; s ) | β = β 0 18 wher e ψ 1 ( t ; s ) = − 1 2 Z n ( x ) ⊥ II x ( z ) ρ ( z − s n ⊥ , − t − s n )) dz ψ 2 ( t ; s ) = − d dt θ H 0 ( t ; s ) = Z n ( x ) ⊥ ρ ( z − s n ⊥ , − t − s n )) dz . Lemma 5.5. L et X b e r -r e gular and x ∈ ∂ X . S upp ose ρ is c on tinuous and has c omp a c t supp ort. L et B ⊆ R d c omp a c t and assume β 0 ∈ (0 , 1) is a r e g ular value for θ H 0 ( · ; s ) for al l s ∈ B . Then the function ( a, s ) 7→ t Q ( a, β 0 ; s ) extends to a wel l- define d C 1 function on ( − ε, ε ) × U for some ε > 0 and U ⊇ B op en so that for al l ( a, s ) ∈ (0 , ε ) × U , t Q ( a, β 0 ; s ) is the uniq ue t w ith θ Q a ( t ; s ) = β 0 . Mor e over, t Q ( a, β 0 ; s ) = t H (0 , β 0 ; s ) + aψ Q ( β 0 ; s ) + o ( a ) and sup s ∈ B | t Q ( a, β 0 ; s ) − t H ( a, β 0 ; s ) | ∈ O ( a ) . Pr o of. First observ e that θ Q a ( t ; s ) = θ H a ( t ; s ) + θ Q \ H a ( t ; s ) − θ H \ Q a ( t ; s ) . Supp ose supp ρ, B ⊆ B ( D ) . W e first rewrite the latter terms: θ Q \ H a ( t ; s ) = Z Q \ H ρ a ( z − ( x + a ( tn + s ))) d z = a − d Z n ⊥ Z 0 ∨  − 1 2 II ( z n ⊥ )  0 ρ ( a − 1 z n ⊥ − s n ⊥ , a − 1 z n − ( t + s n )) dz n dz n ⊥ = a − 1 Z B n ⊥ (2 D ) Z 0 ∨  − a 2 1 2 II ( z n ⊥ )  0 ρ ( z n ⊥ − s n ⊥ , a − 1 z n − ( t + s n )) dz n dz n ⊥ = a Z B n ⊥ (2 D ) Z 0 ∨  − 1 2 II ( z n ⊥ )  0 ρ ( z n ⊥ − s n ⊥ , az n − ( t + s n )) dz n dz n ⊥ where B n ⊥ (2 D ) is the ball in n ⊥ of radius 2 D . Similarly , θ H \ Q a ( t ; s ) = a Z B n ⊥ (2 D ) Z 0 0 ∧  − 1 2 II ( z n ⊥ )  ρ ( z n ⊥ − s n ⊥ , az n − ( t + s n )) dz n dz n ⊥ . This computation sho ws that θ Q a ( t ; s ) extend s con tinuously to a w ell-defined function for all ( a, t, s ) ∈ R 2+ d . Denote this function by β ( a, t ; s ) = θ H 0 ( t ; s ) + a Z B n ⊥ (2 D ) Z − 1 2 II ( z n ⊥ ) 0 ρ ( z n ⊥ − s n ⊥ , az n − ( t + s n )) dz n dz n ⊥ . The a ssumptions on ρ imply that β ( a, t ; s ) is C 1 in ( a, t, s ) and d da β ( a, t ; s ) = − 1 2 Z B n ⊥ (2 D ) II ( z n ⊥ ) ρ ( z n ⊥ − s n ⊥ , − a 1 2 II ( z n ⊥ ) − ( t + s n )) dz n ⊥ d dt β ( a, t ; s ) = d dt θ H 0 ( t ; s ) + a Z B n ⊥ (2 D )  ρ ( z n ⊥ − s n ⊥ , − ( t + s n )) − ρ ( z n ⊥ − s n ⊥ , − a 1 2 II ( z n ⊥ ) − ( t + s n ))  dz n ⊥ . 19 Again, these functions are clearly con tin uous. In particular, at a = 0 w e obtain β (0 , t ; s ) = θ H 0 ( t ; s ) d da β ( a, t ; s ) | a =0 = − 1 2 Z n ⊥ II ( z n ⊥ ) ρ ( z n ⊥ − s n ⊥ , − ( t + s n )) dz n ⊥ d dt β ( a, t ; s ) | a =0 = d dt θ H 0 ( t ; s ) . Since d dt β (0 , t ; s ) | t = t H (0 ,β 0 ; s ) < 0 fo r all s ∈ B b y assumption, the implicit function theorem yield s that in a neigh b orho o d of the compact set { 0 } × B , the solution t to β ( a, t ; s ) = β 0 is giv en b y a C 1 function ( a, s ) 7→ t Q ( a, β 0 ; s ) with t Q ( a, β 0 ; s ) = t H (0 , β 0 ; s ) − a d da β ( a, t H (0 , β 0 ; s ); s ) | a =0 d dt β (0 , t ; s ) | t = t H (0 ,β 0 ; s ) + o ( a ) . The last claim follo ws from the mean v alue theorem, since for 0 ≤ a 0 ≤ ε 2 , | t Q ( a 0 , β 0 ; s ) − t H (0 , β 0 ; s ) | = a 0    d da t Q ( a, β 0 ; s ) | a = a ′    ≤ a 0 sup n    d da t Q ( a, β 0 ; s )       a ∈  0 , ε 2  , s ∈ B o where a ′ ∈ [0 , a 0 ] and the latter suprem um is finite by con tin uit y . Lemma 5.6. L et X b e r -r e gular, x ∈ ∂ X , and B ⊆ R d c omp a c t. L et ρ b e c ontinuous with c omp act supp ort. Then ther e is a function λ ( a ) ∈ o ( a ) such that | θ Q a ( t ; s ) − θ X a ( t ; s ) | ≤ λ ( a ) for al l t ∈ [ − a − 1 r , a − 1 r ] a nd s ∈ B . If, mor e over, β 0 is a r e gular value