Original: S UBPIXEL OBJECT SEGMENTATION USING WAVELETS AND MULTI RESOLUTION ANALYSIS
Revised: S UBPIXEL OBJECT SEGMENTATION USING WAVELETS AND MULTI RESOLUTION ANALYSIS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: We propose a novel deep learning framework for fast prediction of boundaries of two-dimensional simply connected domains using wavelets and Multi Resolution Analysis (MRA).
Revised: We propose a novel deep learning framework for fast prediction of boundaries of two-dimensional simply connected domains using wavelets and Multi Resolution Analysis MRA).

Original: The boundaries are modelled as (piecewise) smooth closed curves using wavelets and the so-called Pyramid Algorithm.
Revised: The boundaries are modelled as (piecewise) smooth closed curves using wavelets and the so-called Pyramid Algorithm.

Original: Our network architecture is a hybrid analog of the U-Net, where the down-sampling path is a two-dimensional encoder with learnable ﬁlters and the upsampling path is a onedimensional decoder which builds curves up from low to high resolution levels.
Revised: Our network architecture is a hybrid analog of the U-Net, where the down-sampling path is a two-dimensional encoder with learnable ﬁlters, and the upsampling path is a onedimensional decoder, which builds curves up from low to high resolution levels.

Original: Any wavelet basis induced by a MRA can be used.
Revised: Any wavelet basis induced by a MRA can be used.

Original: This ﬂexibility allows for incorporation of priors on the smoothness of curves.
Revised: This ﬂexibility allows for incorporation of priors on the smoothness of curves.

Original: The effectiveness of the proposed method is demonstrated by delineating boundaries of simply connected domains (organs) in medical images using Debauches wavelets and comparing performance with a U-Net baseline.
Revised: The effectiveness of the proposed method is demonstrated by delineating boundaries of simply connected domains (organs) in medical images using Debauches wavelets and comparing performance with a U-Net baseline.

Original: Our model demonstrates up to 5x faster inference speed compared to the U-Net, while maintaining similar performance in terms of Dice score and Hausdorff distance.
Revised: Our model demonstrates up to 5x faster inference speed compared to the U-Net while maintaining similar performance in terms of Dice score and Hausdorff distance

Original: Code will be made available in the discussion forum.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Semantic image segmentation is a core component of many medical imaging related tasks.
Revised: Semantic image segmentation is a core component of many medical imaging related tasks.

Original: Both as part of a pipeline to ﬁnd a region of interest, or a task by itself, e.g., for measuring tumor volume.
Revised: Both as part of a pipeline to ﬁnd a region of interest, or a task by itself, e.g., for measuring tumor volume.

Original: Nowadays, almost all segmentation algorithms in medical imaging are replaced by U-Net-like architectures Ronneberger et al.
Revised: Nowadays, almost all segmentation algorithms in medical imaging are replaced by U-Net-like architectures Ronneberger et al.

Original: (2015) combining an encoder and decoder.
Revised: (2015) combining an encoder and decoder.

Original: Typically, the decoder is an upsampling path, and additional skip connections between the encoding and decoding part are added to recover the image’s spatial information.
Revised: Typically, the decoder is an upsampling path, and additional skip connections between the encoding and decoding part are added to recover the image’s spatial information.

Original: While many variants or more exotics methods, such as multi-scale and pyramid based approaches, recurrent networks or generative techniques, can be designed, all of these still yield per-pixel classiﬁcations.
Revised: While many variants or more exotics methods, such as multi-scale and pyramid based approaches, recurrent networks or generative techniques, can be designed, all of these still yield per-pixel classiﬁcations.

Original: In these settings an image is interpreted as a discrete collection of ordered pixels (or voxels in the three-dimensional case), where the task is to assign an appropriate class to each pixel using a probabilistic model.
Revised: In these settings an image is interpreted as a discrete collection of ordered pixels (or voxels in the three-dimensional case), where the task is to assign an appropriate class to each pixel using a probabilistic model.

Original: Typically, the pixels are assumed to be independent so that the joint-likelihood is tractable.
Revised: Typically, the pixels are assumed to be independent so that the joint-likelihood is tractable.

Original: Medical images often represent discrete approximations, measured and stored in  a discrete grid, of inherently continuous objects.
Revised: In practice, e.g., in medical imaging, the boundary of a region is annotated and not the region itself. Hence the raw ground-truth data is a discretization of a closed curve.

Original: The pixel or voxel sizes can be coarse, and each voxel may represent multiple tissue types due to the partial volume effect.
Revised: The main motivation of our paper is to construct a deep learning model which ( i ) may directly use such raw groundtruth data if available ( ii ) is guaranteed to predict smooth planar curves ( iii ) improves inference speed by predicting 1d objects (curves) instead of 2d objects. We argue that in traditional pipelines, where pixel-based predictions are constructed, smooth boundaries are not faithfully represented.

Original: While humans are able to draw continuous contours based on such images, e.g., boundaries of organs, typical deep learning-based segmentation models are not able to take this continuous nature into account. In this work, we propose a novel deep learning framework whichdoes not predict per-pixel segmentation maps, but directly predicts contours as continuous objects.
Revised: In particular, no prior information about the geometry of planar curves is incorporated. In this paper, we present a hybrid analog of the U-Net, where the down-sampling path is a two-dimensional encoder with learnable ﬁlters, and the upsampling path is an one-dimensional decoder, which predicts (smooth) representations of curves.

Original: A fundamental component in the setup of our framework is the decision on how to represent (closed) curves.
Revised: A fundamental component in the setup of our framework is the decision on how to represent (closed) curves.

Original: While the Fourier basis is a natural candidate at ﬁrst glance, its global nature may hamper accurate predictions of curves which exhibit highly localized behavior, requiring accurate estimates of small noisy high-frequency modes.
Revised: While the Fourier basis is a natural candidate at ﬁrst glance, its global nature may hamper accurate predictions of curves which exhibit highly localized behavior, requiring accurate estimates of small noisy high-frequency modes.

Original: For this reason, we have chosen to represent contours using wavelets and Multi Resolution Analysis (MRA) instead.
Revised: For this reason, we have chosen to represent contours using wavelets and Multi Resolution Analysis (MRA) instead.

Original: The main idea is to choose a single map ϕ , the so-called scaling function or father wavelet, and to construct subspaces of functions associated to prescribed resolution levels by taking the span of appropriate dilations and translations of ϕ .
Revised: The main idea is to choose a single map ϕ , the so-called scaling function or father wavelet, and to construct subspaces of functions associated to prescribed resolution levels by taking the span of appropriate dilations and translations of ϕ .

Original: This setup provides an efﬁcient way to decompose and reconstruct contours, from low to high resolution level, using the classical Pyramid Algorithm Mallat (2008).
Revised: This setup provides an efﬁcient way to decompose and reconstruct contours, from low to high resolution level, using the classical Pyramid Algorithm Mallat (2008) as a decoder.

Original: In this paper, we construct a hybrid analog of the U-Net, where the down-sampling path is a twodimensional encoder with learnable ﬁlters, and the upsampling path is an one-dimensional decoder, which builds curves up from low to high resolution levels.

Original: The ﬁlters in the decoder are not learned but uniquely determined by the chosen wavelet basis.
Revised: The ﬁlters in the decoder are not learned but uniquely determined by the chosen wavelet basis.

Original: Any wavelet basis induced by a MRA can be used.
Revised: Any wavelet basis induced by a MRA can be used.

