PDE Evolutions for M-Smoothers in One, Two, and Three Dimensions

Abstract

Local M-smoothers are interesting and important signal and image processing techniques with many connections to other methods. In our paper, we derive a family of partial differential equations (PDEs) that result in one, two, and three dimensions as limiting processes from M-smoothers which are based on local order-p means within a ball the radius of which tends to zero. The order p may take any nonzero value $>-1$, allowing also negative values. In contrast to results from the literature, we show in the space-continuous case that mode filtering does not arise for $p \rightarrow 0$, but for $p \rightarrow -1$. Extending our filter class to p-values smaller than $-1$ allows to include, e.g. the classical image sharpening flow of Gabor. The PDEs we derive in 1D, 2D, and 3D show large structural similarities. Since our PDE class is highly anisotropic and may contain backward parabolic operators, designing adequate numerical methods is difficult. We present an $L^\infty$-stable explicit finite difference scheme that satisfies a discrete maximum–minimum principle, offers excellent rotation invariance, and employs a splitting into four fractional steps to allow larger time step sizes. Although it approximates parabolic PDEs, it consequently benefits from stabilisation concepts from the numerics of hyperbolic PDEs. Our 2D experiments show that the PDEs for $p<1$ are of specific interest: Their backward parabolic term creates favourable sharpening properties, while they appear to maintain the strong shape simplification properties of mean curvature motion.

Appendices

Proofs of PDE Approximation Results

1.1 Proof of Proposition 1

1.1.1 Preliminaries: Some Important Integrals

We start by collecting some definite integrals that will be useful in the following. We define for $\varrho \in (0,1)$ and $q\in \mathbb {R}$

$$\begin{aligned} I_q&:= \int \nolimits _{\sqrt{\varrho }}^1 \sqrt{1-\xi ^2}\,\xi ^q~\mathrm {d}\xi \;, \end{aligned}$$

(59)

$$\begin{aligned} S_q&:= \int \nolimits _{\arcsin \sqrt{\varrho }}^{\pi /2}\sin ^q\varphi ~\mathrm {d}\varphi \;. \end{aligned}$$

(60)

With the additional abbreviation

$$\begin{aligned} R_{q}&:= \varrho ^{q/2}\sqrt{1-\varrho } \end{aligned}$$

(61)

we can derive via substituting $\xi =\sin \varphi$ and integration by parts (integrating $\sin ^q\varphi \cos \varphi$ and differentiating $\cos \varphi$)

$$\begin{aligned} I_q&= \int \nolimits _{\arcsin \sqrt{\varrho }}^{\pi /2} \sin ^q\varphi \cos ^2\varphi ~\mathrm {d}\varphi \nonumber \\*&= \frac{1}{q+1}\bigl [\sin ^{q+1}\varphi \cos \varphi \bigr ]_{\arcsin \sqrt{\varrho }}^{\pi /2} +\frac{1}{q+1}S_{q+2} \nonumber \\&= \frac{-1}{q+1}R_{q+1}+\frac{1}{q+1}S_{q+2} \end{aligned}$$

(62)

for $q\ne -1$. Moreover, we have by $1=\sin ^2\varphi +\cos ^2\varphi$

$$\begin{aligned} S_q&= \int \nolimits _{\arcsin \sqrt{\varrho }}^{\pi /2} \sin ^q\varphi \cos ^2\varphi ~\mathrm {d}\varphi + S_{q+2} \nonumber \\&=\frac{-1}{q+1}R_{q+1}+\frac{q+2}{q+1}S_{q+2} \end{aligned}$$

(63)

for $q\ne -1$ which allows to transform $S_q$ into $S_{q+2}$ and vice versa.

From (62) we can obtain thereby

$$\begin{aligned} I_{p-4}&= \frac{-1}{p-3}R_{p-3} + \frac{-1}{(p-1)(p-3)}R_{p-1} \nonumber \\&\quad +\, \frac{-p}{(p+1)(p-1)(p-3)}R_{p+1} \nonumber \\&\quad +\, \frac{(p+2)p}{(p+1)(p-1)(p-3)}S_{p+2}\;, \end{aligned}$$

(64)

$$\begin{aligned} I_{p-2}&= \frac{-1}{p-1}R_{p-1} + \frac{-1}{(p+1)(p-1)}R_{p+1} \nonumber \\&\quad +\,\frac{p+2}{(p+1)(p-1)}S_{p+2}\;, \end{aligned}$$

(65)

$$\begin{aligned} I_p&= \frac{-1}{p+1}R_{p+1} + \frac{1}{p+1}S_{p+2}\;, \end{aligned}$$

(66)

$$\begin{aligned} I_{p+2}&= \frac{-1}{p+4}R_{p+3} + \frac{1}{p+4}S_{p+2}\;, \end{aligned}$$

(67)

for real p with exception of some odd integers. Note that also for the exceptional values (where some of the denominators become zero) the integrals exist.

1.1.2 Regular Points: Ansatz via Taylor Expansion

Let the image u and mean order p be given as in the proposition. Assume w.l.o.g. that the regular location $\varvec{x}_0$ is (0, 0) with $u(0,0)=0$, and that the gradient of u at (0, 0) is in the positive x direction, i.e. $u_x>0$, $u_y=0$. Let a neighbourhood radius $\varrho >0$ be given, and denote the closed (Euclidean) $\varrho$-neighbourhood of (0, 0) by $\mathrm {D}_{\varrho }$.

Using Taylor expansion of u up to third order, we can write for $(x,y)\in \mathrm {D}_\varrho$

$$\begin{aligned} u(x,y)&= \alpha \bigl (x + \beta x^2 + \gamma xy + \delta y^2 + \varepsilon _0x^3+\varepsilon _1x^2y \nonumber \\&\quad +\,\varepsilon _2xy^2+\varepsilon _3y^3\bigr ) + \mathcal {O}\bigl ((x+y)^4\bigr ) \end{aligned}$$

(68)

where $\alpha =u_x$, $2\beta = u_{xx}/u_x$, $\gamma = u_{xy}/u_x$, $2\delta =u_{yy}/u_x$.

We assume that $\varrho$ is chosen small enough such that $u_x$ is positive everywhere in $\mathrm {D}_\varrho$, each level set of u within the disc $\mathrm {D}_\varrho$ is either a smooth line connecting two points at the circular boundary of the disc, or one of two single points on the boundary of $\mathrm {D}_\varrho$ where u takes its maximum and minimum on $\mathrm {D}_\varrho$, respectively.

The order-p mean of u within $\mathrm {D}_\varrho$ is the minimiser of

$$\begin{aligned} E_0(\mu ) := \mathrm {sgn}\,(p) \iint \nolimits _{\mathrm {D}_\varrho }|u(x,y)-\mu |^p~\mathrm {d}y~\mathrm {d}x \;. \end{aligned}$$

(69)

By some rough estimates one can conclude that for $\varrho \rightarrow 0$, $\mu \sim \varrho ^2$. We substitute therefore

$$\begin{aligned} x = \varrho \xi \;, \quad y = \varrho \eta \;, \quad \mu = \varrho ^2\alpha \kappa \;, \quad u(x,y) = \varrho \alpha \omega (\xi ,\eta ) \end{aligned}$$

(70)

and obtain

$$\begin{aligned} E_0(\mu )&= \mathrm {sgn}\,(p) \varrho ^{p+2}\alpha ^p E(\kappa ) \;, \end{aligned}$$

(71)

$$\begin{aligned} E(\kappa )&= \iint \nolimits _{\mathrm {D}_1}|\omega -\kappa \varrho |^p ~\mathrm {d}\eta ~\mathrm {d}\xi \;, \end{aligned}$$

(72)

$$\begin{aligned} \omega (\xi ,\eta )&=\xi +\beta \xi ^2\varrho +\gamma \xi \eta \varrho +\delta \eta ^2\varrho +\varepsilon _0\xi ^3\varrho ^2+\varepsilon _1\xi ^2\eta \varrho ^2 \nonumber \\&\quad +\,\varepsilon _2\xi \eta ^2\varrho ^2+\varepsilon _3\eta ^3\varrho ^2 +\mathcal {O}\bigl (\varrho ^3(\xi +\eta )\bigr ) \;. \end{aligned}$$

(73)

In the following we focus therefore on finding the extremum of E (minimum for $p>0$, maximum for $p<0$).

1.1.3 Separation of the Integral

The integral E from (72) can be reorganised into a nested integration where the inner integral integrates along a level line of $\omega$ going through $(\xi ,0)$, and the outer integral then integrates along the $\xi$ axis. We have

$$\begin{aligned} E(\kappa )&= \int \nolimits _{-1}^{1} \left( \, \int \nolimits _{\eta ^*_-(\xi )}^{\eta ^*_+(\xi )} \frac{1}{\frac{\partial \omega }{\partial \xi }\bigl (\tilde{\xi }(\eta ),\eta \bigr )} ~\mathrm {d}\eta \right) \nonumber \\&\quad \times |\omega (\xi ,0)-\kappa \varrho |^p \frac{\partial \omega }{\partial \xi }(\xi ,0) ~\mathrm {d}\xi +\mathcal {O}(\varrho ^3) \end{aligned}$$

(74)

where $\tilde{\xi }$ is a function of $\eta$ that describes the level line of $\omega$ that goes through $(\xi ,0)$, and reaches the boundary of $\mathrm {D}_1$ at $\eta ^*_+>0$ and $\eta ^*_-<0$. (Note that the fact that $\omega _\xi$ is positive throughout $\mathrm {D}_1$ implies that the level line through $(\xi ,0)$ can be described in this way.)

The error term $\mathcal {O}(\varrho ^3)$ results from the neglection of those level lines near the maximum and minimum of $\omega$ within $\mathrm {D}_1$ that do not reach the $\xi$ axis within $\mathrm {D}_1$.

In (74), the inner integral

$$\begin{aligned} V(\xi )&:= \int \nolimits _{\eta ^*_-(\xi )}^{\eta ^*_+(\xi )} \frac{1}{\frac{\partial \omega }{\partial \xi }\bigl (\tilde{\xi }(\eta ),\eta \bigr )} ~\mathrm {d}\eta \end{aligned}$$

(75)

measures the density of the value $\omega (\xi ,0)$ in the overall distribution of $\omega$ values within $\mathrm {D}_1$ by integrating along the level line $\tilde{\xi }(\eta )$ with $\eta$ as integration parameter the inverse density of level lines in $\xi$ direction. It is important here that the inverse density of level lines is measured in a direction perpendicular to that of integration. The density of level lines in $\xi$ direction is exactly the derivative $\partial \omega /\partial \xi$ taken at the point $(\tilde{\xi },\eta )$, i.e. the denominator of the integrand.