for θ H 0 ( · ; s ) for al l s ∈ B , ther e is a c onstant M > 0 such that for a l l s ∈ B , | t Q ( a, β 0 ; s ) − t X ( a, β 0 ; s ) | ≤ M λ ( a ) . Pr o of. Supp ose B , supp ρ ⊆ B ( D ) . Observ e that | θ Q a ( t ; s ) − θ X a ( t ; s ) | ≤ Z Q \ X ∪ X \ Q ρ a ( z − ( x + a ( tn + s ))) dz = a Z n ⊥ Z  − 1 2 II ( z n ⊥ )  ∨ a − 2 q ( x,az n ⊥ )  − 1 2 II ( z n ⊥ )  ∧ a − 2 q ( x,az n ⊥ ) ρ ( z n ⊥ − s n ⊥ , az n − ( t + s n )) dz n dz n ⊥ ≤ a sup ρ Z B n ⊥ (2 D ) | 1 2 II ( z n ⊥ ) + a − 2 q ( x, az n ⊥ ) | dz n ⊥ . As | 1 2 II ( z n ⊥ ) + a − 2 q ( x, az n ⊥ ) | is b ounded for z n ⊥ ∈ T 2 D x ∂ X and a ∈ (0 , r 2 D ] b y Lemma 5.2, the same lemma com bined with Lebesgue’s theorem yields that λ ( a ) = a sup ρ Z B n ⊥ (2 D ) | 1 2 II ( z n ⊥ ) + a − 2 q ( x, az n ⊥ ) | dz n ⊥ ∈ o ( a ) . 20 No w supp ose β 0 is a regular v alue for θ H 0 ( · ; s ) fo r all s ∈ B . The function d dt β ( a, t ; s ) from the pro of o f Lemma 5.5 was con tin uous in ( a, t, s ) , so there is a neigh b orho o d of the compact set { ( 0 , t H (0 , β 0 ; s ) , s ) ∈ R 2+ d | s ∈ B } on whic h d dt β ( a, t ; s ) > 0 . In particular, there are constan t s δ, ε, M 1 > 0 suc h that − inf n d dt β ( a, t ; s )    a ∈ [0 , ε ] , s ∈ B , | t − t H (0 , β 0 ; s ) | ≤ δ o = M 1 . Th us for a ∈ (0 , ε ) and t, t + ν ∈ [ t H (0 , β 0 ; s ) − δ , t H (0 , β 0 ; s ) + δ ] , θ Q a ( t + ν ; s ) − θ Q a ( t ; s ) ≤ − M 1 ν. Hence, θ X a ( t + ν ; s ) − θ Q a ( t ; s ) ≤ λ ( a ) − M 1 ν. As lim a → 0 t Q ( a, β 0 ; s ) = t H (0 , β 0 ; s ) uniformly for s ∈ B b y Lemma 5.5, | t Q ( a, β 0 ; s ) − t H (0 , β 0 ; s ) | < 1 2 δ for all s ∈ B and a sufficien tly small. Th us, if θ Q a ( t ; s ) = β 0 , then θ X a ( t + ν ; s ) < β 0 for 1 2 δ ≥ ν > M − 1 1 λ ( a ) . So if a is so small that 1 2 δ > M − 1 1 λ ( a ) , t X ( a, β 0 ; s ) − t Q ( a, β 0 ; s ) ≤ M − 1 1 λ ( a ) for all s ∈ B . The other ineq ualit y is similar. Theorem 5.7. L et X b e a close d r -r e gular se t and A ⊆ R d b e b ound e d me asur able . L et I and J b e no n -empty finite index sets. F or i ∈ I and j ∈ J , let B i , W j ⊆ R d b e non-empty c omp act strictly c on vex sets and let ρ i , ρ j b e c ontinuous PSF ’s with c omp a c t supp ort. Supp ose that β i , ω j ∈ (0 , 1) a r e r e gular values for θ H n ,ρ i 0 ( · ; b ) and θ H n ,ρ j 0 ( · ; w ) , r esp e ctively, for al l n ∈ S d − 1 , b ∈ B i , and w ∈ W j . Then Z ξ − 1 ∂ X ( A ) Y i ∈ I 1  θ X,ρ i a ( x + aB i ) ⊆ ( β i , 1]  Y j ∈ J 1  θ X,ρ j a ( x + aW j ) ⊆ [0 ,ω j ]  dx = a Z ∂ X ∩ A  t H − (0 , β ; B ) − t H + (0 , ω ; W )  + d H d − 1 + a 2 Z ∂ X ∩ A  1 2 T r II  t H − (0 , β ; B ) 2 − t H + (0 , ω ; W ) 2  1  t H − (0 ,β ; B ) >t H + (0 ,ω ; W )  +  min i ∈ I ′ ( n ) { ψ Q,ρ i ( β i , B i ) } − max j ∈ J ′ ( n ) { ψ Q,ρ j ( ω j ; W j  } ) 1  t H − (0 ,β ; B ) >t H + (0 ,ω ; W )  +  min i ∈ I ′ ( n ) { ψ Q,ρ i ( β i , B i ) } − max j ∈ J ′ ( n ) { ψ Q,ρ j ( ω j ; W j ) }  + 1  t H − (0 ,β ; B )= t H + (0 ,ω ; W )   d H d − 1 + o ( a 2 ) . The fo llo wing notation is used in the theorem and its pro of: t X − ( a, β ; B ) = min i ∈ I { t X − ( a, β i ; B i ) } t X + ( a, ω ; W ) = max j ∈ J { t X + ( a, ω j ; W j ) } . 