Original: This ﬂexibility allows for incorporation of priors on the smoothness of curves.
Revised: This ﬂexibility allows for incorporation of priors on the smoothness of curves.

Original: Related work Previous work  Chen et al.
Revised: Related work Previous work by Chen et al.

Original: ; Marcos et al.
Revised: ; Marcos et al.

Original: ; Hatamizadeh et al.
Revised: ; Hatamizadeh et al.

Original: (2019) also proposed models to predict contours by combining Active Contour Models (ACM) with a CNN into an end-to-end model.
Revised: (2020) also proposed models to predict contours by combining Active Contour Models (ACM) with a CNN into an end-to-end model.

Original: In these papers the representation of curves is  based on traditional pixelbased methodology.
Revised: In these papers the representation of curves is ultimately still based on pixel-based computations.

Original: For instance, in Hatamizadeh et al.
Revised: For instance, in Hatamizadeh et al.

Original: (2019) curves are modelled as level sets of distance maps deﬁned on a discretization of the domain of the image.
Revised: (2020) curves are modelled as level sets of distance maps deﬁned on a discretization of the domain of the image.

Original: In Chen et al.
Revised: In Chen et al.

Original: (2019) a similar approach is followed, but a smoothed approximation of an indicator function is used instead of a distance map.
Revised: (2019) a similar approach is followed, but a smoothed approximation of an indicator function is used instead of a distance map.

Original: The work in Marcos et al.
Revised: The work in Marcos et al.

Original: (2018) is perhaps most closely related to ours; they directly construct polygonal approximations of curves and represent them using (pixel) coordinates of the nodes.
Revised: (2018) is perhaps most closely related to ours; they directly construct polygonal approximations of curves and represent them using (pixel) coordinates of the nodes.

Original: The objectives minimized in the cited papers are based on a careful consideration of mean pixel intensities and geometric properties such as area and arc length.
Revised: The objectives minimized in the cited papers are based on a careful consideration of mean pixel intensities and geometric properties such as area and arc length.

Original: These properties are implicitly encoded in an objective function (energy-functional) deﬁned on a space of distance maps
Revised: These properties are implicitly encoded in an objective function (energy-functional) deﬁned on a space of distance maps

Original: Hatamizadeh et al.
Revised: Hatamizadeh et al.

Original:  suitable approximations of indicator functions Chen et al.
Revised: (2020), suitable approximations of indicator functions Chen et al.

Original: (2019), or family of polygons Marcos et al.
Revised: (2019), or family of polygons Marcos et al.

Original: However, in contrast to our approach, the above methods allstill provide pixel-based output.
Revised: However, in contrast to our approach, the above methods all provide pixel-based output.

Original: To the best of our knowledge, our work is the ﬁrst to use MRA and wavelet analysis to construct pixel-independent representations of (closed) curves.
Revised: To the best of our knowledge, our work is the ﬁrst to use MRA and wavelet analysis to construct pixel-independent representations of (closed) curves.

Original: This paper is organized as follows.
Revised: This paper is organized as follows.

Original: In Section 2 we review the mathematical background needed to construct our model.
Revised: In Section 2 we review the mathematical background needed to construct our model.

Original: We explain how contours can be decomposed and reconstructed on different resolution levels.
Revised: We explain how contours can be decomposed and reconstructed on different resolution levels.

Original: The reconstruction algorithm,known as the Pyramid Algorithm, forms a core component of our network architecture.
Revised: The reconstruction algorithm, the Pyramid Algorithm, forms a core component of our network architecture.

Original: In Section 3 we set up a model architecture and loss.
Revised: In Section 3 we set up the model architecture and loss.

Original: Subsequently the datasets, training method and performance measures are described in Section 4.
Revised: Subsequently the datasets, training method and performance measures are described in Section 4.

Original: We end the paper withthe resultsof our method in Section 5 and a brief discussion.
Revised: We end the paper with results and a discussion in Section 5.

Original: A detailed account of all the mathematical details is provided in the appendix.
Revised: Further mathematical details are provided in the appendix.

Original:  B ACKGROUND AND MATHEMATICAL SETUP
Revised:  B ACKGROUND AND MATHEMATICAL SETUP

Original: Let (Ω , Σ , P ) be a probability space and  X : Ω → [0 , 1] n1 × n 2 a random variable whose realizations correspond to two dimensional gray-valued images, e.g., slices of MRI scans of size n1 × n2 , where n1 , n 2 ∈ Nand X := [0 , 1] n 1 × n 2 .
Revised: In this section we describe our mathematical setup and review the theory needed to construct our model. We consider two dimensional gray-valued images x ∈ X := [0 , 1] n × n , e.g., slices of MRI scans of size n × n , where n ∈ N .

Original: We assumewithout loss of generality that the gray-values of the scans are rescaled to [0 , 1] . Commonly, X is paired with a random variable Y : Ω → { 0 , 1 } n 1 × n 2 whose realizations correspond to binary segmentations.
Revised: We assume that each image contains a (uniquely identiﬁable) simply connected region R ( x ) ⊂ R 2 , e.g., an organ, with boundary ∂R ( x ) . It is assumed that ∂R ( x ) can be parameterized by a simple closed continuous curve γ ( x ) .

Original: In this context, realizations of ( X, Y ) correspond to scans with inscribed masks.
Revised: We will develop a deep learning framework for computing such parameterizations γ ( x ) using Multi Resolution Analysis (MRA).

Original: In this paper we assume that the regions approximated by the inscribed masks are almost surely enclosed by simple closed curves . Let ( x , y ) ∈
Revised: Since wavelets play a seminal role in our set up, we ﬁrst review the necessary theory in

Original: X ×{ 0 , 1 } n 1 × n 2 be a realization of ( X, Y ) . The binary mask y constitutes a discretized approximation of the true region of interest R ⊂ R 2 and provides a polygonal approximation of ∂R .In a large variety of situation, however, ∂R is in reality a closed (piecewise) smooth curve. In this paper we develop a deep learning framework for directly parameterizing ∂R
Revised: Sections 2.1 and 2.2 as a subject in its own right in the context of general scalar-valued signals γ .

Original:  = ∂R ( x ) , given an image x , using wavelets and Multi Resolution Analysis (MRA). Since wavelets play a seminal role in our set up, we start in the next section with an extensive review to introduce notation and terminology.
Revised: In the remainder of the paper, we return to the context in which γ = γ ( x ) is interpreted as a periodic (planar) curve parameterizing the boundary of a region.

Original: 2. M ULTI RESOLUTION ANALYSIS
Revised: 2. M ULTI RESOLUTION ANALYSIS

Original: In this section we  reviewthe necessary material from discrete wavelet theory using the framework of a multi resolution analysis (MRA).
Revised: In this section we brieﬂy review wavelet theory using the framework of a multi resolution analysis (MRA).

Original: We closely follow the exposition in Pereyra & Ward (2012) and Montefusco & Puccio (2014).
Revised: We closely follow the exposition in Pereyra & Ward (2012) and Montefusco & Puccio (2014).

Original: The reader is referred to these references for a more comprehensive introduction.
Revised: Throughout this section we denote the space of square integrable functions on R , equipped with standard inner product, by L 2 ( R ) .