Integrating the quantity V multiplied with the penaliser $|\omega -\kappa \varrho |^p$ would directly yield $E(\kappa )$ if the integration were carried out w.r.t. $\omega$. We prefer, however, to keep the integration over $\xi$ in order to avoid plugging in the inverse function of $\omega (\xi )\equiv \omega (\xi ,0)$ everywhere in the expressions. This is compensated by the factor $(\partial \omega /\partial \xi )(\xi ,0)$ placed at the end of the integrand of (74) that represents just the substitution of $\omega$ with $\xi$ (along the $\xi$ axis $\eta =0$) as integration variable.

For ease of evaluation, we combine in the following the substitution factor with the weight $V(\xi )$ in one single expression:

$$\begin{aligned} W(\xi )&:= \frac{\partial \omega }{\partial \xi }(\xi ,0) \, V(\xi ) = \int \nolimits _{\eta ^*_-(\xi )}^{\eta ^*_+(\xi )} \frac{\frac{\partial \omega }{\partial \xi }(\xi ,0)}{\frac{\partial \omega }{\partial \xi }\bigl (\tilde{\xi }(\eta ),\eta \bigr )} ~\mathrm {d}\eta \end{aligned}$$

(76)

1.1.4 Evaluation of the Inner (Weight) Integral

To evaluate (76), we determine first the level line function $\tilde{\xi }(\eta )$ for given $\xi =\tilde{\xi }(0)$ by using the Taylor expansion (73):

$$\begin{aligned}&\omega (\xi ,0) = \omega (\tilde{\xi }(\eta ),\eta ) \end{aligned}$$

(77)

$$\begin{aligned}&\xi +\beta \xi ^2\varrho +\varepsilon _0\xi ^3\varrho ^2 = \tilde{\xi } +\bigl (\beta \tilde{\xi }^2+\gamma \tilde{\xi }\eta +\delta \eta ^2\bigr )\varrho \nonumber \\&\quad +\bigl (\varepsilon _0\tilde{\xi }^3+\varepsilon _1\tilde{\xi }^2\eta +\varepsilon _2\tilde{\xi }\eta ^2+\varepsilon _3\eta ^3\bigr )\varrho ^2 +\mathcal {O}(\varrho ^3) \end{aligned}$$

(78)

$$\begin{aligned}&\tilde{\xi }(\eta ) = \xi - (\gamma \xi +\delta \eta )\eta \varrho \nonumber \\&\quad +\,\bigl ((2\beta \xi +\gamma \eta )(\gamma \xi +\delta \eta ) -\varepsilon _1\xi ^2-\varepsilon _2\xi \eta -\varepsilon _3\eta ^2\big )\eta \varrho ^2 \nonumber \\&\quad +\,\mathcal {O}(\varrho ^3) \;. \end{aligned}$$

(79)

The $\eta$ coordinates $\eta ^*_{\pm }$ of the end points of the level line are obtained from the condition $\tilde{\xi }^2+{\eta ^*}^2=1$ as

$$\begin{aligned} \eta ^*_{\pm }&= \pm \sqrt{1-\xi ^2} +\bigl (\gamma \xi ^2\pm \delta \xi \sqrt{1-\xi ^2}\bigr )\varrho \nonumber \\&\quad +\bigl (\chi (\xi )\pm \psi (\xi )\sqrt{1-\xi ^2}\bigr )\varrho ^2 +\mathcal {O}(\varrho ^3) \end{aligned}$$

(80)

where

$$\begin{aligned} \chi (\xi )&=\chi _0+\chi _1\xi +\chi _2\xi ^2+\chi _3\xi ^3+\chi _4\xi ^4\;, \end{aligned}$$

(81)

$$\begin{aligned} \psi (\xi )&=\psi _0+\psi _1\xi +\psi _2\xi ^2+\psi _3\xi ^3 \end{aligned}$$

(82)

are polynomials in $\xi$ the exact coefficients of which are not further needed.

Based on the Taylor expansion (73) we obtain

$$\begin{aligned} \frac{\partial \omega }{\partial \xi }(\xi ,\eta )&=1+\bigl (2\beta \xi +\gamma \eta \bigr )\varrho \nonumber \\&\quad +\bigl (3\varepsilon _0\xi ^2+2\varepsilon _1\xi \eta +\varepsilon _2\eta ^2)\varrho ^2 +\mathcal {O}(\varrho ^3) \;, \end{aligned}$$

(83)

$$\begin{aligned} \frac{\partial \omega }{\partial \xi }(\xi ,0)&=1+2\beta \xi \varrho +3\varepsilon _0\xi ^2\varrho ^2 +\mathcal {O}(\varrho ^3) \;, \end{aligned}$$

(84)

and with (79)

$$\begin{aligned} \frac{\partial \omega }{\partial \xi }(\tilde{\xi },\eta )&=1+\bigl (2\beta \xi +\gamma \eta \bigr )\varrho +\bigl (-2\beta \gamma \xi \eta -2\beta \delta \eta \nonumber \\&\quad +3\varepsilon _0\xi ^2+2\varepsilon _1\xi \eta +\varepsilon _2\eta ^2\bigr )\varrho ^2 +\mathcal {O}(\varrho ^3) \;. \end{aligned}$$

(85)

Combining (84) and (85) we have

$$\begin{aligned} \frac{\frac{\partial \omega }{\partial \xi }(\xi ,0)}{\frac{\partial \omega }{\partial \xi }(\tilde{\xi },\eta )}&= 1-\gamma \eta \varrho +\bigl (4\beta \gamma \xi \eta +2\beta \delta \eta ^2+\gamma ^2\eta ^2 \nonumber \\&\quad -\,2\varepsilon _1\xi \eta -\varepsilon _2\eta ^2\bigr )\varrho ^2 +\mathcal {O}(\varrho ^3) \end{aligned}$$

(86)

and therefore

$$\begin{aligned} W(\xi )&= \int \nolimits _{\eta ^*_-}^{\eta ^*_+} ~\mathrm {d}\eta +\bigl (-\gamma \varrho +4\beta \gamma \xi \varrho ^2-2\varepsilon _1\xi \varrho ^2\bigr ) \int \nolimits _{\eta ^*_-}^{\eta ^*_+} \eta ~\mathrm {d}\eta \nonumber \\&\quad +\,\bigl (2\beta \delta +\gamma ^2-\varepsilon _2\bigr )\varrho ^2 \int \nolimits _{\eta ^*_-}^{\eta ^*_+} \eta ^2 ~\mathrm {d}\eta +\mathcal {O}(\varrho ^3)\nonumber \\&= \bigl (\eta ^*_+-\eta ^*_-\bigr ) +\frac{1}{2} \bigl (-\gamma +4\beta \gamma \xi \varrho -2\varepsilon _1\xi \varrho \bigr )\varrho \bigl ({\eta ^*_+}^2-{\eta ^*_-}^2\bigr ) \nonumber \\&\quad +\,\frac{1}{3} \bigl (2\beta \delta +\gamma ^2-\varepsilon _2\bigr )\varrho ^2 \bigl ({\eta ^*_+}^3-{\eta ^*_-}^3\bigr ) +\mathcal {O}(\varrho ^3)\;. \end{aligned}$$

(87)

From (80) one sees that

$$\begin{aligned} \eta ^*_+-\eta ^*_-&= 2\sqrt{1-\xi ^2}+2\delta \xi \sqrt{1-\xi ^2}\varrho \nonumber \\&\quad +\,2\psi (\xi )\sqrt{1-\xi ^2}\varrho ^2 +\mathcal {O}(\varrho ^3)\;, \end{aligned}$$

(88)

$$\begin{aligned} {\eta ^*_+}^2-{\eta ^*_-}^2&= 4\gamma \xi ^2\sqrt{1-\xi ^2}\varrho +\mathcal {O}(\varrho ^2)\;, \end{aligned}$$

(89)

$$\begin{aligned} {\eta ^*_+}^3-{\eta ^*_-}^3&= 2(1-\xi ^2)^{3/2} +\mathcal {O}(\varrho )\;, \end{aligned}$$

(90)

which allows to continue (87) into

$$\begin{aligned} W(\xi )&= 2\sqrt{1-\xi ^2}+2\delta \xi \sqrt{1-\xi ^2}\varrho +2\psi (\xi )\sqrt{1-\xi ^2}\varrho ^2 \nonumber \\&\quad -\, 2\gamma ^2\xi ^2\sqrt{1-\xi ^2}\varrho ^2 \nonumber \\&\quad +\, \frac{2}{3}\bigl (2\beta \delta +\gamma ^2-\varepsilon _2\bigr )(1-\xi ^2)^{3/2} \varrho ^2 +\mathcal {O}(\varrho ^3) \nonumber \\&= \left( 2 +\left( \frac{4}{3}\beta \delta +\frac{2}{3}\gamma ^2-\frac{2}{3}\varepsilon _2 +2\psi _0 \right) \varrho ^2 \right) \sqrt{1-\xi ^2} \nonumber \\&\quad +\, \left( 2\delta +2\psi _1\varrho \right) \varrho \xi \sqrt{1-\xi ^2} \nonumber \\&\quad +\, \left( -2\gamma ^2 -\frac{4}{3}\beta \delta -\frac{2}{3}\gamma ^2+\frac{2}{3}\varepsilon _2 +2\psi _2 \right) \nonumber \\&\quad \times \varrho ^2 \xi ^2\sqrt{1-\xi ^2} \nonumber \\&\quad +\, 2\psi _3\varrho ^2 \xi ^3\sqrt{1-\xi ^2} +\mathcal {O}(\varrho ^3) \nonumber \\&= \Bigl ( (w_{0,0}+w_{0,2}\varrho ^2) +w_1\varrho \xi +w_2\varrho ^2\xi ^2 \nonumber \\&\quad +\,w_3\varrho ^2\xi ^3\Bigr ) \sqrt{1-\xi ^2} +\mathcal {O}(\varrho ^3) \end{aligned}$$

(91)

with

$$\begin{aligned} \left. \begin{aligned} w_{0,0}&= 2\;, \\ w_{0,2}&= \frac{4}{3}\beta \delta +\frac{2}{3}\gamma ^2-\frac{2}{3}\varepsilon _2 +2\psi _0\;, \\ w_1&= 2\delta +2\psi _1\varrho \;, \\ w_2&= -2\gamma ^2 -\frac{4}{3}\beta \delta -\frac{2}{3}\gamma ^2+\frac{2}{3}\varepsilon _2 +2\psi _2 \;, \\ w_3&= 2\psi _3 \;. \end{aligned} \quad \right\} \end{aligned}$$

(92)

1.1.5 Domain Splitting of the Outer Integral

The outer integral of (74), i.e. the integration of $W(\xi )$ with the penaliser function $|\omega -\kappa \varrho |^p$, is now split into four parts.