21 Moreo v er, I ′ , J ′ are the index sets I ′ ( n ) = { i 0 ∈ I | t H n − (0 , β ; B ) = t H n − (0 , β i 0 ; B i 0 ) } J ′ ( n ) = { j 0 ∈ J | t H n + (0 , ω ; W ) = t H n + (0 , ω j 0 ; W j 0 ) } and ψ Q ( x ) ,ρ i ( β i ; B i ) = ψ Q ( x ) ,ρ i ( β i ; b i ( n )) ψ Q ( x ) ,ρ j ( ω j ; W j ) = ψ Q ( x ) ,ρ j ( ω j ; w j ( n )) where b i ( n ) ∈ B i and w j ( n ) ∈ W j are unique with h ( B i , n ) = h b i ( n ) , n i and h ( ˇ W j , n ) = −h w j ( n ) , n i , resp ectiv ely . Pr o of. F or an r - regular set X , the formula (3.1) simplifi es for 2 aD < r to the W eyl tub e form ula Z R d 1 ξ − 1 ∂ X ( A ) f X a ( x ) g X a ( x ) dx = d X m =1 Z ∂ X ∩ A Z r − r t m − 1 f X a ( x + tn ) g X a ( x + tn ) dts m − 1 ( k ) H d − 1 ( dx ) where D is c hosen as in the the pro of of The orem 3.3 and s m ( k ) denotes the m th symmetric p olynomial in the principal curv atures. Again a − 2 Z r − r t m − 1 f X a ( x + tn ) g X a ( x + tn ) dt ≤ m − 1 a m − 2 (2 D ) m and hence Leb esgue’s theorem yiel ds lim a → 0 a − 2 d X m =2 Z ∂ X Z r − r t m − 1 f X a ( x + tn ) g X a ( x + tn ) dts k − 1 ( k ) H d − 1 ( dx ) = Z ∂ X  lim a → 0 Z 2 D − 2 D tf X a ( x + atn ) g X a ( x + atn ) dt  s 1 ( k ) H d − 1 ( dx ) if the limit of the inner in tegral exists. The m = 1 term will b e treated separately . Again we g et the inequalities (3.2) fo r all a small enough and th us 1 2 (( t B in − ( a, β ; B ) + ) 2 − ( t R d \ B out + ( a, ω ; W ) + ) 2 ) 1  t B in − ( a,β ; B ) >t R d \ B out + ( a,ω ; W )  = Z 2 D 0 tf R d \ B out a ( x + atn ) g B in a ( x + atn ) dt ≤ Z 2 D 0 tf X a ( x + atn ) g X a ( x + atn ) dt (5.1) ≤ Z 2 D 0 tf B in a ( x + atn ) g R d \ B out a ( x + atn ) dt = 1 2 (( t R d \ B out − ( a, β ; B ) + ) 2 − ( t B in + ( a, ω ; W ) + ) 2 ) 1  t R d \ B out − ( a,β ; B ) >t B in + ( a,ω ; W )  22 so (3.5) forces the middle in tegral to conv erge to 1 2 (( t H − (0 , β ; B ) + ) 2 − ( t H + (0 , ω ; W ) + ) 2 ) 1 { t H − (0 ,β ; B ) >t H + (0 ,ω ; W ) } . The inte gration ov er [ − 2 D, 0] is similar, except t he inequalities in (5.1) are switc hed. It remains to determine the asymptotics of a − 1 Z ∂ X ∩ A Z 2 D − 2 D  f X a ( x + atn ) g X a ( x + atn ) dt − ( t H − (0 , β ; B ) − t H + (0 , ω ; W )) +  H d − 1 ( dx ) . (5.2) The pro of of Theorem 3.3 yields M , ε > 0 dep ending only on r , ρ , and D suc h that ( t H − (0 , β , B ) − t H + (0 , ω ; W ) − 2 M a ) + ≤ Z D − D f X a ( x + atn ) g X a ( x + atn ) dt ≤ ( t H − (0 , β ; B ) − t H + (0 , ω ; W ) + 2 M a ) + for all a < ε . Hence a − 1  Z 2 D − 2 D f X a ( x + atn ) g X a ( x + atn ) dt − ( t H − (0 , β ; B ) − t H + (0 , ω ; W )) +  ≤ 2 M , (5.3) allo wing us to apply Leb esgue’s theorem to (5.2). Since θ X a ( · ; s ) is decreasing b y Lemma 5.3, Z 2 D − 2 D f X a ( x + atn ) g X a ( x + atn ) dt = ( t X − ( a, β ; B ) − t X + ( a, ω ; W )) + . By Lemm a 5.6,    t X − ( a, β i ; B ) − t X + ( a, ω j ; W )  + −  t Q − ( a, β i ; B ) − t Q + ( a, ω j ; W )  +   ≤   t Q − ( a, β i ; B ) − t X − ( a, β i ; B )   +   t Q + ( a, ω j ; W ) − t X + ( a, ω j ; W )   ≤ max i ∈ I sup b ∈ B i {| t Q ( a, β i ; b ) − t X ( a, β i ; b ) | } + max j ∈ J sup w ∈ W j {| t Q ( a, ω j ; w ) − t X ( a, ω j ; w ) |} ≤ 2 M λ ( a ) . Hence the limit of (5.3) equals the limit of a − 1 (( t Q − ( a, β ; B ) − t Q + ( a, ω ; W )) + − ( t H − (0 , β ; B ) − t H + (0 , ω ; W )) + ) . (5.4) The last part of Lemma 5.5 yields t ha t || t Q − ( a, β ; B ) − t Q + ( a, ω ; W ) | − | t H − (0 , β ; B ) − t H + (0 , ω ; W ) || ≤ | t Q − ( a, β ; B ) − t H − (0 , β ; B ) | + | t Q + ( a, ω ; W ) − t H + (0 , ω ; W ) | ≤ 2 M a so that (5.4) equals a − 1  ( t Q − ( a, β ; B ) − t H − (0 , β ; B ) − ( t Q + ( a, ω ; W ) − t H + (0 , ω ; W ))) 1 { t H − (0 ,β ; B ) >t H + (0 ,ω ; W ) } + ( t Q − ( a, β ; B ) − t H − (0 , β ; B ) − ( t Q + ( a, ω ; W ) − t H + (0 , ω ; W ))) + 1 { t H − (0 ,β ; B )= t H + (0 ,ω ; W ) }  23 for suffi cien tly small a . As B i is strictly conv ex, there is a unique b i ∈ B i with h ( B i , n ) = h b i , n i . In particular, min i ∈ I inf b ∈ B i { t H (0 , β i ; b ) } = min i ∈ I t H (0 , β i ; b i ) = t H (0 , β i 0 ; b i 0 ) (5.5) for all i 0 ∈ I ′ ( n ) . Since b 7→ t Q ( a, β i ; b ) is con tin uous and B i is compact, there is a b i ( a ) ∈ B i for ev ery a suc h that inf b ∈ B i { t Q ( a, β i ; b ) } = t Q ( a, β i ; b i ( a )) . On the other hand, Lemm a 5.5 yield s an M > 0 suc h that for all b ∈ B i , | t Q ( a, β i ; b ) − t H (0 , β i ; b ) | ≤ M a. Th us t Q ( a, β i ; b i ( a )) ≤ t Q ( a, β i ; b i ) implies that 0 ≤ t H (0 , β i ; b i ( a )) − t H (0 , β i ; b i ) = h b i , n i − h b i ( a ) , n i ≤ 2 M a. Strict con vex ity and compactness of B i th us implies that lim a → 0 b i ( a ) = b i and again con tin uit y yields lim a → 0 t Q − ( a ; β i ; B i ) = lim a → 0 t Q ( a, β i ; b i ( a )) = t Q (0 , β i ; b i ) = t H − (0 , β i ; B i ) . Using (5.5), this yields t Q − ( a, β i ; B ) − t H − (0 , β i ; B ) = min i ∈ I ′ ( n ) { t Q − ( a, β i ; B i ) − t H − (0 , β i ; B i ) } for a sufficien tly small. On the other hand, Lemm a 5.5 sho ws that there are a ′ , a ′′ ∈ [0 , a ] suc h that t Q ( a, β i ; b i ) − t H (0 , β i ; b i ) = a d da t Q ( a, β i ; b i ) | a = a ′ t Q ( a, β i ; b i ( a )) − t H (0 , β i ; b i ( a )) = a d da t Q ( a, β i ; b i ( a )) | a = a ′′ . Subtracting thes e equations and using t H (0 , β i ; b i ) ≤ t H (0 , β i ; b i ( a )) yields 0 ≤ a − 1 ( t Q ( a, β i , b i ) − t Q ( a, β i ; b i ( a ))) ≤ d da t Q ( a, β i ; b i ) | a = a ′ − d da t Q ( a, β i ; b i ( a )) | a = a ′′ . The right hand side g o es to zero for a → 0 by con tin uit y of ( a, b ) 7→ d da t Q ( a, β i ; b ) , so lim a → 0 a − 1 ( t Q − ( a, β i ; B ) − t H − (0 , β i ; B )) = min i ∈ I ′ ( n ) { lim a → 0 a − 1 ( t Q − ( a, β i ; B i ) − t H − (0 , β i ; B i )) } = min i ∈ I ′ ( n ) { lim a → 0 a − 1 ( t Q ( a, β i ; b i ( a )) − t H (0 , β i ; b i )) } = min i ∈ I ′ ( n ) { lim a → 0 a − 1 ( t Q ( a, β i ; b i ) − t H (0 , β i ; b i )) } = min i ∈ I ′ ( n ) { ψ Q,ρ i ( β i ; b i ) } . The W terms in (5.4) are handled similarly , completing the proof . 24 The next theorem is a mo dification intende d for estimators of the t yp e (2.2). F or n ∈ S d − 1 , let ν n b e the signed measure ν n = ν 1 n − ν 2 n where ν 1 n is the Leb esgue–Stieltjes measure on the interv a l (0 , 1) defined by the function β 7→ − 1 2 ( ϕ ρ ( β , n ) + ) 2 and ν 2 n the Leb esgue–Stieltjes measure defined by the function β 7→ 1 2 ( ϕ ρ ( β , n ) − ) 2 . Theorem 5.8. L et X b e a c omp a c t r - r e gular set. L et ρ b e c ontinuous with c omp act supp ort such that al l β ∈ (0 , 1) ar e r e gular values for θ H n 0 ( · ; 0) for al l n ∈ S d − 1 . L et f : [0 , 1] → R have supp f ⊆ [ β , ω ] for some β , ω ∈ (0 , 1) and supp ose f is C 1 on ( β , ω ) w i th f ′ b ounde d and that f + ( β ) = lim x → β + f ( x ) a n d f − ( ω ) = lim x → ω − f ( x ) exist. Then Z R d f ◦ θ X a d H d = a Z (0 , 1) f dµ X + a 2 Z ∂ X  T r II Z (0 , 1) f dν n − 1 2 Z R f ′ ( θ H n 0 ( t ; 0 )) Z n ⊥ II ( z ) ρ ( z − tn ) dz dt + f + ( β ) ψ Q ( β ; 0) − f − ( ω ) ψ Q ( ω ; 0 )  d H d − 1 + o ( a 2 ) . Pr o of. Since | f ◦ θ X a | ≤ M 1 ∂ X ⊕ aB ( D ) for some M > 0 if supp ρ ⊆ B ( D ) , w e still ha ve the form ula a − 2 Z R d f ◦ θ X a ( z ) d z = a − 2 d X m =1 Z ∂ X Z aD − aD t m − 1 f ◦ θ X a ( x + tn ) dts