Original: Multi Resolution Analysis It is a well known-fact from Fourier analysis  that a signal γ ∈ L 2 ( R ) cannot be simultaneously localized in the time and frequency domain, see Pereyra & Ward (2012).
Revised: The uncertainty principle in Fourier analysis dictates that a signal γ ∈ L 2 ( R ) cannot be simultaneously localized in the time and frequency domain, see Pereyra & Ward (2012).

Original: Multi resolution analysis aims to address this shortcoming by decomposing a signal on different discrete resolution levels.
Revised: Multi resolution analysis aims to address this shortcoming by decomposing a signal on different discrete resolution levels.

Original: The idea is to construct subspaces V j ⊂ L 2 ( R ) , associated to various resolution levels j ∈ Z , spanned by  shifts of a localized mapping ϕ j .
Revised: The idea is to construct subspaces V j ⊂ L 2 ( R ) , associated to various resolution levels j ∈ Z , spanned by integer shifts of a localized mapping ϕ j .

Original: The level of localization associated to V j is determined by taking an appropriate dilation of a prescribed map ϕ ; the socalled scaling function .
Revised: The level of localization associated to V j is determined by taking an appropriate dilation of a prescribed map ϕ ; the so-called scaling function .

Original: In the MRA framework the dilation factors are chosen to be powers of two.
Revised: In the MRA framework the dilation factors are chosen to be powers of two.

Revised: Formally, we require that ( ϕ jk )

Original: Altogether, this yields an increasing sequence of closed subspaces V j ⊂ L 2 ( R ) comprising all of
Revised: k ∈ Z is an orthonormal basis for V j , where ϕ jk ( t ) := 2 j 2 φ (2 j t − k ) , see Figures 1a and [Table]

Revised: [Table]

Revised: [Table]

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Revised: [Table]

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Revised: [Table]

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Revised: [Table]

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Original: L 2 ( R ) .
Revised: [Table] k 2 j <latexit sha1_base [Table] M7UKKJnEQHA3s72OH

Original: In particular, V j +1 ⊃ V j is the next level up in resolution after V j . The “view” of γ on resolution level j is then deﬁned as the orthogonal projection of γ onto V j . The formal deﬁnition of a MRA is given below in Deﬁnition 2.1. Deﬁnition 2.1 (Formal deﬁnition MRA) .
Revised: [Table]

Original: Let T k : L 2 ( R ) → L 2 ( R ) and D j : L 2 ( R ) → L 2 ( R ) denote the translation and normalized dilation operator, respectively, deﬁned by T k γ ( x ) =
Revised: t <latexit sha1_base64="/r8pzfVx4NPn5wchjI9gPOjn8Gs=">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</latexit>

Original: [-γ ( x − k ) and D j γ ( t ) = 
-]
Revised: [-γ ( x − k ) and D j γ ( t ) = 
-][Figure]

Original: j 2 γ (2 j t ) for γ ∈ L 2 ( R ) ∩ C ∞ 0 ( R ) and j, k ∈ Z . A multi-resolution analysis of L 2 ( R ) is an increasing subsequence of subspaces ( V j ) j ∈ Z , such that ( i ) (cid:84) j ∈ Z V j = { 0 } , ( ii ) (cid:83) j ∈ Z V j is dense in L 2 ( R ) , ( iii )
Revised: 1b.

Original: γ ∈ V j if and only if D 1 γ ∈ V j +1 , ( iv ) V 0 is invariant under translations, ( v ) ∃ ϕ ∈ L 2 ( R ) such that { T k ϕ } k ∈ Z is an orthonormal basis for V 0 . Conditions ( iii ) , ( iv ) and ( v ) imply that each subspace V j is closed and invariant under integer shifts. Condition ( v ) formalizes the idea that the subspaces are spanned by translations and dilations of a single map ϕ ; the so-called scaling function or father wavelet .
Revised: Altogether, this yields an increasing sequence of closed subspaces V j ⊂ V j +1 ⊂

Original: Indeed, it is straightforward to show that { ϕ jk : k ∈ Z } is an orthonormal basis for V j , where ϕ jk : = T k D j ϕ . Approximation coefﬁcients Let γ j ∈ V 
Revised: L 2 ( R ) in L 2 ( R ) , where V j +1 is the next level up in resolution after V j .

Original: j denote the best approximation of γ at resolution level j , i.e., γ j :=
Revised: The representation of γ at resolution level j , denoted by γ j :=

Original: P j γ , where P j : L 2 ( R ) → V j is the orthogonal projection onto V j .
Revised: P j γ , is its orthogonal projection onto V j . Here P j denotes the orthogonal projection onto V j .

Original: Then there exists unique coefﬁcients  a j ( γ ) =
Revised: The coefﬁcients of γ j with respect to the basis for V j , denoted by a j ( γ ) =

Original: ( a jk ( γ ))
Revised: ( a jk ( γ ))

Original: k ∈ Zsuch that γ j = (cid:80) k ∈ Z a jk ( γ ) ϕ jk , where a jk =
Revised: k ∈ Z , are called the approximation coefﬁcients .

Original: (cid:104) γ, ϕ jk (cid:105) , since { ϕ jk : k ∈ Z } is an orthonormal basis for V j .

Original: Thecoefﬁcients a j ( γ ) are referred to as the approximation coefﬁcients of γ at resolution level j . The associated subspaces V j are referred to as the approximation subspaces. We will write a j ( γ ) = a j whenever it is clear from the context that the coefﬁcients are associated to γ .
Revised: The associated subspaces V j are referred to as the approximation subspaces (see Figure 1b).

Original: Detail coefﬁcients To study the information that is lost when a signal in V j +1 is projected onto V j , we consider the operator Q j :=
Revised: To study the information that is lost when a signal in V j +1 is projected onto V j , we consider the operator Q j :=

Original: P j +1 − P j .
Revised: P j +1 − P j .

Original: Note that Q j γ ∈ W j :=

Original: V ⊥ j ∩ V j+1 , since P j P j +1 = P j and V j ⊂ V j +1 . The subspace W j  is called the detail subspace at level j  .
Revised: The range of Q j , denoted by W j , is referred to as the detail subspace at level j ; it is the orthogonal complement of V j in V j +1 .

Original: The detail subspaces ( W j ) j ∈ Z are mutually disjoint and orthogonal by construction.
Revised: The detail subspaces ( W j ) j ∈ Z are mutually disjoint and orthogonal by construction.

Original: Furthermore, since

Original: V j = V j − 1 ⊕ W j − 1 for any j ∈ Z , it follows that V j = V j 0 ⊕ (cid:76) j − 1 l = j 0 W l , for all j > j 0 ,

Original: where j, j 0 ∈ Z .

Original: This decomposition shows that a signal on resolution level j can be reconstructed from any lower level j 0 if all the details in between are known and that L 2 ( R ) =

Original: (cid:76) j ∈ Z W j .

Original: A fundamental result, known as Mallat’s Theorem, states that the subspaces W j can too be spanned by dilating and shifting a single map.
Revised: A fundamental result, known as Mallat’s Theorem, states that the subspaces W j can too be spanned by dilating and shifting a single map.

Original: More precisely, there exists a map ψ ∈ W 0 , the so-called mother wavelet , such that { ψ jk : k ∈ Z } is an orthonormal basis for W j , where ψ jk :
Revised: More precisely, there exists a map ψ ∈ W 0 , the so-called mother wavelet , such that ( ψ jk )

Original: = T k D j ψ . The wavelet constructed in Mallat’s Theorem is often referred to as the mother wavelet . The proof is based on a very careful analysis of the deﬁning properties of a scaling function in the frequency domain.
Revised: k ∈

Original: The reader is referred to Pereyra & Ward (2012) for the details.
Revised: Z is an orthonormal basis for W j , see Pereyra & Ward (2012).