First, we split the integration interval at $\xi =\nu \varrho$ where $\omega (\nu \varrho )=\kappa \varrho$ to reduce $|\omega -\kappa \varrho |$ to either $\omega -\kappa \varrho$ or $-\omega +\kappa \varrho$ in each subinterval. By (73) one has $\nu =\kappa +\mathcal {O}(\varrho ^2)$.

Second, the density term $W(\xi )$ contains $\sqrt{1-\xi ^2}$ which is not differentiable at $\pm 1$, precluding Taylor expansion of this term near the outer interval boundaries. On the other hand, the p-th power penaliser is for $p\le 1$ not differentiable at 0 and can therefore not be treated by Taylor expansion at the boundary $\nu$ between the two integration intervals. For this reason, we split each of the two intervals again at $|\xi |=\sqrt{\varrho }$. This allows to simplify the integrals in later steps by applying Taylor expansion to either $W(\xi )$ or the penaliser function, safely avoiding the critical regions of each.

As a result, we have

$$\begin{aligned} E(\kappa )&= F_-(\kappa )+G_-(\kappa )+G_+(\kappa )+F_+(\kappa ) + \mathcal {O}(\varrho ^3) \;, \end{aligned}$$

(93)

$$\begin{aligned} F_-(\kappa )&= \int \nolimits _{-1}^{-\sqrt{\varrho }} W(\xi )\,\bigl (-\omega (\xi )+\kappa \varrho \bigr )^p ~\mathrm {d}\xi \nonumber \\&= \int \nolimits _{\sqrt{\varrho }}^1 W(-\xi )\,\bigl (-\omega (-\xi )+\kappa \varrho \bigr )^p ~\mathrm {d}\xi \;, \end{aligned}$$

(94)

$$\begin{aligned} G_-(\kappa )&= \int \nolimits _{-\sqrt{\varrho }}^{\nu \varrho } W(\xi )\,\bigl (-\omega (\xi )+\kappa \varrho \bigr )^p ~\mathrm {d}\xi \nonumber \\&= \int \nolimits _0^{\sqrt{\varrho }+\nu \varrho } W(-(\xi -\nu \varrho ))\nonumber \\&\quad \times \bigl (-\omega (-(\xi -\nu \varrho ))+\kappa \varrho \bigr )^p ~\mathrm {d}\xi \;, \end{aligned}$$

(95)

$$\begin{aligned} G_+(\kappa )&= \int \nolimits _{\nu \varrho }^{\sqrt{\varrho }} W(\xi )\,\bigl (\omega (\xi )-\kappa \varrho \bigr )^p ~\mathrm {d}\xi \nonumber \\&= \int \nolimits _0^{\sqrt{\varrho }-\nu \varrho } W(\xi +\nu \varrho ) \, \bigl (\omega (\xi +\nu \varrho )-\kappa \varrho \bigr )^p ~\mathrm {d}\xi \;, \end{aligned}$$

(96)

$$\begin{aligned} F_+(\kappa )&= \int \nolimits _{\sqrt{\varrho }}^1 W(\xi )\,\bigl (\omega (\xi )-\kappa \varrho \bigr )^p ~\mathrm {d}\xi \;. \end{aligned}$$

(97)

1.1.6 Evaluation of the Outer Integral I

We start by evaluating the integrals $F_{\mp }$. In the following the upper signs refer to $F_-$, the lower ones to $F_+$. In expanding the power $(1+\ldots )^p$ by a Taylor series, it is important to note that $\varrho /\xi$ is of order $\mathcal {O}\bigl (\sqrt{\varrho }\bigr )$ due to the lower integral bound.

$$\begin{aligned} F_{\mp }&= \int \nolimits _{\sqrt{\varrho }}^1 W(\mp \xi )\,\bigl (\mp \omega (\mp \xi )\pm \kappa \varrho \bigr )^p ~\mathrm {d}\xi \nonumber \\&= \int \nolimits _{\sqrt{\varrho }}^1 W(\mp \xi ) \, \bigl (\xi \mp \beta \xi ^2\varrho \pm \kappa \varrho +\varepsilon _0\xi ^3\varrho ^2+\mathcal {O}(\varrho ^3\xi )\bigr )^p ~\mathrm {d}\xi \nonumber \\&= \int \nolimits _{\sqrt{\varrho }}^1 W(\mp \xi ) \xi ^p\,\left( 1\pm \kappa \frac{\varrho }{\xi }\mp \beta \xi \varrho +\varepsilon _0\xi ^2\varrho ^2+\mathcal {O}(\varrho ^3)\right) ^p ~\mathrm {d}\xi \nonumber \\&= \int \nolimits _{\sqrt{\varrho }}^1 W(\mp \xi )\,\xi ^p \, \left( 1 \pm p\kappa \frac{\varrho }{\xi }\mp p\beta \xi \varrho +p\varepsilon _0\xi ^2\varrho ^2 \right. \nonumber \\&\quad \left. +\left( {\begin{array}{c}p\\ 2\end{array}}\right) \kappa ^2\frac{\varrho ^2}{\xi ^2}-2\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta \kappa \varrho ^2 +\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta ^2\xi ^2\varrho ^2 \right. \nonumber \\&\quad \left. \mp \left( {\begin{array}{c}p\\ 3\end{array}}\right) \kappa ^3\frac{\varrho ^3}{\xi ^3} +\left( {\begin{array}{c}p\\ 4\end{array}}\right) \kappa ^4\frac{\varrho ^4}{\xi ^4} +\mathcal {O}(\varrho ^{5/2}) \right) ~\mathrm {d}\xi \nonumber \\&= \int \nolimits _{\sqrt{\varrho }}^1 \Bigl ( (w_{0,0}+w_{0,2}\varrho ^2) \mp w_1\varrho \xi +w_2\varrho ^2\xi ^2 \nonumber \\&\quad \mp w_3\varrho ^2\xi ^3 + \mathcal {O}(\varrho ^3) \Bigr ) \xi ^p\sqrt{1-\xi ^2} \nonumber \\&\quad \times \left( 1 \pm p\kappa \frac{\varrho }{\xi } \mp p\beta \xi \varrho +\left( {\begin{array}{c}p\\ 2\end{array}}\right) \kappa ^2\frac{\varrho ^2}{\xi ^2} \mp \left( {\begin{array}{c}p\\ 3\end{array}}\right) \kappa ^3\frac{\varrho ^3}{\xi ^3} \right. \nonumber \\&\quad \left. +\,p\varepsilon _0\xi ^2\varrho ^2 -2\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta \kappa \varrho ^2 +\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta ^2\xi ^2\varrho ^2 \right. \nonumber \\&\quad \left. +\,\left( {\begin{array}{c}p\\ 4\end{array}}\right) \kappa ^4\frac{\varrho ^4}{\xi ^4} + \mathcal {O}(\varrho ^{5/2}) \right) ~\mathrm {d}\xi \;. \end{aligned}$$

(98)

This gives

$$\begin{aligned} F_-+F_+&= 2\int \nolimits _{\sqrt{\varrho }}^1 w_{0,0} \xi ^p\sqrt{1-\xi ^2} \left( 1+\left( {\begin{array}{c}p\\ 2\end{array}}\right) \kappa ^2\frac{\varrho ^2}{\xi ^2} \right. \nonumber \\&\quad \left. +\,p\varepsilon _0\xi ^2\varrho ^2 -2\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta \kappa \varrho ^2 +\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta ^2\xi ^2\varrho ^2 \right. \nonumber \\&\quad \left. +\left( {\begin{array}{c}p\\ 4\end{array}}\right) \kappa ^4\frac{\varrho ^4}{\xi ^4} + \mathcal {O}(\varrho ^{5/2}) \right) ~\mathrm {d}\xi \nonumber \\&\quad +\,2\int \nolimits _{\sqrt{\varrho }}^1 \Bigl ( w_{0,2} +w_2\xi ^2 \Bigr ) \varrho ^2 \xi ^p\sqrt{1-\xi ^2} \nonumber \\&\quad \times \left( 1 + \mathcal {O}(\varrho ^{1/2}) \right) ~\mathrm {d}\xi \nonumber \\&\quad +\,2\int \nolimits _{\sqrt{\varrho }}^1 \Bigl ( w_1\varrho \xi + \mathcal {O}(\varrho ^2) \Bigr ) \xi ^p\sqrt{1-\xi ^2} \nonumber \\&\quad \times \left( -p\kappa \frac{\varrho }{\xi } +p\beta \xi \varrho + \mathcal {O}(\varrho ^{3/2}) \right) ~\mathrm {d}\xi \nonumber \\&= 2w_{0,0}\left( {\begin{array}{c}p\\ 4\end{array}}\right) \kappa ^4\varrho ^4 I_{p-4} +2 w_{0,0}\left( {\begin{array}{c}p\\ 2\end{array}}\right) \kappa ^2\varrho ^2 I_{p-2} \nonumber \\ {}&\quad {} +2\left( w_{0,0} - 2w_{0,0}\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta \kappa \varrho ^2 + w_{0,2}\varrho ^2 \right. \nonumber \\&\quad \left. -\, w_1p\kappa \varrho ^2 \right) I_p \nonumber \\&\quad +\,2\left( w_{0,0}p\varepsilon _0\varrho ^2+w_{0,0}\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta ^2\varrho ^2 + w_2\varrho ^2 \right. \nonumber \\&\quad \left. +w_1p\beta \varrho ^2 \right) I_{p+2} + \mathcal {O}(\varrho ^{5/2}) \end{aligned}$$

(99)

and by (64)–(67) we obtain

$$\begin{aligned} F_-+F_+&= 2w_{0,0}\left( {\begin{array}{c}p\\ 4\end{array}}\right) \kappa ^4\varrho ^4 \left( \frac{-R_{p-3}}{p-3} + \frac{-R_{p-1}}{(p-1)(p-3)} \right. \nonumber \\&\quad \left. {} +\, \frac{-p\,R_{p+1}}{(p+1)(p-1)(p-3)} \right. \nonumber \\&\quad \left. {} +\, \frac{(p+2)p\,S_{p+2}}{(p+1)(p-1)(p-3)}\right) \nonumber \\&\quad +\,2\left( w_{0,0}\left( {\begin{array}{c}p\\ 2\end{array}}\right) \kappa ^2\varrho ^2 \right) \left( \frac{-R_{p-1}}{p-1} \right. \nonumber \\&\quad \left. {} +\, \frac{-R_{p+1}}{(p+1)(p-1)} +\frac{(p+2)\,S_{p+2}}{(p+1)(p-1)}\right) \nonumber \\&\quad +\,2\left( w_{0,0} - 2w_{0,0}\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta \kappa \varrho ^2 + w_{0,2}\varrho ^2 \right. \nonumber \\&\quad \left. -\, w_1p\kappa \varrho ^2 \right) \left( \frac{-R_{p+1}}{p+1} + \frac{S_{p+2}}{p+1}\right) \nonumber \\&\quad {} +\,2\left( w_{0,0}p\varepsilon _0\varrho ^2+w_{0,0}\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta ^2\varrho ^2 + w_2\varrho ^2 \right. \nonumber \\&\quad \left. +\,w_1p\beta \varrho ^2 \right) \left( \frac{-R_{p+4}}{p+4} + \frac{S_{p+2}}{p+4}\right) \nonumber \\&\quad +\,\mathcal {O}(\varrho ^{5/2}) \end{aligned}$$