m − 1 ( k ) H d − 1 ( dx ) and the same argumen ts a s in the pro of of Theorem 5.7 sh ow that Leb esgue’s the- orem can b e applied to determine the limit of terms with m ≥ 2 and that a ll terms with m ≥ 3 v anish a symptotically . F or a Bor el set A ⊆ (0 , 1) , let ν 1 n,a ( A ) = Z D 0 t 1 A ∩ ( β ,ω ) ( θ X a ( x + atn )) dt. This defines a measure concen trated on ( β , ω ) where it coincides with the Leb esgue- Stieltjes measure determined b y the function α 7→ − ( t X ( a, α ; 0) + ) 2 . It follo ws from the pro of of Theorem 5 .7 and the same ar g umen ts as in the pro of of C orollary 4.7 that ν 1 n,a con v erges we akly to ν 1 n and hence lim a → 0 Z D 0 tf ◦ θ X a ( x + atn ) dt = lim a → 0 Z (0 , 1) f dν 1 n,a = Z (0 , 1) f dν 1 n . The integration ov er [ − r , 0 ] is handled similarly , sho wing that lim a → 0 Z D − D tf ◦ θ X a ( x + atn ) dt = Z (0 , 1) f dν n . 25 It remains to consider the m = 1 term. The assumptions on ϕ ( · , n ) − 1 = θ H 0 ( · ; 0) ensures Z (0 , 1) f dµ n = Z 1 0 f ( β ) d dβ ϕ ( β , n ) dβ = Z D − D f ◦ θ H 0 ( t ; 0 ) dt. Th us we m ust determine the limit of a − 1 Z ∂ X Z D − D ( f ◦ θ X a ( t ; 0 ) − f ◦ θ H 0 ( t ; 0 )) dtd H d − 1 . Note that | f ◦ θ X a ( t ; 0 ) − f ◦ θ H 0 ( t ; 0 ) | ≤ sup | f ′ || θ X a ( t ; 0 ) − θ H 0 ( t ; 0 ) | 1 A 1 + sup | f | 1 R \ A 1 ∪ A 2 where A 1 = { t ∈ ( − D, D ) | θ X a ( t ; 0 ) , θ H 0 ( t ; 0 ) ∈ ( β , ω ) } A 2 = { t ∈ ( − D, D ) | θ X a ( t ; 0 ) , θ H 0 ( t ; 0 ) / ∈ ( β , ω ) } . By the pro o f of Theorem 3.3, H 1 ( R \ A 1 ∪ A 2 ) = | t X ( a, β ; 0) − t H (0 , β ; 0) | + | t X ( a, ω ; 0) − t H (0 , ω ; 0) | ≤ M 1 a where M 1 can be c hosen indep enden tly of x b y r -regularity . Moreo v er, | θ X a ( t ; 0 ) − θ H 0 ( t ; 0 ) | ≤ max { | θ B in a ( t ; 0 ) − θ H 0 ( t ; 0 ) | , | θ R d \ B out a ( t ; 0 ) − θ H 0 ( t ; 0 ) |} ≤ a − d sup ρ Z aB d − 1 ( D ) ( r − p r 2 − | z | 2 ) dz ≤ M 2 a where M 2 is again uniform b y r - regularit y . Henc e a − 1     Z D − D ( f ◦ θ X a ( t ; 0 ) − f ◦ θ H 0 ( t ; 0 )) dt     ≤ 2 D sup | f ′ | M 2 + sup | f | M 1 and t hus w e can apply Leb esgue’s theorem to the m = 1 term as well if only w e can determine the limit of a − 1 Z D − D ( f ◦ θ X a ( t ; 0 ) − f ◦ θ H 0 ( t ; 0 )) dt. (5.6) By Lemm a 5.6, a − 1 Z D − D | f ◦ θ X a ( t ; 0 ) − f ◦ θ Q a ( t ; 0 ) | dt ≤ a − 1 Z D − D (sup | f ′ || θ X a ( t ; 0 ) − θ Q a ( t ; 0 ) | 1 B 1 + sup | f | 1 R \ ( B 1 ∪ B 2 ) ) dt ≤ 2 D M 3 a − 1 λ ( a ) + a − 1 sup | f |H 1 ( R \ ( B 1 ∪ B 2 )) ≤ 2 D M 3 a − 1 λ ( a ) + M 4 a − 1 λ ( a ) . 26 for some M 3 , M 4 > 0 where B 1 = { t ∈ ( − D , D ) | θ X a ( t ; 0 ) , θ Q a ( t ; 0 ) ∈ ( β , ω ) } B 2 = { t ∈ ( − D , D ) | θ X a ( t ; 0 ) , θ Q a ( t ; 0 ) / ∈ ( β , ω ) } . Hence the limit of (5.6) equals the limit of a − 1 Z D − D ( f ◦ θ Q a ( t ; 0 ) − f ◦ θ H 0 ( t ; 0 )) dt. As in the pro of of Lemma 5.5, θ Q a ( t ; 0 ) extends to a C 1 function β ( a, t ; 0) for ( a, t ) ∈ R 2 . Introduce the follo wing sets: C 1 ( a ) = { t ∈ ( − D , D ) | β ( a, t ; 0) , β (0 , t ; 0) ∈ ( β , ω ) } C 1 β ( a ) = { t ∈ ( − D , D ) | β (0 , t ; 0 ) ≤ β < β ( a, t ; 0) } C 2 β ( a ) = { t ∈ ( − D , D ) | β ( a, t ; 0) ≤ β < β (0 , t ; 0) } . First consider | f ◦ β ( a, t ; 0) − f ◦ β (0 , t ; 0) | 1 C 1 ( a ) ≤ a sup | f ′ | sup ( a,t ) ∈ B n d da β ( a, t ; 0) o 1 C 1 ( a ) where B is a compact neigh b orho o d of { 0 } × ϕ ([ ω , β ] , n ) . It fo llows that lim a → 0 a − 1 Z C 1 ( a ) ( f ◦ θ Q a ( t ; 0 ) − f ◦ θ H 0 ( t ; 0 )) dt = Z R lim