Original: Now, using Mallat’s mother wavelet, we may write Q j γ = (cid:80) k ∈ Z d jk ( γ ) ψ jk for any γ ∈ L 2 ( R ) , where d j ( γ ) :=
Revised: The coefﬁcients of Q j γ with respect to this basis, denoted by d j ( γ ) :

Original:  ( d jk ( γ ))
Revised: = ( d jk ( γ ))

Original: k ∈ Z ∈ (cid:96) 2 ( Z ) . The coefﬁcients d j ( γ ) are referred to as the detail coefﬁcients of γ at resolution level j .We will suppress the dependence on γ and write d j ( γ ) = d j when there is no chance of confusion.
Revised: k ∈ Z , are referred to as the detail coefﬁcients of γ at resolution level j .

Original: The detail coefﬁcients store the information needed to go back one level up in resolution, since P j +1 = P j + Q j by construction.
Revised: The detail coefﬁcients store the information needed to go back one level up in resolution, since P j +1 = P j + Q j by construction.

Original: In general, given approximation coefﬁcients a j0 ∈ (cid:96) 2 ( Z ) at level j 0 and detail coefﬁcients d l ∈ (cid:96) 2 (Z ) at levels j 0 ≤ l ≤ j − 1 , we can reconstruct the approximation at level j from j 0 via γ j =(cid:80) k ∈ Z a j 0 k ϕ j 0 k
Revised: We often write a j ( γ ) =

Original: + (cid:80) j 0 ≤ l ≤ j − 1 (cid:80) k ∈
Revised: a j and d j ( γ ) =

Original: Z d lk ψ lk .
Revised: d j for brevity.

Original: Galerkin projections In practice, we only approximate a ﬁnite number of approximation and detail coefﬁcients.
Revised: In practice, we only approximate a ﬁnite number of approximation and detail coefﬁcients, see Section 2.3.

Original: For this purpose, we introduce some additional notation.

Original: We deﬁne the ﬁnite dimensional subspaces (cid:98) V j := span { ϕ jk : − 2 j − 1 ≤ k ≤ 2 j − 1 − 1 } ⊂ V j and (cid:99) W j := span { ψ jk : − 2 j − 1 ≤ k ≤ 2 j − 1 − 1 } ⊂ W j .

Original: We shall frequently identify (cid:98) V j (cid:39) R 2 j and (cid:99) W j (cid:39) R 2 j .

Original: 2. T HE D ISCRETE W AVELET T RANSFORM
Revised: 2. T HE D ISCRETE W AVELET T RANSFORM

Original: In this section we recall how to compute the approximation and detail coefﬁcients given a prescribed scaling function ϕ .
Revised: In this section we describe how to compute the approximation and detail coefﬁcients given a prescribed scaling function ϕ .

Original: Many fundamental aspects of MRA’s, both theoretical and computational, can be traced back to the followingsimple but key observation.
Revised: Many fundamental aspects of MRA’s, both theoretical and computational, can be traced back to the following key observation.

Original: Since V 0 ⊂ V 1 , there must exist coefﬁcients h ∈ (cid:96) 2 ( Z )  such that ϕ
Revised: Since V 0 ⊂ V 1 , there must exist coefﬁcients h = ( h k ) k ∈ Z such that ϕ

Original: = (cid:80) l ∈
Revised: = (cid:80) k ∈

Original: Z h l ϕ 1 l .
Revised: Z h k ϕ 1 k .

Original: This equation is referred to as the scaling equation ; one of the fundamental properties of a scaling function.
Revised: This equation is referred to as the scaling equation ; one of the fundamental properties of a scaling function.

Original: The coefﬁcients h  completely characterize the scaling function.Therefore, to deﬁne a MRA, one only needs to specify an appropriate sequence h ∈ (cid:96) 2 ( Z ) . Throughout this paper we assume that an appropriate choice for h is given. Decomposition
Revised: The sequence h , the so-called lowpass ﬁlter, completely characterizes the scaling function.

Original: [-The scaling equation can be used to derive an efﬁcient scheme for computing lower order approximation coefﬁcients (of any order) given an initial approximation a j +1 ∈ (cid:96) 
-]
Revised: [-The scaling equation can be used to derive an efﬁcient scheme for computing lower order approximation coefﬁcients (of any order) given an initial approximation a j +1 ∈ (cid:96) 
-]

Original: ( Z ) .

Original: [-Deﬁne C ˜ h : (cid:96) 2 ( Z ) → (cid:96) 2 ( Z ) and M − : (cid:96) 
-]
Revised: [-Deﬁne C ˜ h : (cid:96) 2 ( Z ) → (cid:96) 2 ( Z ) and M − : (cid:96) 
-]

Original: [-( Z ) → (cid:96) 
-]
Revised: [-( Z ) → (cid:96) 
-]

Original: ( Z ) by C ˜ h ( a ) = a ∗ ˜ h and ( M − a ) k = a 2 k , respectively, where ˜ h ∈ [-(cid:96) 
-]
Revised: Similarly, since ψ ∈ [-(cid:96) 
-]W 0 ⊂ V 1 , there exist coefﬁcients g

Original: ( Z ) is deﬁned by ˜ h

Original: [-k = h − k , and ∗ : (cid:96) 
-]
Revised: [-k = h − k , and ∗ : (cid:96) 
-]

Original: [-( Z ) × (cid:96) 2 ( Z ) → (cid:96) 
-]
Revised: [-( Z ) × (cid:96) 2 ( Z ) → (cid:96) 
-]

Original: ( Z ) is the twosided discrete convolution.

Original: Then a j =M − C ˜ h a j +1 for any j ∈ Z . Furthermore, one can show, by analyzing the Fourier transform of the scaling equation, that C ˜ h cuts off relatively high frequencies. For this reason, h is often referred to as the low-pass ﬁlter associated to the MRA, see Pereyra &
Revised: =

Original: Ward (2012); Montefusco & Puccio (2014)

Original: Similarly, since ψ ∈ W 0 ⊂ V 1 , there exist coefﬁcients g  ∈[-(cid:96) 
 (-] Z ) such that ψ = (cid:80) k ∈
Revised: ( g k ) k ∈[-(cid:96) 
 (-] Z , the so-called high-pass ﬁlter associated to h , such that ψ = (cid:80) k ∈

Original: Z g k ϕ 1 k .
Revised: Z g k ϕ 1 k .

Original: For Mallat’s mother wavelet, we have g k
Revised: For Mallat’s mother wavelet, we have g k

Original: =
Revised: =

Original: ( − 1) k − 1 h 1 − k .
Revised: ( − 1) k − 1 h 1 − k .

Original: An analogous argument shows that d j = M − C ˜ g a j +1 for all j ∈ Z . Moreover, it can be shown that C ˜ g cuts of relatively low frequencies, and is therefore referred to as the high pass ﬁlter associated to the MRA.
Revised: Altogether, in order to deﬁne a MRA, one only needs to specify an appropriate low-pass ﬁlter h . The scaling equations can be used to derive an efﬁcient scheme for computing lower order approximation coefﬁcients (of any order) given an initial approximation a j +1 .