(100)

$$\begin{aligned}&= w_{0,0}\frac{2}{p+1} S_{p+2} + \left( w_{0,0}\frac{(p+2)p}{(p+1)}\kappa ^2 \right. \nonumber \\&\quad \left. -\,w_{0,0}\frac{2p(p-1)}{p+1}\beta \kappa +w_{0,2}\frac{1}{p+1} -w_1\frac{p}{p+1}\kappa \right. \nonumber \\&\quad \left. {} +\,w_{0,0}\frac{2p}{p+4}\varepsilon _0 +w_{0,0}\frac{p(p-1)}{(p+4)}\beta ^2 +w_2\frac{2}{p+4} \right. \nonumber \\&\quad \left. +\,w_1\frac{2p}{p+4}\beta \right) \varrho ^2 S_{p+2} \nonumber \\&\quad -\, w_{0,0}\frac{2}{p+1} \varrho ^{(p+1)/2}\sqrt{1-\varrho } \nonumber \\&\quad -\, w_{0,0}p\kappa ^2 \varrho ^{(p+3)/2}\sqrt{1-\varrho } \nonumber \\&\quad -\, \left( w_{0,0}\frac{p(p-1)(p-2)}{12}\kappa ^4 +w_{0,0}\frac{p}{(p+1)}\kappa ^2 \right. \nonumber \\&\quad \left. -w_{0,0}\frac{2p(p-1)}{p+1}\beta \kappa \right) \varrho ^{(p+5)/2}\sqrt{1-\varrho } \nonumber \\&\quad +\,\mathcal {O}(\varrho ^{5/2})\;. \end{aligned}$$

(101)

In the intermediate step (100) the factors $p-1$, $p-3$ occur in the denominators of some terms, which would necessitate the exclusion of $p=1$ and $p=3$. However, we see in (99) that the coefficient $\left( {\begin{array}{c}p\\ 4\end{array}}\right)$ in front of $I_{p-4}$ vanishes for $p=1$ and $p=3$, and similarly $\left( {\begin{array}{c}p\\ 2\end{array}}\right)$ in front of $I_{p-2}$ vanishes for $p=1$, thus sparing the expansion of the respective integrals via (64) and (65). With this consideration, (101) can be obtained also in these cases.

1.1.7 Evaluation of the Outer Integral II

We turn now to evaluating $G_\mp$. After expanding $\omega$ in the penaliser function and cancelling terms due to $\nu =\kappa +\mathcal {O}(\varrho ^2)$ we substitute $\xi =\sqrt{\varrho }\,\zeta$. Using furthermore the Taylor expansion of $\omega$ in $\xi$ direction around $\nu \varrho$,

$$\begin{aligned} \omega (\nu \varrho +\xi )&=\kappa \varrho +(1+2\,\beta \nu \varrho ^2)\xi +\beta \varrho \xi ^2 +\mathcal {O}(\varrho ^3\xi )\;, \end{aligned}$$

(102)

we obtain

(103)

$$\begin{aligned} G_-+G_+&= 2\varrho ^{(p+1)/2} \left( \frac{2}{p+1} +p\nu ^2\varrho - \frac{1}{p+3} \varrho \right) \nonumber \\&\quad +\mathcal {O}(\varrho ^{(p+5)/2})\;. \end{aligned}$$

(104)

1.1.8 Extremum of the Combined Integral

Combining (93), (101) and (104), applying (92) and $\nu =\kappa +\mathcal {O}(\varrho ^2)$ we obtain

$$\begin{aligned} E(\kappa )&= \frac{4}{p+1} S_{p+2} + \left( 2\frac{(p+2)p}{p+1}\kappa ^2 -2\frac{2p(p-1)}{p+1}\beta \kappa \right. \nonumber \\&\qquad \left. {} +\left( \frac{4}{3}\beta \delta +\frac{2}{3}\gamma ^2-\frac{2}{3}\varepsilon _2+2\psi _0\right) \frac{1}{p+1} \right. \nonumber \\&\qquad \left. {} -2\delta \frac{p}{p+1}\kappa +2\frac{2p}{p+4}\varepsilon _0 +2\frac{p(p-1)}{p+4}\beta ^2 \right. \nonumber \\&\qquad \left. {} +\left( -2\gamma ^2-\frac{4}{3}\beta \delta -\frac{2}{3}\gamma ^2+\frac{2}{3}\varepsilon _2 +2\psi _2\right) \frac{2}{p+4} \right. \nonumber \\&\qquad \left. {} +2\delta \frac{2p}{p+4}\beta \right) \varrho ^2 S_{p+2} \nonumber \\&\qquad {} - \frac{4}{p+1} \varrho ^{(p+1)/2}\sqrt{1-\varrho } - 2p\kappa ^2 \varrho ^{(p+3)/2}\sqrt{1-\varrho } \nonumber \\&\qquad {} +2\varrho ^{(p+1)/2} \left( \frac{2}{p+1} +p\kappa ^2\varrho - \frac{1}{p+3} \varrho \right) \nonumber \\&\qquad {} +\mathcal {O}(\varrho ^{5/2}) +\mathcal {O}(\varrho ^{(p+5)/2}) \nonumber \\&= \mathrm {const}(\kappa ) \nonumber \\&\qquad + \left( -\frac{-4p(p-1)}{p+1}\beta \varrho ^2S_{p+2} -\frac{2p}{p+1}\delta \varrho ^2S_{p+2} \right) \kappa \nonumber \\&\qquad + \left( \frac{2(p+2)p}{p+1}\varrho ^2S_{p+2} -2p\varrho ^{(p+3)/2}\sqrt{1-\varrho } \right. \nonumber \\&\qquad \left. +2p\varrho ^{(p+3)/2} \right) \kappa ^2 + \mathcal {O}\bigl (\varrho ^{\min \{(p+5)/2,5/2\}}\bigr ) \nonumber \\&= \mathrm {const}(\kappa ) + \mathcal {O}\bigl (\varrho ^{\min \{(p+5)/2,5/2\}}\bigr ) \nonumber \\&\qquad + \left( -\frac{-4p(p-1)}{p+1}\beta \varrho ^2S_{p+2} -\frac{2p}{p+1}\delta \varrho ^2S_{p+2} \right) \kappa \nonumber \\&\qquad + \left( \frac{2(p+2)p}{p+1}\varrho ^2S_{p+2} \right) \kappa ^2 \;. \end{aligned}$$

(105)

It is worth noting that the $\varrho ^{(p+3)/2}$ contributions cancelling in the last step belong to the integration boundaries of $F_{\mp }$ and $G_{\mp }$ at $\pm \sqrt{\varrho }$.

For $\varrho \rightarrow 0$, the last expression (105) is a quadratic function of $\kappa$ with its apex at

$$\begin{aligned} \kappa&= - \frac{-\frac{-4p(p-1)}{p+1}\beta \varrho ^2S_{p+2} -\frac{2p}{p+1}\delta \varrho ^2S_{p+2}}{2\frac{2(p+2)p}{p+1}\varrho ^2S_{p+2}} \nonumber \\&\quad + \mathcal {O}(\varrho ^{\min \{(p+1)/2,1/2\}}) \nonumber \\&= \frac{p-1}{p+2}\beta + \frac{1}{p+2}\delta + \mathcal {O}(\varrho ^{\min \{(p+1)/2,1/2\}}) \;. \end{aligned}$$

(106)

Due to the sign of the $\kappa ^2$ coefficient in (105) the apex is a minimum for $p>0$ and a maximum for $p<0$. The $\mathrm {sgn}\,(p)$ factor in the original energy function $E_0$ compensates for this such that $E_0$ is always minimised.

1.1.9 Conclusion for Regular Points

From (106) the claim of the proposition for regular points follows by substituting back $\kappa \alpha \varrho ^2=\mu$, $\alpha \beta =u_{xx}/2$, $\alpha \delta =u_{yy}/2$, and noticing that by our ansatz $u_x>0$, $u_y=0$ the coordinates x, y coincide with the geometric coordinates $\eta$, $\xi$ as used in the proposition.

1.1.10 Critical Points

The inequalities for local minima (maxima) are obvious consequences of the fact that for any $\varrho >0$ the mean-p filter value is in the convex hull of values $u(\varvec{x})$, $\varvec{x}\in \mathrm {D}_\varrho (\varvec{x}_0)$. $\square$

1.2 Proof of Proposition 2

With the same substitutions as in the previous proof, the mode of $\omega$ is given by the maximiser of $V(\xi )$. By a slight modification of the calculations of the previous proof one finds

$$\begin{aligned} V(\xi ) = 2\, (1+\delta \xi \varrho -2\beta \xi \varrho )\sqrt{1-\xi ^2} +\mathcal {O}(\xi ^2\varrho ^2) \,. \end{aligned}$$

(107)

Equating $V'(\xi )$ to zero yields $\omega (\xi )=(\delta -2\beta )\varrho +\mathcal {O}(\varrho ^2)$ for the mode. For local minima (maxima), the same reasoning as in the previous proof applies. $\square$

1.3 Proof of Proposition 3

We proceed largely analogous to the proof of Proposition 1 in Appendix A.1. However, the integral decomposition gets simpler since no infinite ascents of the weighting at the integral boundaries $\pm 1$ need to be controlled.

1.3.1 Regular Points: Ansatz via Taylor Expansion

Let the signal u and mean order p be given as in the proposition. Assume w.l.o.g. that the regular location $x_0$ is 0 with $u(0)=0$, and that the derivative of u at 0 is positive, $u_x(0)>0$. Let a neighbourhood radius $\varrho$ be given.

Using Taylor expansion of u up to third order, we obtain for $-\varrho \le x\le \varrho$ the following expression:

$$\begin{aligned} u(x)&= \alpha (x + \beta x^2 + \varepsilon x^3) + \mathcal {O}(x^4) \end{aligned}$$

(108)

where $\alpha =u_x$, $2\beta = u_{xx}/u_x$.