a → 0 1 C 1 ( a ) a − 1 ( f ◦ β ( a, t ; 0) − f ◦ β (0 , t ; 0)) dt = Z R 1 C 1 (0) d da f ◦ β ( a, t ; 0) | a =0 dt = − 1 2 Z ϕ (( β ,ω ) ,n ) f ′ ( θ H 0 ( t ; 0 )) Z n ⊥ II ( z ) ρ ( z − tn ) dz d t. Next C 1 β ( a ) ∪ C 2 β ( a ) is the in terv al [ t H (0 , β ; 0) 1 C 1 β ( a ) + t Q ( a, β ; 0) 1 C 2 β ( a ) , t H (0 , β ; 0) 1 C 2 β ( a ) + t Q ( a, β ; 0) 1 C 1 β ( a ) ] and 1 C 1 β ( a ) ∪ C 2 β ( a ) ( f ◦ β ( a, t ; 0) − f ◦ β (0 , t ; 0)) = f ◦ β ( a, t ; 0) 1 C 1 β ( a ) − f ◦ β (0 , t ; 0) 1 C 2 β ( a ) . Let ε > 0 and t ∈ C 1 β ( a ) . Then | t − t H (0 , β ; 0) | ≤ | t H (0 , β ; 0) − t Q ( a, β ; 0) | ≤ M 4 a and hence | f ◦ β ( a, t ; 0) − f ◦ β (0 , t ; 0) − f + ( β ) | = | f ◦ β ( a, t ; 0) − f + ( β ) | ≤ ε M 4 27 for a sufficien tly small, since β is contin uous. A similar argumen t for t ∈ C 2 β ( a ) yields a − 1     Z C 1 β ( a ) ∪ C 2 β ( a ) ( f ◦ β ( a, t ; 0) − f ◦ β (0 , t ; 0) − f + ( β )( 1 C 1 β ( a ) ( t ) − 1 C 2 β ( a ) ( t ))) dt     = a − 1     Z t Q ( a,β ;0) t H (0 ,β ;0) ( f ◦ β ( a, t ; 0) + f ◦ β ( 0 , t ; 0) − f + ( β )) dt     ≤ a − 1     Z t Q ( a,β ;0) t H (0 ,β ;0) ε M 4 dt     ≤ ε for all a sufficien tly small. It follo ws that lim a → 0 a − 1 Z C 1 ( a ) ∪ C 2 ( a ) ( f ◦ β ( a, t ; 0) − f ◦ β (0 , t ; 0)) dt = lim a → 0 a − 1 Z t Q ( a,β ;0) t H (0 ,β ;0) f + ( β ) dt = lim a → 0 a − 1 ( t Q ( a, β ; 0) − t H (0 , β ; 0)) f + ( β ) = ψ Q ( β ; 0) f + ( β ) . The ω terms are handled similarly . 6 Applications of the second order form ula 6.1 Thresholding In the case where a grey-scale image is thresholded at lev el β , Theorem 5.7 reduces to: Corollary 6.1. L e t ( B l , W l ) b e a c onfigur ation. L et X and ρ b e as in The o r em 5.7 and let β ∈ (0 , 1) b e a r e gular value for θ H n 0 ( · ; c ) for al l c ∈ B l ∪ W l and n ∈ S d − 1 . Then E N ( β ) a L c l ( X ) = a Z ∂ X ( − h ( B l ⊕ ˇ W l , n )) + d H d − 1 + a 2 Z ∂ X   1 2 (( ϕ ρ ( β , n ) − h ( B l , n )) 2 − ( ϕ ρ ( β , n ) + h ( ˇ W l , n )) 2 ) T r II + min b ∈ B + l ( n ) { ψ Q ( β ; b ) } − max w ∈ W − l ( n ) { ψ Q ( β ; w ) }  1  h ( B l ⊕ ˇ W l ,n ) < 0  +  min b ∈ B + l ( n ) { ψ Q ( β ; b ) } − max w ∈ W − l ( n ) { ψ Q ( β ; w ) }  + 1  h ( B l ⊕ ˇ W l ,n )=0   d H d − 1 + o ( a 2 ) . Here S ± ( n ) is short f o r the supp ort set { s ∈ S | h ( S, ± n ) = h s, ± n i} . Comparing with the form ula [14, Theorem 4.3] for the blac k-a nd- white case, the first order term is the same, whereas the second order term no w depends on ρ and β . Ho w eve r, if ρ is reflection in v ariant a nd β = 1 2 , then ϕ ρ ( β , n ) = 0 and the firs t line in the second order term is the same as in the blac k-and-white case. 28 If supp ρ ⊆ B ( D ) and c ∈ B l ∪ W l , then ϕ ρ ( β , n ) ∈ ( − D , D ) a nd henc e | ψ Q ( x ) ,ρ ( β ; c ) + 1 2 II x ( c ) | ≤ 1 2 r − 1 ( D 2 + 2 d n 2 D ) . Th us, if ρ is concen t r a ted near 0 , so that θ X ( z ) appro ximates the Dirac measure δ z ( X ) , the form ula is close to the correspo nding formul a in the blac k-and-white case. 6.2 First order bias of surface area estimato rs F or surface ar ea estimators, Theorem 5.8 yields: Corollary 6.2. L et X , f , and ρ b e as in Th e or em 5.8. T h en a first or der exp ansion of E ˆ V ( f ) a L c d − 1 ( X ) = a − 1 Z R d f ◦ θ X a d H d − 1 is given by The or em 5.8. In particular if ρ is reflection inv ariant, then 1 − θ H n 0 ( t ; 0 ) = θ H n 0 ( − t ; 0) . If, more- o v er, f satisfies f ( x ) = f (1 − x ) , Z (0 , 1) f dν n = 0 f ′ ( θ H n 0 ( t ; 0 )) = − f ′ ( θ H