Original: Reconstruction The orthogonal decomposition
Revised: Conversely, the orthogonal decomposition

Original: V j +

Revised: V j +

Original: =
Revised: =

Original: V j ⊕ W j can be used to reconstruct a j +1∈ V j +1 given the approximation and detail coefﬁcients a j∈ V j and d j ∈ W j at resolution level j .
Revised: V j ⊕ W j can be used to reconstruct a j +1 given the approximation and de- tail coefﬁcients a j and d j , respectively, at resolution level j .

Original: [-More precisely, substitution of the scaling equations for ϕ and ψ into this decomposition yields a j +1 = C h M + a j + C g M + d j , where M + : (cid:96) 
 ( Z ) → (cid:96) 
 ( Z ) is a right inverse of M − deﬁned by ( M + a ) k :-]
Revised: [-More precisely, substitution of the scaling equations for ϕ and ψ into this decomposition yields a j +1 = C h M + a j + C g M + d j , where M + : (cid:96) 
 ( Z ) → (cid:96) 
 ( Z ) is a right inverse of M − deﬁned by ( M + a ) k :-]The computations are summarized in

Original: = a k 2 if k is even and zero otherwise.
Revised: Figure 2.

Original: The reconstruction and decomposition formulae together form the well-known PyramidAlgorithm
Revised: The reconstruction and decomposition formulae together form the well-known Pyramid

Original:  Mallat (2008).
Revised: Algorithm Mallat (2008).

Original: In practice, our signals are periodic and do not directly ﬁt into the MRA framework, since non-zero periodic signals are not elements in L 2 ( R ) .
Revised: In practice, our signals are periodic and do not directly ﬁt into the MRA framework, since non-zero periodic signals are not elements in L 2 ( R ) .

Original: We will address this issue in the next section by using an appropriate cut-off.
Revised: We will address this issue in the next section by using an appropriate cut-off.

Original: Here we only remark that the periodicity needs to be carefully taken into account in the decomposition and reconstruction formulae, see Appendix A.2. 
Revised: Here we only remark that the periodicity needs to be carefully taken into account in the decomposition and reconstruction formulae, see Appendix A.2. [Table] [Table] d j

Revised: [Table]

Revised: a

Original: 2. W AVELET REPRESENTATION OF PERIODIC CURVES
Revised: [Table] + [Table] HxrbbD++uPZvvXwBMt4xp16gXVsYBNj1fcJn/El+Br8CH4Gv85Sg067o4/wz+l2fwPw66tl</latexit> a j + <latexit sha1_base64="+f3+H5d1VSXBWTnXjS3e7/surv0=">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</latexit> a

Revised: <latexit sha1_

Revised: [Table]

Revised: ="3Z+Px9GqgjMzIdbzOKO8uk2VK4I=">

Revised: [Table]

Revised: d

Revised: [Table]

Revised: [Table]

Revised: Reconstruction

Revised: [Table]

Original: {+//A
+}
Revised: {+//A
+}

Revised: +7r54=</latexit>

Revised: [Figure]

Revised: W AVELET REPRESENTATION OF PERIODIC CURVES

Revised: t <latexit sha1_base64="/r8pzfVx4NPn5wchjI9gPOjn8Gs=">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</latexit> <latexit sha1_base64="ZEtWR83MugYGN4gLCH8eIsigBmI=">AAADSnicjVLLThRRED3ToDx8ga6Mm44TEzdOug1GlyQkhCUaB0iQmO7LHezQr9y

Revised: [Table]

Revised: /MBheXK+K2ig9UUVamM04qnSa5HpiE5vqzdLoKItTvRHvP3T+jZfaVEmRP7XzUu9k0V6ezBIVWULTwfL2zESqCf3b/viZXC+mTdi2bTNup4OVYBTI8Y8bYWesrD64+vnTu+HbtWLYu4Zt7KKAQo0MGjks7RQRKv62ECJASWwHDTFDKxG/RotFcmtGaUZERPf53eNrq0Nzvl3OStiKVVL+DZk+bpJTMM7QdtV88deS2aF/yt1ITtfbnHfc5cqIWjwn+i/eYeT/8pwWixnui4aEmkpBnDrVZallKq5z/xdVlhlKYs7epd/QVsI8nLMvnEq0u9lG4v8mkQ51b9XF1vjedWkE0XglE8tEQ84KDX0zUel6mRPR8nYztezU+SsiR3H/L9oVszndjvdzctyy8OhOHTfWx6PwzujuE67bo9eQs4Bl3MAt7tQ9rOIx1jBhhTd4jw/42P/qLXmXvSsHof3ewY1L

Original:  Let j 1 ∈ N be a prescribed initial resolution-level on which a periodic signal γ is to be approximated.
Revised: O14138AyXe1pQ In this section we explain how a scalar-valued periodic signal γ with period l > 0 can be approximated using a MRA. Let j 1 ∈ N be a desired resolution-level.

Original: We re-parameterize γ to have period 1 , so that we can conveniently choose the number of approximation coefﬁcients to be a power of two (see the discussion in Appendix A.1).
Revised: First, we re-parameterize γ to have period 1 . The reason for this is rather technical and we refer the reader to Appendix A.1. In essence, we require the number of approximation coefﬁcients to be a power of two and this parameterization ﬁts nicely with this requirement.

Original: Furthermore, to address the issue that periodic signals are not contained in L 2 ( R ) , we restrict the re-parameterized curve to [ − 1 , 1] , i.e., set γ ∗ ( t ) :
Revised: Next, to address the issue that periodic signals are not contained in L 2 ( R ) , we restrict the re-parameterized curve to [ − 1 , 1] , i.e., set γ ∗ ( t ) :

Original: = γ ( lt ) 1 [ − 1 , 1] ( t ) .
Revised: = γ ( lt ) 1 [ − 1 , 1] ( t ) .

Original: In general, this will introduce discontinuities at the boundary points − l and l .
Revised: In general, this will introduce discontinuities at the boundary points − 1 and 1 .

Original: However, this issue is easily dealt with, as described below, since we only need information about γ  on a strict subset [ I 0 , I 1 ] ⊂ [ − l, l ] of length l .
Revised: This is, however, not an issue, since we only need information about γ ∗ on a strict subset [ I 0 , I 1 ] ⊂ [ − 1 , 1] of length 1 .

Revised: It is shown in Lemma A.1 how (and which) approximation coefﬁcients can be related to the sample values of γ .

Original: If j 1 is sufﬁciently large, the approximations coefﬁcients needed to (approximately) cover [ − 1 , 1] are ( a j 1 k ( γ ∗ ))
Revised: In particular, if j 1 is sufﬁciently large, the approximation coefﬁcients needed to (approximately) cover [ − 1 , 1] are ( a j 1 k ( γ ∗ ))

Original: (cid:98) 

Revised: (cid:98)

Original: j 1 − β (cid:99) k
Revised: j 1 − β (cid:99) k

Original: = − 2 j 1 .
Revised: = − 2 j 1 .

Original: Here β > 0 is the support of the underlying wavelet.
Revised: Here β > 0 is the support of the underlying wavelet.

Original: These coefﬁcients will be close to the (scaled) sample values of γ on { k 2 − j 1 : − 2 j 1 ≤ k ≤ (cid:98) 2 j 1 − β (cid:99)} .
Revised: These coefﬁcients will be close to the (scaled) sample values of γ on { k 2 − j 1 : − 2 j 1 ≤ k ≤ (cid:98) 2 j 1 − β (cid:99)} .