We assume that $\varrho$ is chosen small enough so $u_x$ is positive throughout $[-\varrho ,\varrho ]$, i.e. u is strictly monotonic within this interval. The order-p mean of u within $[-\varrho ,\varrho ]$ is the minimiser of

$$\begin{aligned} E_0(\mu ) := \mathrm {sgn}\,(p) \int \nolimits _\varrho ^\varrho |u(x)-\mu |^p~\mathrm {d}x \;. \end{aligned}$$

(109)

By rough estimates one can again conclude that for $\varrho \rightarrow 0$, $\mu \sim \varrho ^2$. We substitute therefore

$$\begin{aligned} x = \varrho \xi \;, \quad \mu = \varrho ^2\alpha \kappa \;, \quad u(x) = \varrho \alpha \omega (\xi ) \end{aligned}$$

(110)

and obtain

$$\begin{aligned} E_0(\mu )&= \mathrm {sgn}\,(p) \varrho ^{p+1}\alpha ^p E(\kappa ) \;, \end{aligned}$$

(111)

$$\begin{aligned} E(\kappa )&= \int \nolimits _\varrho ^\varrho |\omega -\kappa \varrho |^p ~\mathrm {d}\xi \;, \end{aligned}$$

(112)

$$\begin{aligned} \omega (\xi )&=\xi +\beta \xi ^2\varrho +\varepsilon \xi ^3\varrho ^2 +\mathcal {O}(\varrho ^3\xi ) \;. \end{aligned}$$

(113)

In the following we focus therefore on finding the extremum of E (minimum for $p>0$, maximum for $p<0$).

1.3.2 Domain Splitting of the Integral

We split the integral (112) into two parts, using again the location $\xi =\nu \varrho$ where $\omega (\nu \varrho )=\kappa \varrho$ as splitting point. By (113), one has $\nu =\kappa +\mathcal {O}(\varrho ^2)$. We have then

$$\begin{aligned} E(\kappa )&= F_-(\kappa )+F_+(\kappa )\;, \end{aligned}$$

(114)

$$\begin{aligned} F_-(\kappa )&= \int \nolimits _{-1}^{\nu \varrho } \bigl (\kappa \varrho -\omega (\xi )\bigr )^p~\mathrm {d}\xi \;, \end{aligned}$$

(115)

$$\begin{aligned} F_+(\kappa )&= \int \nolimits _{\nu \varrho }^1 \bigl (\omega (\xi )-\kappa \varrho \bigr )^p~\mathrm {d}\xi \;. \end{aligned}$$

(116)

By substituting the integration variables, one obtains

$$\begin{aligned} F_{\mp }&= \int \nolimits _0^{1\pm \nu \varrho } \bigl (\mp \omega (\mp \xi +\nu \varrho )\pm \kappa \varrho \bigr )^p~\mathrm {d}\xi \end{aligned}$$

(117)

where again the upper and lower signs refer to $F_-$ and $F_+$, respectively.

1.3.3 Evaluation of the Integrals

The Taylor expansion for $\omega$ around $\nu \varrho$ is identical with (102). Inserting this into (117), we have further

$$\begin{aligned} F_{\mp }&= \int \nolimits _0^{1\pm \nu \varrho } \bigl ((1+2\,\beta \nu \varrho ^2)\xi \mp \beta \varrho \xi ^2+\mathcal {O}(\varrho ^3\xi ) \bigr )^p~\mathrm {d}\xi \nonumber \\&= \int \nolimits _0^{1\pm \nu \varrho } \xi ^p\bigl (1+2\,\beta \nu \varrho ^2\mp \beta \varrho \xi +\mathcal {O}(\varrho ^3) \bigr )^p~\mathrm {d}\xi \nonumber \\&= \int \nolimits _0^{1\pm \nu \varrho } \xi ^p\biggl (1+2\,p\beta \nu \varrho ^2\mp p\beta \varrho \xi + \frac{p(p-1)}{2}\beta ^2\varrho ^2\xi ^2 \nonumber \\&\qquad +\mathcal {O}(\varrho ^3) \biggr )~\mathrm {d}\xi \nonumber \\&= \bigl (1+2\,p\beta \nu \varrho ^2+\mathcal {O}(\varrho ^3)\bigr ) \int \nolimits _0^{1\pm \nu \varrho } \xi ^p~\mathrm {d}\xi \nonumber \\&\qquad \mp p\beta \varrho \int \nolimits _0^{1\pm \nu \varrho } \xi ^{p+1}~\mathrm {d}\xi \nonumber \\&\qquad + \frac{p(p-1)}{2}\beta ^2\varrho ^2 \int \nolimits _0^{1\pm \nu \varrho } \xi ^{p+2}~\mathrm {d}\xi \;, \end{aligned}$$

(118)

from which by evaluating the standard integrals, adding $F_-$ and $F_+$ and inserting $\nu =\kappa +\mathcal {O}(\varrho ^2)$ we reach

$$\begin{aligned} E(\kappa )&= F_-(\kappa )+F_+(\kappa ) \nonumber \\&= \mathrm {const}(\kappa ) + p\varrho ^2\left( \kappa ^2-2\frac{p-1}{p+1}\beta \kappa \right) + \mathcal {O}(\varrho ^3) \;. \end{aligned}$$

(119)

The extremum of E is again found as the apex of the quadratic function on the r.h.s., from which the claim for regular points follows.

For critical points, the reasoning from Appendix A.1 applies. $\square$

1.4 Proof of Proposition 4

Assuming again that the regular location for the signal u is $x_0=0$, and u is strictly monotonically increasing and Lipschitz within $[-\varrho ,\varrho ]$, the density of each value u(x) for $-\varrho \le x\le \varrho$ is proportional to $1/u'(x)$. The maximum of these values is reached at $u(-\varrho )$ if u is convex, or $u(\varrho )$ if u is concave. This proves the claim for regular points. If $x_0$ is a local extremum, the density has a pole at $u(x_0)$ and is finite for all other values, making $u(x_0)$ the mode. $\Box$

1.5 Proof of Proposition 5

1.5.1 Regular Points: Ansatz via Taylor Expansion

Let the volume image u and mean order p be given as in the proposition. Assume w.l.o.g. that the regular location $\varvec{x}_0$ is (0, 0, 0) with $u(0,0,0)=0$, and that the gradient of u at (0, 0, 0) is in the positive x direction, i.e. $u_x>0$, $u_y=u_z=0$. Let a neighbourhood radius $\varrho >0$ be given, and denote the closed (Euclidean) $\varrho$-neighbourhood of (0, 0, 0) by $\mathrm {B}_{\varrho }$.

Using Taylor expansion of u up to third order, we can write for $(x,y,z)\in \mathrm {B}_\varrho$ the ansatz

$$\begin{aligned} u(x,y,z)&= \alpha \bigl (x + \beta x^2 + \gamma _0 xy + \gamma _1 xz + \delta _0 y^2 + \delta _1 yz + \delta _2 z^2 \nonumber \\&\quad {} + \varepsilon _0x^3+\varepsilon _{10}x^2y+\varepsilon _{01}x^2z +\varepsilon _{20}xy^2+\varepsilon _{11}xyz \nonumber \\&\quad {} +\varepsilon _{02}xz^2 +\varepsilon _{30}y^3+\varepsilon _{21}y^2z+\varepsilon _{12}yz^2 +\varepsilon _{03}z^3\bigr ) \nonumber \\&\quad {} + \mathcal {O}\bigl ((x+y+z)^4\bigr ) \;. \end{aligned}$$

(120)

We assume that $\varrho$ is chosen small enough such that $u_x$ is positive everywhere in $\mathrm {B}_\varrho$, each level set of u within the ball $\mathrm {B}_\varrho$ is either a smooth surface patch bounded by a closed regular curve on the boundary of the ball, or one of two single points on the boundary of $\mathrm {B}_\varrho$ where u takes its maximum and minimum on $\mathrm {B}_\varrho$, respectively.

The order-p mean of u within $\mathrm {B}_\varrho$ is the minimiser of

$$\begin{aligned} E_0(\mu ) := \mathrm {sgn}\,(p) \iiint \nolimits _{\mathrm {B}_\varrho }|u(x,y,z)-\mu |^p~\mathrm {d}z~\mathrm {d}y~\mathrm {d}x \;. \end{aligned}$$

(121)

Rough estimates again ensure $\mu \sim \varrho ^2$ for $\varrho \rightarrow 0$. Combining an appropriate rescaling with a transition to cylindrical coordinates with the axis in gradient (x) direction, we substitute

$$\begin{aligned}&x = \varrho \xi \;, \quad y = \varrho \eta \cos \varphi \;, \quad z = \varrho \eta \sin \varphi \;, \quad \mu = \varrho ^2\alpha \kappa \;,\nonumber \\ \end{aligned}$$

(122)

$$\begin{aligned}&u(x,y,z) = \varrho \alpha \omega (\xi ,\eta ,\varphi ) \end{aligned}$$

(123)

and obtain

$$\begin{aligned} E_0(\mu )&= \mathrm {sgn}\,(p) \varrho ^{p+3}\alpha ^p E(\kappa ) \;, \end{aligned}$$

(124)

$$\begin{aligned} E(\kappa )&= \iint \nolimits _{\mathrm {D}_1}\int \nolimits _{0}^{2\pi } |\omega (\xi ,\eta ,\varphi )-\kappa \varrho |^p \eta ~\mathrm {d}\varphi ~\mathrm {d}\eta ~\mathrm {d}\xi \;, \end{aligned}$$

(125)

where the integration in cylindrical coordinates has been written using the disc $\mathrm {D}_1$ for the $\xi$, $\eta$ coordinates. The Taylor expansion of u transfers to

$$\begin{aligned} \omega (\xi ,\eta ,\varphi )&=\xi +\beta \xi ^2\varrho +\gamma (\varphi )\xi \eta \varrho +\delta (\varphi )\eta ^2\varrho \nonumber \\&\quad {} +\varepsilon _0\xi ^3\varrho ^2+\varepsilon _1(\varphi )\xi ^2\eta \varrho ^2 +\varepsilon _2(\varphi )\xi \eta ^2\varrho ^2 \nonumber \\&\quad {} +\varepsilon _3(\varphi )\eta ^3\varrho ^2 +\mathcal {O}\bigl (\varrho ^3(\xi +\eta )\bigr ) \;, \end{aligned}$$

(126)

$$\begin{aligned} \gamma (\varphi )&:=\gamma _0\cos \varphi +\gamma _1\sin \varphi \;, \end{aligned}$$

(127)

$$\begin{aligned} \delta (\varphi )&:=\delta _0\cos ^2\varphi +\delta _1\cos \varphi \sin \varphi +\delta _2\sin ^2\varphi \;, \end{aligned}$$

(128)

$$\begin{aligned} \varepsilon _1(\varphi )&:=\varepsilon _{10}\cos \varphi +\varepsilon _{01}\sin \varphi \;, \end{aligned}$$

(129)

$$\begin{aligned} \varepsilon _2(\varphi )&:=\varepsilon _{20}\cos ^2\varphi +\varepsilon _{11}\cos \varphi \sin \varphi +\varepsilon _{02}\sin ^2\varphi \;, \end{aligned}$$

(130)

$$\begin{aligned} \varepsilon _3(\varphi )&:=\varepsilon _{30}\cos ^3\varphi +\varepsilon _{21}\cos ^2\varphi \sin \varphi \nonumber \\&\quad {} +\varepsilon _{12}\cos \varphi \sin ^2\varphi +\varepsilon _{03}\sin ^3\varphi \;. \end{aligned}$$

(131)

We aim again at finding the extremum of E.