n 0 ( − t ; 0)) f + ( β ) ψ Q ( x ) ( β ; 0) = f − (1 − β ) ψ Q ( x ) (1 − β ; 0) . Th us the second o rder term in The orem 5.8 v anishes. This yields: Corollary 6.3. F o r X , f , and ρ as in Th e or em 5.8 with ρ r efle ction invariant and f ( x ) = f (1 − x ) , E ˆ V ( f ) a L c d − 1 ( X ) = Z (0 , 1) f dµ X + o ( a ) . Recall that the condition f ( x ) = f (1 − x ) w as already justified b y Corollary 4.9 in order to minimiz e the asymptotic bias. Example 6.4. Assume ρ is rota tion inv ariant. Under t he assumptions of Theo- rem 5.8, Corollary 6 .3 sho ws that for the asymptotically unbiase d estimators (4.3), c ho osing ω = 1 − β yields the best appro ximations in finite high resolution. These estimators ta k e the form ϕ ρ ( β ) − 1 N a L c ( β , 1 − β ) ( X ) . 6.3 Estimation of the in tegrated mean curv ature Similarly , for estimators for V d − 2 , w e obtain: 29 Corollary 6.5. L e t X , f , and ρ b e as in The or em 5 .8. Then E ˆ V ( f ) a L c d − 2 ( X ) = a − 1 Z (0 , 1) f dµ X + Z ∂ X  T r II Z (0 , 1) f dν n − 1 2 Z R f ′ ( θ H n 0 ( t ; 0 )) Z n ⊥ II ( z ) ρ ( z − tn ) dz d t + f + ( β ) ψ Q ( β ; 0) − f − ( ω ) ψ Q ( ω ; 0 )  d H d − 1 + o (1 ) . In p articular, lim a → 0 E ˆ V ( f ) a L c d − 2 ( X ) exists if ρ is r efle ction invariant and f satisfies f ( x ) = − f (1 − x ) . Supp ose ρ is rotation in v arian t and f is as in Theorem 5.8 with f ( x ) = − f (1 − x ) . In particular, ω = 1 − β for some β ∈ (0 , 1 2 ) . Then w e ha v e Z ϕ ( β ) ϕ (1 − β ) f ′ ( θ H 0 ( t ; 0 )) Z n ⊥ II ( z ) ρ ( z − tn ) dz dt = Z ϕ ( β ) − ϕ ( β ) f ′ ( θ H 0 ( t ; 0 )) Z ∞ 0 Z S d − 2 II ( u ) r 2 ρ t ( r ) r d − 2 H d − 2 ( du ) dr dt = κ d − 1 T r II Z ϕ ( β ) − ϕ ( β ) f ′ ( θ H 0 ( t ; 0 )) Z n ⊥ | z | 2 ρ ( z − tn ) dz dt where for a fixed t ∈ R , ρ t is the function ρ ( z − tn ) = ρ t ( | z | ) f or z ∈ n ⊥ . Moreo ve r, f + ( β ) ψ Q ( β ; 0) − f − (1 − β ) ψ Q (1 − β ; 0) = 2 f + ( β ) ψ Q ( β ; 0) = f + ( β ) ϕ ′ ( β ) Z ∞ 0 Z S d − 2 II ( u ) r 2 ρ ϕ ( β ) ( r ) r d − 2 H d − 2 ( du ) dr = κ d − 1 T r II f + ( β ) ϕ ′ ( β ) Z n ⊥ | z | 2 ρ ( z − ϕ ( β ) n ) dz . In tro ducing the constants c 1 = Z (0 , 1) f dν n = Z ϕ ( β ) − ϕ ( β ) tf ◦ θ H 0 ( t ; 0 ) dt c 2 = κ d − 1 f + ( β ) ϕ ′ ( β ) Z n ⊥ | z | 2 ρ ( z − ϕ ( β ) n ) dz c 3 = − κ d − 1 2 Z ϕ ( β ) − ϕ ( β ) f ′ ( θ H 0 ( t ; 0 )) Z n ⊥ | z | 2 ρ ( z − tn ) dz dt, w e obtain: Corollary 6.6. F or X , f , an d ρ as in The or em 5.8 with ρ r otation invariant and f ( x ) = − f (1 − x ) , lim a → 0 E ˆ V ( f ) a L c d − 2 ( X ) = ( c 1 + c 2 + c 3 ) Z ∂ X T r II d H d − 1 = 2 π ( c 1 + c 2 + c 3 ) V d − 2 ( X ) In p articular, ˆ V ( f ) d − 2 is asymptotic al ly unbiase d if and only if c 1 + c 2 + c 3 = (2 π ) − 1 . 30 Example 6.7. Let f ( x ) = ( x − 1 2 ) 1 ( β , 1 − β ) . Then c 1 = Z ϕ ( β ) − ϕ ( β ) t ( θ H 0 ( t ; 0 ) − 1 2 ) dt = Z ϕ ( β ) − ϕ ( β ) t Z − t −∞ Z n ⊥ ρ ( z + sn ) dz dsdt = Z ϕ ( β ) −∞ Z ( − s ) ∧ ϕ ( β ) − ϕ ( β ) tdt Z n ⊥ ρ ( z + sn ) dz ds = Z ϕ ( β ) − ϕ ( β ) 1 2 ( s 2 − ϕ ( β ) 2 ) Z n ⊥ ρ ( z + sn ) dz ds = 1 2 Z ϕ ( β ) − ϕ ( β ) Z n ⊥ s 2 ρ ( z − sn ) d z ds + ϕ ( β ) 2 ( β − 1 2 ) . It fo llows that c 1 + c 2 + c 3 = d 1 ( ϕ ( β )) d ′ 1 ( ϕ ( β )) − 1 d ′ 2 ( ϕ ( β )) − d 2 ( ϕ ( β )) where d 1 ( t ) = 1 2 ( θ H 0 ( t ; 0 ) − θ H 0 ( − t ; 0)) d 2 ( t ) = 1 2 Z t − t Z n ⊥ ( κ d − 1 | z | 2 − s 2 ) ρ ( z − sn ) d z ds. Ho w ev er, d ′ 2 (0) > 0 and d ′ 2 ( t ) < 0 for t 2 ≥ D 2 1+ κ d − 1 where supp ρ ⊆ B ( D ) . By con tin uit y and the fact that d 2 (0) = 0 , d 2 m ust hav e a lo cal maxim um at some t 0 ∈  0 , D √ 1+ κ d − 1  with d 2 ( t 0 ) > 0 . Hence c 1 + c 2 + c 3 6 = 0 for β in some neigh b orho o d of β 0 = θ H 0 ( t 0 ; 0 ) . It follo ws that the function f ( x ) = (2 π ( c 1 + c 2 + c 3 )) − 1  x − 1 2  1 ( β 0 , 1 − β 0 ) ( x ) yields an asymptotically un biased es timator for V d − 2 . If ρ is kno wn, the constan ts c 1 , c 2 , c 3 , and β 0 can b e determined directly b y the a b o v e, o t herwise these constan ts could be determine d e xp erimen tally . Example 6.8. A similar argumen t sho ws that also the estimator N ( β , 1 2 ) − N ( 1 2 , 1 − β ) is asymptotically un biased up to some constan t f a ctor whic h is non-zero fo r a suitable β ∈ (0 , 1 2 ) . This estimator has the same adv an tag e as (4.3) that it can be applie d ev en if the grey-v a lues are only kno wn discretely . 7 Discussion T o judge from the results of this pap er, it see ms that the blurring of digital images should b e considered a help rather than an obstacle to the estimation of in trinsic v ol- umes. The biasedne ss of lo cal algorithms in the blac k-and-white case can be vie w ed 31 as a consequence of the ro t a tional asymm etry of the n × · · · × n pixel configurations when n > 1 . F or n = 1 the re is only one estimator, namel y the v o lume es timator, whic h is well known to be unb iased. In the grey-scale setting, c ho osing n = 1 , th us a v oiding the a symmetry , leads t o a wide range of estimators, allo wing instead an ex- ploitation of the symmetry of a rotation in v ariant PSF to obtain information ab out the lo we r in trinsic v olumes. One should k eep in mind, ho w ev er, that the results of this paper are only asymp- totic and say nothing ab out how the suggested algorithms work in finite resolution. Especially b ecause of the assumptions on the asymptotic b eha viour of the PSF. Moreo v er, it is not p o ssible to sa y muc h fro m the asymptotic results ab out whic h algorithms w ork b est in practice. F or instance, it is not clear how to choose β b est p ossible for the estimator N ( β , 1 − β ) . Th us lo cal grey-scale algorithms should b e care- fully stud ied and teste d in finite res olution b efore b eing tak en in to use. In some practical applications it may b e p ossible to adjust the PSF, for instance if the PSF has the form ρ B . The results of this pa p er could b e used to design measuremen ts suc h that the suggested algorithms apply , for instance b y c ho osing a PSF of the form ρ B with B rotation inv ariant rat her than the classical ρ C 0 . F rom the mathematical viewpo int, the prov en existence of asymptotically un bi- ased estimators for intrins ic v o lumes V q with q = d, d − 1 , d − 2 naturally ra ises the question whether it stops here or generalizes to the remaining V q with q < d − 2 . A pro of w ould probably require some stronger smoothness a ssumptions on b oth X , ρ , and f and maybe a whole differen t approach . A ckno wledgemen ts The author w as suppo r ted b y the Cen tre for Sto c hastic Geometry and A dv anced Bioimaging, funded b y t he Villum F oundation. The author is wishes to thank Markus Kiderlen for helping with the set-up of this researc h pro ject and for useful input along the w ay . References [1] F ederer, H.: Curv ature measures. T r ans. Amer. Math. So c. 93 , 418–491 (1959) [2] Hall, P ., Molc hano v, I.: Corrections for systematic b oundary effects in pixel-based area coun ts. 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