Original: We refer to Appendix A.1 for an explanation. Motivated by this observation,  we use the coefﬁcients ( a j 1 k ( γ ∗ ))
Revised: Motivated by this observation, and the fact that we only need γ on [ − 12 , 12 ] , we use the coefﬁcients ( a j 1 k ( γ ∗ ))

Original: 2 j 1 − 1 − 1 k = − 2 j 1 −only, which cover [ − 12 , 1 − 2 1 − j 1] approximately.
Revised: 2 j 1 − 1 − 1 k = − 2 j 1 − 1 only, which cover [ − 12 , 1 − 2 1 − j 1] approximately, see Figure 3.

Original: To ensure that 2 j 1 − 1 − 1 < (cid:98) 2 j 1 − β (cid:99) , we require that j 1 ≥ (cid:108) log( β − 1) log(2)
Revised: To ensure that 2 j 1 − 1 − 1 < (cid:98) 2 j 1 − β (cid:99) , we require that j 1 ≥ (cid:108) log( β − 1) log(2)

Original: + 1 (cid:109) .
Revised: + 1 (cid:109) .

Original: This quantity is well-deﬁned for all bases considered in this paper, except the Haar-basis.
Revised: This quantity is well-deﬁned for all bases considered in this paper, except the Haar-basis.

Original: Formally, the approximation coefﬁcients at a prescribed level j need to be computed by evaluating the integrals (cid:104) γ ∗ , ϕ jk (cid:105) .

Original: Typically, in signal analysis one uses appropriately rescaled samples of γ ∗ instead.

Original: This approach is justiﬁed for signals sampled on a sufﬁciently high resolution, since the family ( ϕ j 0 ) j ∈ Z behaves as an approximate identity as j → ∞ .

Original: Such an initialization scheme may be highly inaccurate, however, when the signal is not sampled on a sufﬁciently ﬁne grid.

Original: In the appendix, we propose an alternative to the usual initialization scheme by exploiting the fact that we are dealing with periodic signals; see Lemma A.1.

Original:  M ODEL
Revised:  M ODEL

Original: In this section we formulatethe objective of our model and present a suitable architecture.
Revised: In this section we formulate our objective and network architecture.

Original: Furthermore, we set up a loss function for determining network-parameters.

Original: We will slightly abuse notation from the previous sections. A contour associated to an image x ∈ X will be denoted by γ ( x )and is henceforth assumed to have two componentsdenoted by [ γ ( x )] 1 and [ γ ( x )] 2 . 
Revised: Henceforth γ = γ ( x ) is associated to an image x and has two components [ γ ( x )] 1 and [ γ ( x )] 2 . operations from the previous sections are understood to be out component wise

Original: In general, for any quantity that depends on x , we will frequently denote its dependency on x in a similar way.
Revised: [Table] [Equation] [Table] [Equation]

Original: Finally, all operations from the previous sections are understood to be carried out component-wise.
Revised: [Table]

Original: 3. O BJECTIVE
Revised: [Equation] [Table] [Equation] [Equation]

Revised: [Table]

Revised: [Table]

Revised: [Equation]

Revised: [Table]

Revised: [Equation]

Original: Let y ∈ Y be a mask associated to an image x ∈ X , where we regard y as a realization of
Revised: [Table] [Equation] [Equation] [Table]

Original: Y | X = x .
Revised: [Equation]

Original: We assume that the boundary ∂R ( x ) of the region R ( x ) encoded in y is parameterized by a closed curve γ ( x ) ∈ C 2 per ([0 , l ( x )]; R 2 ) . Here l ( x ) > 0 denotes the arc-length of γ ( x ) . The objective is to compute the approximation coefﬁcients of γ ∗ ( x ) , where γ ∗ ( x ) denotes the truncated re-parameterization of γ ( x ) .
Revised: [Equation]

Original: More precisely, let j 0 , j 1 , j 2 ∈ N be resolution levels, where j 0 ≤ j 1 ≤ j 2 . We will construct a convolutional neural network F : X × Θ → (0 , 1) × (cid:81) j 2 j = j 0 (cid:98) V j × (cid:98) V j , where F ( x, θ ) = ( p ( x, θ ) , f j 0 ( x, θ ) , . . .
Revised: [Table]

Original: f j 2 ( x, θ )) .

Original: Here p ( x, θ ) is the conditional probability that R ( x ) (cid:54) = ∅ , f j ( x, θ ) is a prediction for the approximation coefﬁcients of γ ∗ ( x ) at level j , and θ ∈ Θ ⊂ R n are the network parameters. The goal is to ﬁnd θ ∈ Θ such that f jk ( x, θ ) ≈ ( a jk ([ γ ∗ ( x )] 1 ) , a jk ([ γ ∗ ( x )] 2 )) for − 2 j − 1 ≤ k ≤ 2 j − 1 − 1 , j 0 ≤ j ≤ j 2 and “most” realizations of X . Henceforth we will omit the dependence of F and its components on θ .
Revised: +Zqufg7yczon3IvJSf3tsC/SrlqoJHeAP0X7zzyf3msJdKMnogGA02tIKxOpSydTIU7z39RFZGhBcb2FH4PWwnzfM65cIJo59mW4v8qkYzyXaXYjr6lLr0gmt7KxGrR0KDCEr6ZqOReFkC03HmmEZ2yPwC5iOd/0a6QjXUz7+fksGXjizt12djfHo13Ro9ebA93n6V9W6V7dJ8eYqce0y49pz2aoMI7+kAf6VP/S7aSrWcbP0L7vcS5Tb+d7M533hevrg==</latexit>

Original: 3. N ETWORK ARCHITECTURE
Revised: 3. N ETWORK ARCHITECTURE

Revised: To formulate our objective consider an image x ∈ X with an associated boundary ∂R ( x ) of a simply connected region R ( x ) .

Revised: We assume that ∂R ( x ) is parameterized by a closed continuous curve γ ( x ) of length l ( x ) > 0 .

Revised: The objective is to compute the relevant approximation coefﬁcients of γ ∗ ( x ) , see the explanation in Section 2.3, where γ ∗ ( x ) denotes the cut-off re-parameterization of γ ( x ) , using a suitable neural network.

Original: Our network architecture consists of an encoder, bottleneck and decoder with skip-connections in between.
Revised: Our network is a hybrid analog of the U-net. It consists of an encoder, bottleneck and decoder with skip-connections in between.

Original: Only the approximation coefﬁcients at the lowest resolution-level j 0 are “directly” computed by the network (in the bottleneck).
Revised: Only the approximation coefﬁcients at the lowest resolution-level jare “directly” computed by the network (in the bottleneck).

Original: Afterwards, the Pyramid Algorithm takes over to compute approximation coefﬁcients on higher resolution levels (the decoder).
Revised: Afterwards, the Pyramid Algorithm takes over to compute approximation coefﬁcients on higher resolution levels (the decoder).

Original: In practice, the detail coefﬁcients are negligible on sufﬁciently high resolution levels.
Revised: In practice, the detail coefﬁcients are negligible on sufﬁciently high resolution levels.

Original: For this reason, we only predict detail coefﬁcients up to a prescribed level j 1 .
Revised: For this reason, we only predict detail coefﬁcients up to a prescribed level j 1 .