1.5.2 Separation of the Integral

Similar to Appendix A.1.3, the integral E from (125) can be reorganised into a nested integration where the inner double integral (in polar coordinates) integrates over a level surface of $\omega$ going through $(\xi ,0,0)$, and the outer integral then integrates along the $\xi$ axis. We have

$$\begin{aligned} E(\kappa )&= \int \nolimits _{-1}^{1} \left( \, \int \nolimits _0^{2\pi } \int \nolimits _0^{\eta ^*(\varphi )} \frac{\eta }{\frac{\partial \omega }{\partial \xi } \bigl (\tilde{\xi }(\eta ),\eta ,\varphi \bigr )} ~\mathrm {d}\eta ~\mathrm {d}\varphi \right) \nonumber \\&\quad {}\times |\omega (\xi ,0,0)-\kappa \varrho |^p \frac{\partial \omega }{\partial \xi }(\xi ,0,0) ~\mathrm {d}\xi +\mathcal {O}(\varrho ^3) \end{aligned}$$

(132)

where $\tilde{\xi }$ is a function of $\eta ,\varphi$ that describes the level set of $\omega$ which goes through $(\xi ,0,0)$, and reaches the boundary of $\mathrm {B}_1$ at $(\eta ^*(\varphi ),\varphi )$. (Note that our initial assumptions on u ensure that the level set can be described in this way.)

Analogously to Appendix A.1.3 we rewrite (132) as

$$\begin{aligned} E(\kappa )&= \int \nolimits _{-1}^1W(\xi )~\mathrm {d}\xi \;, \end{aligned}$$

(133)

$$\begin{aligned} W(\xi )&:= \int \nolimits _0^{2\pi } \int \nolimits _0^{\eta ^*(\varphi )} \eta \, \frac{\frac{\partial \omega }{\partial \xi }(\xi ,0,0)}{\frac{\partial \omega }{\partial \xi } \bigl (\tilde{\xi }(\eta ),\eta ,\varphi \bigr )} ~\mathrm {d}\eta ~\mathrm {d}\varphi \;. \end{aligned}$$

(134)

1.5.3 Evaluation of the Weight Integral

Within any axial plane ($\varphi =\mathrm {const}$), (126) is exactly (73). We can therefore transfer verbatim the analysis from Appendix A.1.4, which leads to the expression (79) for $\tilde{\xi }$, the expression for $\eta ^*_+$ from (80) for $\eta ^*(\varphi )$, and (86) for $\frac{\partial \omega }{\partial \xi }(\xi ,0,0)/\frac{\partial \omega }{\partial \xi } (\tilde{\xi },\eta ,\varphi )$.

Inserting (86) into the inner integral of (134) leads to

$$\begin{aligned} W(\xi ,\varphi )&:= \int \nolimits _0^{\eta ^*(\varphi )} \eta \, \frac{\frac{\partial \omega }{\partial \xi }(\xi ,0,0)}{\frac{\partial \omega }{\partial \xi } \bigl (\tilde{\xi }(\eta ),\eta ,\varphi \bigr )} ~\mathrm {d}\eta \nonumber \\&= \int \nolimits _0^{\eta ^*(\varphi )} \eta ~\mathrm {d}\eta \nonumber \\&\quad +\, \bigl (-\gamma (\varphi )\varrho +2\beta \gamma (\varphi )\xi \varrho ^2 -2\varepsilon _1(\varphi )\xi \varrho ^2\bigr ) \int \nolimits _0^{\eta ^*}\eta ^2~\mathrm {d}\eta \nonumber \\&\quad +\, \bigl (2\beta \delta (\varphi )+\gamma (\varphi )^2-\varepsilon _2(\varphi )\bigr ) \varrho ^2 \int \nolimits _0^{\eta ^*}\eta ^3~\mathrm {d}\eta \nonumber \\&\quad +\, \mathcal {O} (\varrho ^3) \nonumber \\&= \frac{1}{2} \eta ^*(\varphi )^2 \nonumber \\&\quad + \frac{1}{6} (-\gamma _0+2\beta \gamma _0\xi \varrho -2\varepsilon _{10}\xi \varrho ) \varrho \eta ^*(\varphi )^3\cos \varphi \nonumber \\&\quad +\, \frac{1}{6} (-\gamma _1+2\beta \gamma _1\xi \varrho -2\varepsilon _{01}\xi \varrho ) \varrho \eta ^*(\varphi )^3\sin \varphi \nonumber \\&\quad +\, \frac{1}{4} (2\beta \delta _0+\gamma _0^2-\varepsilon _{20})\varrho ^2 \eta ^*(\varphi )^4\cos ^2\varphi \nonumber \\&\quad +\, \frac{1}{2} (2\beta \delta _1+\gamma _0\gamma _1-\varepsilon _{11})\varrho ^2 \eta ^*(\varphi )^4\cos \varphi \sin \varphi \nonumber \\&\quad +\, \frac{1}{4} (2\beta \delta _2+\gamma _1^2-\varepsilon _{02})\varrho ^2 \eta ^*(\varphi )^4\sin ^2\varphi \nonumber \\&\quad +\, \mathcal {O} (\varrho ^3) \;. \end{aligned}$$

(135)

To finally obtain $W(\xi )$, the latter expression needs to be integrated over $\varphi$. From (80) one obtains by lengthy but straightforward calculation

$$\begin{aligned}&\int \nolimits _0^{2\pi }\eta ^*(\varphi )^2~\mathrm {d}\varphi \nonumber \\&= \pi \Bigl ( 2(1-\xi ^2) +2(\delta _0+\delta _2)(1-\xi ^2)\xi \varrho +(\gamma _0^2+\gamma _1^2)\xi ^4\varrho ^2 \nonumber \\&\quad +\,\tfrac{1}{4}(3\delta _0^2+\delta _1^2+3\delta _2^2+2\delta _0\delta _2) \nonumber \\&\quad +\,2\varPsi (\xi )(1-\xi ^2)\varrho ^2 ) \Bigr ) +\mathcal {O}(\varrho ^3)\;, \end{aligned}$$

(136)

where $\varPsi$ is a third-order polynomial in $\xi$ obtained by integrating (82) w.r.t. $\varphi$,

$$\begin{aligned} \varPsi (\xi )&=\frac{1}{\pi }\int \nolimits _0^{2\pi }\psi (\xi ,\varphi )~\mathrm {d}\varphi =\varPsi _0+\varPsi _1\xi +\varPsi _2\xi ^2+\varPsi _3\xi ^3 \;. \end{aligned}$$

(137)

Analogously one obtains

$$\begin{aligned}&\int \nolimits _0^{2\pi }\eta ^*(\varphi )^3\cos \varphi ~\mathrm {d}\varphi =3\pi \gamma _0(1-\xi ^2)\xi ^2\varrho +\mathcal {O}(\varrho ^2) \;, \end{aligned}$$

(138)

$$\begin{aligned}&\int \nolimits _0^{2\pi }\eta ^*(\varphi )^3\sin \varphi ~\mathrm {d}\varphi =3\pi \gamma _1(1-\xi ^2)\xi ^2\varrho +\mathcal {O}(\varrho ^2) \;, \end{aligned}$$

(139)

$$\begin{aligned}&\int \nolimits _0^{2\pi }\eta ^*(\varphi )^4\cos ^2\varphi ~\mathrm {d}\varphi =\pi (1-\xi ^2)^2 +\mathcal {O}(\varrho ) \;, \end{aligned}$$

(140)

$$\begin{aligned}&\int \nolimits _0^{2\pi }\eta ^*(\varphi )^4\cos \varphi \sin \varphi ~\mathrm {d}\varphi =0 +\mathcal {O}(\varrho ) \;, \end{aligned}$$

(141)

$$\begin{aligned}&\int \nolimits _0^{2\pi }\eta ^*(\varphi )^4\sin ^2\varphi ~\mathrm {d}\varphi =\pi (1-\xi ^2)^2 +\mathcal {O}(\varrho ) \;. \end{aligned}$$

(142)

Inserting (135)–(142) into (134) yields after sorting terms, similarly to (91), (92),

$$\begin{aligned} W(\xi )&= \Bigl ( (w_{0,0} + w_{0,2}\varrho ^2) +(w_{1,1} + w_{1,2}\varrho )\varrho \xi \nonumber \\&\quad {} +(w_{2,0} + w_{2,2}\varrho ^2)\xi ^2 +(w_{3,1} + w_{3,2}\varrho )\varrho \xi ^3 \nonumber \\&\quad {} +w_{4,2}\varrho ^2\xi ^4 +w_{5,2}\varrho ^2\xi ^5 \Bigr )\pi +\mathcal {O}(\varrho ^3) \end{aligned}$$

(143)

with

$$\begin{aligned} \left. \begin{aligned} w_{0,0}&= 1 \;, \\ w_{0,2}&= \tfrac{1}{2}\beta (\delta _0+\delta _2) + \tfrac{1}{4}(\gamma _0^2+\gamma _1^2) \\ {}&\quad {} - \tfrac{1}{4}(\varepsilon _{20}+\varepsilon _{02}) + \varPsi _0 \;, \\ w_{1,1}&= \delta _0+\delta _2 \;, \\ w_{1,2}&= \varPsi _1 \;, \\ w_{2,0}&= - 1 \;, \\ w_{2,2}&= - \beta (\delta _0+\delta _2) - \tfrac{3}{2}(\gamma _0^2+\gamma _1^2) + \tfrac{1}{2}(\varepsilon _{20}+\varepsilon _{02}) \\ {}&\quad {} + \tfrac{1}{8}(3\delta _0^2+\delta _1^2+3\delta _2^2+2\delta _0\delta _2) + \varPsi _2 - \varPsi _0 \;, \\ w_{3,1}&= - (\delta _0+\delta _2) \;, \\ w_{3,2}&= \varPsi _3 - \varPsi _1 \;, \\ w_{4,2}&= \tfrac{1}{2}\beta (\delta _0+\delta _2) + \tfrac{9}{4}(\gamma _0^2+\gamma _1^2) - \tfrac{1}{4}(\varepsilon _{20}+\varepsilon _{02}) \\ {}&\quad {} - \tfrac{1}{8}(3\delta _0^2+\delta _1^2+3\delta _2^2+2\delta _0\delta _2) - \varPsi _2 \;, \\ w_{5,2}&= - \varPsi _3 \;. \end{aligned} \quad \right\} \end{aligned}$$

(144)

1.5.4 Domain Splitting of the Outer Integral

As the outer integral of (132) has the same structure as in the 2D case, we use the same domain splitting (93).