Original: The predictions at higher resolution levels j 1 < j ≤ j 2 are computed without detail coefﬁcients.
Revised: The predictions at higher resolution levels j 1 < j ≤ j 2 are computed without detail coefﬁcients.

Original: The encoder consists of ﬁve down-sampling blocks.

Original: Each block consists of six (convolutional) residual blocks, using ReLU-activation and kernels of size 3 × 3 , followed by an average-pooling layer of size 2 × 2 .

Original: The number of ﬁlters used in the ﬁve blocks, from top to bottom, is 32 , 32 , 64 , 64 , 128 , respectively.

Original: The encoder is followed by a a bottleneck.

Original: The ﬁrst layer in the bottleneck is fully connected and transforms the output of the encoder to a vector in R 128 .

Original: [-Attached to this layer are ﬁve branches to predict the probability p ( x ) ∈ (0 , 1) , components [ f j 0 ( x )] s ∈ (cid:98) V j 
-]
Revised: [-Attached to this layer are ﬁve branches to predict the probability p ( x ) ∈ (0 , 1) , components [ f j 0 ( x )] s ∈ (cid:98) V j 
-]

Original: and [ h j 0 ( x )] s ∈ (cid:99) W j 0 , where s ∈ { 1 , 2 } . The latter quantity [ h j 0 ( x )] s is a prediction for the detail coefﬁcients at level j 0 . Each branch consists of four fully-connected layers. The ﬁrst three layers map from R 128 to itself with
Revised: The full architecture is summarized in Figure 4.

Original: ReLu-activation and residual connections in between.

Original: The ﬁnal fourth layer transforms the Rdimensional output to an element in R n , where n = 2 j 0 for the branches associated to the approximation and detail coefﬁcients, and n = 1 for the probability.

Original: The branch associated to the probability is followed by a sigmoid-activation.

Original: The detail coefﬁcients at levels j 0 ≤ j < j 1 are predicted using skip-connections.

Original: More precisely, the feature map from the encoder path, that corresponds to the same “spatial level”, is compressed to a feature map with four channels using a 1 × 1 convolution.

Original: Subsequently, a branch of the above form is used to predict detail coefﬁcients in R 2 j (cid:39) (cid:99) W j .

Original: Altogether, the predicted approximation coefﬁcients at level j 0 and detail coefﬁcients at levels j 0 ≤ j ≤ j 1 − 1 are used as input to the

Original: Pyramid algorithm and together form the upsampling path.

Original: 3. L OSS
Revised: 3. L OSS

Original: The loss L consists of two parts: ordinary cross-entropy for optimizing the likelihood of p ( x ) and a part corresponding to the L 2 -error between observed and predicted curves on different resolution levels.
Revised: The loss L consists of two parts: ordinary cross-entropy for optimizing the likelihood of p ( x ) and a part corresponding to the L 2 -error between observed and predicted curves on different resolution levels.

Original: More precisely, set r ( x ) = 

Revised: More precisely, set r ( x ) =

Original: if R ( x ) (cid:54) = ∅ and r ( x ) = 

Revised: if R ( x ) (cid:54) = ∅ and r ( x ) =

Original: otherwise, then
Revised: otherwise, then

Original: L ce ( p ( x ) , r ( x )) :
Revised: L ce ( p ( x ) , r ( x )) :

Original: = r ( x ) log p ( x ) + [Equation]
Revised: = r ( x ) log p ( x ) + (1 − r ( x ))

Original: 1 − r ( x )) log(1 − p ( x )) .
Revised: log(1 − p ( x )) .

Original: Next, suppose a ([ γ ∗ ( x )]) =
Revised: Next, suppose a ([ γ ∗ ( x )]) =

Original: ( a j 0 ([ γ ∗ ( x )]) , . . .
Revised: ( a j 0 ([ γ ∗ ( x )]) , . . .

Original: , a j 2 ([ γ ∗ ( x )]))∈ (cid:81) j 2 j = j 0 (cid:98) V j × (cid:98) V j are the approximation coefﬁcients of γ ∗ ( x ) on resolution levels j 0 ≤ j ≤ j 2 .
Revised: , a j 2 ([ γ ∗ ( x )])) are the approximation coefﬁcients of γ ∗ ( x ) on resolution levels j 0 ≤ j ≤ j 2 .

Original: Deﬁne L js ( f j ( x ) , a j ( γ ∗ ( x ))) :
Revised: Deﬁne L js ( f j ( x ) , a j ( γ ∗ ( x ))) :

Original: = (cid:107) [ f j ( x )] s − a j ([ γ ∗ ( x )] s ) (cid:107) 2 for s ∈ { 1 , 2 }  .
Revised: = (cid:107) [ f j ( x )] s − a j ([ γ ∗ ( x )] s ) (cid:107) 2 for s ∈ { 1 , 2 } and j ∈ { j 0 , j 2 } .

Revised: Here f j ( x ) are the network predictions of the relevant approximation coefﬁcients at level j .

Original: This quantity corresponds to the L 2 -error on resolution level j between the curves with approximation coefﬁcients [ f j ( x )] s and a j ([ γ ∗ ( x )] s ) .
Revised: This loss-term corresponds to the L 2 -error on resolution level j between the curves with approximation coefﬁcients [ f j ( x )] s and a j ([ γ ∗ ( x )] s ) .

Original: Finally, deﬁne the total loss by
Revised: Finally, deﬁne the total loss by

Original: [Equation]
Revised: [Equation]

Original: where w > 0 is a weight.
Revised: where w > 0 is a weight.

Original: In practice, we set w = 1 .

Original: Notice that L measures the discrepancies between observed and predicted curves on the lowest and highest resolution levels only.
Revised: Notice that L measures the discrepancies between observed and predicted curves on the lowest and highest resolution levels only.

Original: In practice, this enforces the approximation and detail coefﬁcients at intermediate levels to agree as well; see the experiments in Section 5.
Revised: In practice, this enforces the approximation and detail coefﬁcients at intermediate levels to agree as well; see the experiments in Section 5.

Original:  T RAINING
Revised:  T RAINING

Original: In this section we describe the dataset on which we test our method.
Revised: In this section we describe the dataset on which we test our method.

Original: In addition, we provide details about preprocessing steps and model development (training).
Revised: In addition, we provide details about preprocessing steps and model development (training).

Original: 4. D ATASETS
Revised: 4. D ATASETS

Original: Toy dataset The main purpose of the toy-example is to create a setting in which the annotated contours differ substantially from annotations conﬁned to a grid.
Revised: Toy dataset The main purpose of the toy-example is to create a setting in which the annotated contours differ substantially from annotations conﬁned to a grid.

Original: For this purpose, we consider piecewise smooth curves having a ﬁnite number of non-differentiable points.
Revised: For this purpose, we consider piecewise smooth curves having a ﬁnite number of non-differentiable points.

Original: The toy-dataset consists of hypocycloids, up to an Euclidian motion and scaling, deﬁned by
Revised: The toy-dataset consists of hypocycloids, up to an Euclidian motion and scaling, deﬁned by

Original: [Equation]
Revised: [Equation]

Original: where r 1 > r 2 and t ∈ R .
Revised: where r 1 > r 2 and t ∈ R .

Original: If r 1 r 2 ∈ N , then η is closed and has exactly r 1 r 2 cusps (non-differentiable points).
Revised: If r 1 r 2 ∈ N , then η is closed and has exactly r 1 r 2 cusps (non-differentiable points).