1.5.5 Evaluation of the Outer Integral I

The first steps in evaluating the integrals $F_{\mp }$ are as in the 2D case. In (98), the longer expansion (143) has to be used for $W(\mp )$, which then leads to

$$\begin{aligned} F_{\mp }&= \int \nolimits _{\sqrt{\varrho }}^1 \Bigl ( (w_{0,0} + w_{0,2}\varrho ^2) +(w_{1,1} + w_{1,2}\varrho )\varrho \xi \nonumber \\&\quad {} +(w_{2,0} + w_{2,2}\varrho ^2)\xi ^2 +(w_{3,1} + w_{3,2}\varrho )\varrho \xi ^3 \nonumber \\&\quad {} +w_{4,2}\varrho ^2\xi ^4 +w_{5,2}\varrho ^2\xi ^5 +\mathcal {O}(\varrho ^3) \Bigr )\pi \xi ^p \nonumber \\&\quad {}\times \left( 1 \pm p\kappa \frac{\varrho }{\xi } \mp p\beta \xi \varrho +\left( {\begin{array}{c}p\\ 2\end{array}}\right) \kappa ^2\frac{\varrho ^2}{\xi ^2} \mp \left( {\begin{array}{c}p\\ 3\end{array}}\right) \kappa ^3\frac{\varrho ^3}{\xi ^3} \right. \nonumber \\&\quad \left. {} +p\varepsilon _0\xi ^2\varrho ^2 -2\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta \kappa \varrho ^2 +\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta ^2\xi ^2\varrho ^2 \right. \nonumber \\&\quad \left. {} +\left( {\begin{array}{c}p\\ 4\end{array}}\right) \kappa ^4\frac{\varrho ^4}{\xi ^4} + \mathcal {O}(\varrho ^{5/2}) \right) ~\mathrm {d}\xi \;, \end{aligned}$$

(145)

which yields

$$\begin{aligned} F_-+F_+&= 2\pi \int \nolimits _{\sqrt{\varrho }}^1 (w_{0,0} + w_{2,0}\xi ^2) \xi ^p \left( 1+\left( {\begin{array}{c}p\\ 2\end{array}}\right) \kappa ^2\frac{\varrho ^2}{\xi ^2} \right. \nonumber \\&\quad \left. {} +p\varepsilon _0\xi ^2\varrho ^2 -2\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta \kappa \varrho ^2 +\left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta ^2\xi ^2\varrho ^2 \right. \nonumber \\&\quad \left. {} +\left( {\begin{array}{c}p\\ 4\end{array}}\right) \kappa ^4\frac{\varrho ^4}{\xi ^4} + \mathcal {O}(\varrho ^{5/2}) \right) ~\mathrm {d}\xi \nonumber \\&\quad {} +2\pi \int \nolimits _{\sqrt{\varrho }}^1 ( w_{0,2} + w_{2,2} \xi ^2 + w_{4,2} \xi ^4 ) \varrho ^2 \xi ^p \nonumber \\&\quad {}\times \left( 1 + \mathcal {O}(\varrho ^{1/2}) \right) ~\mathrm {d}\xi \nonumber \\&\quad {} +2\pi \int \nolimits _{\sqrt{\varrho }}^1 ( w_{1,1} \xi + w_{3,1} \xi ^3 ) \varrho \xi ^p \nonumber \\&\quad {}\times \left( -p\kappa \frac{\varrho }{\xi } +p\beta \xi \varrho + \mathcal {O}(\varrho ^{3/2}) \right) ~\mathrm {d}\xi \nonumber \\&\quad {} +2\pi \int \nolimits _{\sqrt{\varrho }}^1 ( w_{1,2} \xi + w_{3,2} \xi ^3 + w_{5,2} \xi ^5 ) \varrho ^2 \xi ^p \nonumber \\&\quad {}\times \left( -p\kappa \frac{\varrho }{\xi } + \mathcal {O}(\varrho ^{1/2}) \right) ~\mathrm {d}\xi \;. \end{aligned}$$

(146)

Using the abbreviation

$$\begin{aligned} J_q&:= \int \nolimits _{\sqrt{\varrho }}^1\xi ^q~\mathrm {d}\xi \;, \end{aligned}$$

(147)

we can sort this into

$$\begin{aligned} F_-+F_+&= 2\pi \Biggl ( w_{0,0} \left( {\begin{array}{c}p\\ 2\end{array}}\right) \kappa ^2 \varrho ^2 J_{p-2} \nonumber \\&\quad {} + \biggl ( w_{0,0} + \Bigl ( - 2w_{0,0} \left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta \kappa + w_{2,0} \left( {\begin{array}{c}p\\ 2\end{array}}\right) \kappa ^2 \nonumber \\&\quad {} + w_{0,2} - w_{1,1} p\kappa \Bigr ) \varrho ^2 \biggr ) J_p \nonumber \\&\quad {} + \biggl ( w_{2,0} + \Bigl ( w_{0,0} p\varepsilon _0 + w_{0,0} \left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta ^2 \nonumber \\&\quad {} - 2w_{2,0} \left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta \kappa + w_{2,2} + w_{1,1} p\beta \nonumber \\&\quad {} - w_{3,1} p\kappa \Bigr ) \varrho ^2 \biggr ) J_{p+2} \nonumber \\&\quad {} + \biggl ( w_{2,0} p\varepsilon _0 + w_{2,0} \left( {\begin{array}{c}p\\ 2\end{array}}\right) \beta ^2 + w_{4,2}\nonumber \\&\quad {} + w_{3,1} p\beta \biggr ) \varrho ^2 J_{p+4} \Biggr ) + \mathcal {O}(\varrho ^{5/2}) \;, \end{aligned}$$

(148)

which by $J_q= \tfrac{1}{q+1}(1-\varrho ^{(q+1)/2})$ for $q\ne -1$ (the special case $J_{-1}$ only occurs as $J_{p-2}$ for $p=1$ and has then a vanishing coefficient) yields

$$\begin{aligned} F_-+F_+&= 2\pi \Biggl ( w_{0,0}\frac{1}{p+1} + w_{2,0}\frac{1}{p+3} \nonumber \\&\quad {} + \biggl ( w_{0,0} \frac{p}{2}\kappa ^2 - 2w_{0,0} \frac{p(p-1)}{2(p+1)}\beta \kappa \nonumber \\&\quad {} + w_{2,0} \frac{p(p-1)}{2(p+1)}\kappa ^2 + w_{0,2} - w_{1,1} \frac{p}{p+1}\kappa \nonumber \\&\quad {} + w_{0,0} \frac{p}{p+3}\varepsilon _0 + w_{0,0} \frac{p(p-1)}{2(p+3)}\beta ^2 \nonumber \\&\quad {} - 2w_{2,0} \frac{p(p-1)}{2(p+3)}\beta \kappa + w_{2,2}\frac{1}{p+3} \nonumber \\&\quad {} + w_{1,1} \frac{p}{p+3}\beta - w_{3,1} \frac{p}{p+3}\kappa \nonumber \\&\quad {} + w_{2,0} \frac{p}{p+5}\varepsilon _0 + w_{2,0} \frac{p(p-1)}{2(p+5)}\beta ^2 \nonumber \\&\quad {} + w_{4,2}\frac{1}{p+5} + w_{3,1} \frac{p}{p+5}\beta \biggr ) \varrho ^2 \nonumber \\&\quad {} - w_{0,0}\frac{1}{p+1} \varrho ^{(p+1)/2} \nonumber \\&\quad {} - \biggl ( w_{0,0} \frac{p}{2}\kappa ^2 + w_{2,0}\frac{1}{p+3} \biggr ) \varrho ^{(p+3)/2} \Biggr ) \nonumber \\&\quad {} + \mathcal {O}(\varrho ^{5/2}) + \mathcal {O}(\varrho ^{(p+5)/2})\;. \end{aligned}$$

(149)

1.5.6 Evaluation of the Outer Integral II

Starting with the same substitution $\xi =\sqrt{\varrho }\,\zeta$ and Taylor expansion of $\omega$ in $\xi$ direction as in Appendix A.1.7, we evaluate

$$\begin{aligned}&G_\mp {\mathop {=}\limits ^{(143)}} \pi \varrho ^{(p+1)/2} \int \nolimits _0^{1\pm \nu \sqrt{\varrho }} \bigl ( w_{0,0} + w_{2,0}\varrho \zeta ^2 \mp w_{1,1}\varrho ^{3/2}\zeta \nonumber \\&\quad \mp 2w_{2,0}\nu \varrho ^{3/2}\zeta +\mathcal {O}(\varrho ^2) \bigr ) \nonumber \\&\quad {}\times \left( 1\mp \beta \zeta \varrho ^{3/2} +\mathcal {O}(\varrho ^2) \right) ^p \zeta ^p ~\mathrm {d}\zeta \nonumber \\&= \pi \varrho ^{(p+1)/2}w_{0,0} \int \nolimits _0^{1\pm \nu \sqrt{\varrho }}\zeta ^p~\mathrm {d}\zeta \nonumber \\&\quad \mp \pi \varrho ^{(p+4)/2} (w_{1,1}+2w_{2,0}\nu +w_{0,0}p\beta ) \int \nolimits _0^{1\pm \nu \sqrt{\varrho }}\zeta ^{p+1}~\mathrm {d}\zeta \nonumber \\&\quad +\pi \varrho ^{(p+3)/2}w_{2,0} \int \nolimits _0^{1\pm \nu \sqrt{\varrho }}\zeta ^{p+2}~\mathrm {d}\zeta +\mathcal {O}(\varrho ^{(p+5)/2}) \nonumber \\&= \frac{\pi \varrho ^{(p+1)/2}}{p+1} w_{0,0} (1\pm \nu \sqrt{\varrho })^{p+1} \nonumber \\&\quad {} \mp \frac{\pi \varrho ^{(p+4)/2}}{p+2} (w_{1,1}+2w_{2,0}\nu +w_{0,0}p\beta ) (1\pm \nu \sqrt{\varrho })^{p+2} \nonumber \\&\quad {} +\frac{\pi \varrho ^{(p+3)/2}}{p+3} w_{2,0} (1\pm \nu \sqrt{\varrho })^{p+3} +\mathcal {O}(\varrho ^{(p+5)/2}) \nonumber \\&= \frac{\pi \varrho ^{(p+1)/2}}{p+1}w_{0,0}(1+\nu ^2\varrho ) \nonumber \\&\quad {} \mp \frac{\pi \varrho ^{(p+4)/2}}{p+2}(w_{1,1}+2w_{2,0}\nu +w_{0,0}p\beta ) \nonumber \\&\quad {} +\frac{\pi \varrho ^{(p+3)/2}}{p+3}w_{2,0} +\mathcal {O}(\varrho ^{(p+5)/2}) \;, \end{aligned}$$

(150)

$$\begin{aligned}&G_-+G_+ = 2\pi \varrho ^{(p+1)/2} \left( \frac{1}{p+1}w_{0,0} +\frac{p}{2}\nu ^2\varrho w_{0,0} \right. \nonumber \\&\quad \qquad \qquad \left. {} +\frac{1}{p+3}\varrho w_{2,0} \right) +\mathcal {O}(\varrho ^{(p+5)/2}) \;. \end{aligned}$$

(151)

1.5.7 Extremum of the Combined Integral

When we finally combine (93), (149) and (151) and apply (144) and $\nu =\kappa +\mathcal {O}(\varrho ^2)$, we observe as in the 2D case than all terms originating from $G_-+G_+$ (151) cancel, and it remains

$$\begin{aligned} E(\kappa )&= \mathrm {const}(\kappa ) + \mathcal {O}\bigl (\varrho ^{\min \{(p+5)/2,5/2\}}\bigr ) \nonumber \\&\quad {} + \left( p - \frac{p(p-1)}{p+1} \right) \pi \varrho ^2\kappa ^2 \nonumber \\&\quad {} + \left( - \frac{2p(p-1)}{(p+1)} \beta + \frac{2p(p-1)}{(p+3)}\beta \right. \nonumber \\&\quad \left. - \frac{2p}{p+1} (\delta _0+\delta _2) + \frac{2p}{p+3} (\delta _0+\delta _2) \right) \pi \varrho ^2\kappa \nonumber \\&= \mathrm {const}(\kappa ) + \mathcal {O}\bigl (\varrho ^{\min \{(p+5)/2,5/2\}}\bigr ) \nonumber \\&\quad +\frac{2p}{p+1}\pi \varrho ^2\kappa ^2 -\left( \frac{4p(p-1)}{(p+1)(p+3)}\beta \right. \nonumber \\&\quad \left. {} +\frac{4p}{(p+1)(p+3)}(\delta _0+\delta _2)\right) \pi \varrho ^2\kappa \;. \end{aligned}$$

(152)

For $\varrho \rightarrow 0$, the extremum of $E(\kappa )$ can again be found as the apex of the quadratic function in (152), which yields

$$\begin{aligned} \kappa&= \frac{ \frac{4p(p-1)}{(p+1)(p+3)}\beta +\frac{4p}{(p+1)(p+3)}(\delta _0+\delta _2)}{\frac{4p}{p+1}\pi \varrho ^2} \nonumber \\&\quad {} + \mathcal {O}(\varrho ^{\min \{(p+1)/2,1/2\}})\ \nonumber \\&= \frac{p-1}{p+3}\beta +\frac{1}{p+3}(\delta _0+\delta _2) \nonumber \\&\quad {} + \mathcal {O}(\varrho ^{\min \{(p+1)/2,1/2\}})\ \;. \end{aligned}$$

(153)

1.5.8 Conclusion of the Proof

From (153) the claim of the proposition for regular points follows by substituting back $\kappa \alpha \varrho ^2=\mu$, $\alpha \beta =u_{xx}/2$, $\alpha \delta _0=u_{yy}/2$, $\alpha \delta _2=u_{zz}/2$ and noticing that by our ansatz $u_x>0$, $u_y=u_z=0$ the coordinates x, y, z coincide with the geometric coordinates $\eta$, $\xi$, $\chi$ as used in the proposition.

For critical points, the reasoning from Appendix A.1 applies. $\square$

1.6 Proof of Proposition 6

Analogous to Appendix A.2, we calculate

$$\begin{aligned} V(\xi ) = \pi (1+(\delta _0+\delta _2)\xi \varrho -2\beta \xi \varrho )(1-\xi ^2) +\mathcal {O}(\xi ^2\varrho ^2) \,. \end{aligned}$$

(154)

The relevant solution of $V'(\xi )=0$ yields up to higher order terms $\omega (\xi )=\tfrac{1}{2}(\delta _0+\delta _2-2\beta )\varrho$. $\square$

Continuous Order-p Means and Mode: A Toy Example

To understand the behaviour of order-p mean filters for $p>-1$, $p\ne 0$ and their relation to the mode of a continuous density, we consider the following simple example. Let z be a real random variable with (non-normalised) density

$$\begin{aligned} \gamma (z) = {\left\{ \begin{array}{ll} 1-\lambda (z-m)^2\;, &{}-1\le z\le 1\;,\\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

(155)

Here, $0<m<<1$ is a fixed parameter, and $0<\lambda \le (1+m)^{-2}$ to ensure that $\gamma (z)\ge 0$ for all z. Obviously, the mode of z is the maximum of $\gamma$, i.e. m.

For any $p>-1$, $p\ne 0$ the order-p mean of z is given by the minimiser of $E(\mu )$ where

$$\begin{aligned}&\mathrm {sgn}\,(p) E(\mu ) = \int \nolimits _{-1}^1 \gamma (z)\,|z-\mu |^p~\mathrm {d}z \nonumber \\&= \int \nolimits _{-1}^\mu \bigl (1-\lambda (z-m)^2\bigr ) (-z+\mu )^p~\mathrm {d}z \nonumber \\&\quad {} +\int \nolimits _\mu ^1 \bigl (1-\lambda (z-m)^2\bigr ) (z-\mu )^p~\mathrm {d}z \nonumber \\&= \int \nolimits _0^{1+\mu } \bigl (1-\lambda (z-\mu +m)^2\bigr )\, z^p~\mathrm {d}z \nonumber \\&\quad {} + \int \nolimits _0^{1-\mu } \bigl (1-\lambda (z+\mu -m)^2\bigr )\, z^p~\mathrm {d}z \nonumber \\&= \bigl (1-\lambda (\mu -m)^2\bigr ) \left( \int \nolimits _0^{1+\mu } z^p ~\mathrm {d}z +\int \nolimits _0^{1-\mu } z^p ~\mathrm {d}z \right) \nonumber \\&\quad {} + 2\lambda (\mu -m) \left( \int \nolimits _0^{1+\mu } z^{p+1} ~\mathrm {d}z -\int \nolimits _0^{1-\mu } z^{p+1} ~\mathrm {d}z \right) \nonumber \\&\quad {} - \lambda \left( \int \nolimits _0^{1+\mu } z^{p+2} ~\mathrm {d}z +\int \nolimits _0^{1-\mu } z^{p+2} ~\mathrm {d}z \right) \nonumber \\&= \frac{1}{p+1} \, \bigl (1-\lambda (\mu -m)^2\bigr ) \left( (1+\mu )^{p+1} + (1-\mu )^{p+1} \right) \nonumber \\ {}&\quad {} + \frac{2\lambda }{p+2} \, (\mu -m) \left( (1+\mu )^{p+2} - (1-\mu )^{p+2} \right) \nonumber \\&\quad {} - \frac{\lambda }{p+3} \left( (1+\mu )^{p+3} + (1-\mu )^{p+3} \right) \nonumber \\&=\frac{2}{p+1} \, \bigl (1-\lambda (\mu -m)^2\bigr ) \left( 1 + \left( {\begin{array}{c}p+1\\ 2\end{array}}\right) \mu ^2 + \mathcal {O}(\mu ^4) \right) \nonumber \\&\quad {} + \frac{4\lambda }{p+2} \, (\mu -m) \left( (p+2) \mu + \mathcal {O}(\mu ^3) \right) \nonumber \\&\quad {} - \frac{2\lambda }{p+3} \left( 1 + \left( {\begin{array}{c}p+3\\ 2\end{array}}\right) \mu ^2 + \mathcal {O}(\mu ^4) \right) \nonumber \\&= \frac{2}{p+1} + p\mu ^2 - \frac{2\lambda }{p+1} \mu ^2 + \frac{4\lambda }{p+1} m\mu - \frac{2\lambda }{p+1} m^2 \nonumber \\&\quad {} - p\lambda m^2\mu ^2 + 4\lambda \mu ^2 - 4\lambda m\mu - \frac{2\lambda }{p+3} - (p+2)\lambda \mu ^2 \nonumber \\ {}&\quad {} + \mathcal {O}(\mu ^3) \nonumber \\&= \mathrm {const}(\mu ) + \left( \frac{4\lambda }{p+1} -4\lambda \right) m \mu \nonumber \\&\quad {} + \left( p - \frac{2\lambda }{p+1} - p\lambda m^2 + 4\lambda - (p+2)\lambda \right) \mu ^2 + \mathcal {O}(\mu ^3) \nonumber \\&= \mathrm {const}(\mu ) + \frac{-4p\lambda }{p+1} m \mu \nonumber \\&\quad {} + \left( p - \frac{p(p-1)}{p+1}\lambda - p\lambda m^2 \right) \mu ^2 + \mathcal {O}(\mu ^3) \end{aligned}$$

(156)

from which the minimiser $\mu ^*$ of $E(\mu )$ can be read off as the apex of the quadratic function of $\mu$ as

$$\begin{aligned} \mu ^*&= -\frac{-2p\lambda }{p+1}\,m\bigg / \left( p - \frac{p(p-1)}{p+1}\lambda - p\lambda m^2 \right) \nonumber \\&= \frac{2\lambda }{(p+1)-(p-1)\lambda }\,m + \mathcal {O}(m^3) \;. \end{aligned}$$

(157)

For any fixed $\lambda \in \bigl (0,(1+m)^{-2}\bigr )$, the minimiser $\mu ^*$ goes to m for $p\rightarrow -1$, but approaches $2\lambda m / (1+\lambda )<m-m^2/3$ for $p\rightarrow 0$. Moreover, if we send $\lambda$ to 0, making the density more and more uniform, for any $p>-1$ we have $\mu ^*\rightarrow 0$ which comes as no surprise as for a flattening out density, any penalisation where the penaliser increases with distance will end up in the symmetry centre of the support interval $[-1,1]$.