Original: To easily control the number of cusps, we ﬁx r 2 = 1 and vary r 1 .
Revised: To easily control the number of cusps, we ﬁx r 2 = 1 and vary r 1 .

Original: Note that in this case η has r 1 cusps and period 2 π .
Revised: Note that in this case η has r 1 cusps and period 2 π .

Original: We construct curves and binary masks of various sizes, orientation and positions, by sampling a radius r 1 ∈ { 3 , 4 , 5 , 6 } from U ( { 3 , 4 , 5 , 6 } ) , angle θ from U ([ − π 2 , π 2 ]) , components q 1 , q 2 from
Revised: We construct curves and binary masks of various sizes, orientation and positions, by sampling a radius r 1 ∈ { 3 , 4 , 5 , 6 } from U ( { 3 , 4 , 5 , 6 } ) , angle θ from U ([ − π 2 , π 2 ]) , components q 1 , q 2 from

Original: U ([ − 80 , 80]) for a shift, and a scaling factor κ from U ([10 , 20]) .
Revised: U ([ − 80 , 80]) for a shift, and a scaling factor κ from U ([10 , 20]) .

Original: Here U ( I ) denotes a random variable uniformly distributed on I , where I is an interval of ﬁnite length or a discrete ﬁnite set.
Revised: Here U ( I ) denotes a random variable uniformly distributed on I , where I is an interval of ﬁnite length or a discrete ﬁnite set.

Original: Next, we evaluate the curve κ (cid:16) R ( θ ) η
Revised: Next, we evaluate the curve κ (cid:16) R ( θ ) η

Original: + [160 + q160 + q 2 ] T (cid:17) on an equispaced grid of [0 , 2 π ] of size 512 .
Revised: + [160 + q160 + q 2 ] T (cid:17) on an equispaced grid of [0 , 2 π ] of size 512 .

Original: Here R ( θ ) corresponds to an anti-clockwise rotation around the origin with angle θ .
Revised: Here R ( θ ) corresponds to an anti-clockwise rotation around the origin with angle θ .

Original: Finally, the discretized curve is used to construct a binary mask of size 320 × 320 using SKIMAGE .
Revised: Finally, the discretized curve is used to construct a binary mask of size 320 × 320 using SKIMAGE .

Original: Medical decathlon The data used to evaluate the performance of our model consists of MRI images of the prostate  and CT scans of the spleen.
Revised: Medical decathlon The data used to evaluate the performance of our model consists of MRI images of the prostate central gland, henceforth abbreviated as just the prostate, and CT scans of the spleen.

Original: The datasets are part of a public dataset made available for the Medical Decathlon Contest Simpson et al.
Revised: The datasets are part of a public dataset made available for the Medical Decathlon Contest Simpson et al.

Original: The dataset for the prostate consists of T 2 -weighted MRI images of size 320 × 320 , which were cropped to size 224 × 224 .
Revised: The dataset for the prostate consists of T 2 -weighted MRI images of size 320 × 320 , which were cropped to size 224 × 224 .

Original: The dataset for the spleen consists of CT scans of size 512 × 512 and was cropped to size 256 × 256 .
Revised: The dataset for the spleen consists of CT scans of size 512 × 512 and was cropped to size 256 × 256 .

Original: The cropping was based on constructing bounding boxes of the form [ u min − δ p , u max + δ p ] × [ v min − δ p , v max + δ p ] for the training set, where u min , v min , u max and v max are the minimal and maximal coordinates of the segmentation in each direction, respectively, using an offset of δ p = 65 pixels.
Revised: The cropping was based on constructing bounding boxes of the form [ u min − δ p , u max + δ p ] × [ v min − δ p , v max + δ p ] for the training set, where u min , v min , u max and v max are the minimal and maximal coordinates of the segmentation in each direction, respectively, using an offset of δ p = 65 pixels.

Original: A residual CNN (encoder of ﬁve blocks) was trained (and validated) on the training set to regress the corner and center points of the bounding boxes using a RMSE-loss.
Revised: A residual CNN (encoder of ﬁve blocks) was trained (and validated) on the training set to regress the corner and center points of the bounding boxes using a RMSE-loss.

Original: This rather crude approach is not meant to produce tight bounding boxes, but serves as a (rough) localization step to improve performance, and allows us to focus on the task of shape-prediction only.
Revised: This rather crude approach is not meant to produce tight bounding boxes, but serves as a rough necessary localization step to improve performance, and allows us to focus on the task of shape-prediction only.

Original: 4. C ONSTRUCTION GROUND - TRUTH
Revised: 4. C ONSTRUCTION GROUND - TRUTH

Original: In this section we describe how the ground-truth data is generated using the Pyramid Algorithm and
Revised: In this section we describe how the ground-truth data is generated using the Pyramid Algorithm and

Original: Lemma A.1.
Revised: Lemma A.1.

Original: Let ( x, u ) ∈
Revised: Let ( x, u ) ∈

Original: X × R n s × n p be an image (slice) - contour pair, where x is a slice of the
Revised: X × R n s × n p be an image (slice) - contour pair, where x is a slice of the

Original: CT or MRI scan, u is a ﬁnite sequence of points approximating a closed curve and n s = 2 is the number of spatial components.
Revised: CT or MRI scan, u is a ﬁnite sequence of points approximating a closed curve and n s = 2 is the number of spatial components.

Original: Since we only have access to binary masks (for the public datasets), and not to the raw annotations themselves, we extract u using OPENCV .
Revised: Since we only have access to binary masks (for the public datasets), and not to the raw annotations themselves, we extract u using OPENCV .

Original: While not ideal, we stress that u contains “subpixel” information and is not constrainted to an integer-valued grid.
Revised: While not ideal, we stress that u contains “subpixel” information and is not constrainted to an integer-valued grid.

Original: Fourier coefﬁcients To initialize the Pyramid Algorithm, we compute the approximation coefﬁcients at level j 1 using Lemma A.1.
Revised: Fourier coefﬁcients To initialize the Pyramid Algorithm, we compute the approximation coefﬁcients at level j 1 using Lemma A.1.

Original: To accomplish this, we need to compute a Fourier expansion for u .
Revised: To accomplish this, we need to compute a Fourier expansion for u .

Original: First, we parameterize the contour by arc length.
Revised: First, we parameterize the contour by arc length.

Original: The arc length l is approximated by summing up the Euclidian distances between subsequent points in u .
Revised: The arc length l is approximated by summing up the Euclidian distances between subsequent points in u .

Original: The Fourier coefﬁcients are then computed by evaluating the contour on an equispaced grid of [0 , l ] of size 2 N − 1 , where N ∈ N , using linear interpolation and the Discrete Fourier Transform.
Revised: The Fourier coefﬁcients are then computed by evaluating the contour on an equispaced grid of [0 , l ] of size 2 N − 1 , where N ∈ N , using linear interpolation and the Discrete Fourier Transform.

Original: Since the contours are real-valued, we only store the Fourier coefﬁcients (˜ γ m ) N − 1 m =0 ∈
Revised: Since the contours are real-valued, we only store the Fourier coefﬁcients (˜ γ m ) N − 1 m =0 ∈

Original: ( C n s ) N .
Revised: ( C n s ) N .

Original: Fourier coefﬁcients that are too small, i.e., have no relevant contribution, are set to zero; see Appendix B.1 for the details.
Revised: Fourier coefﬁcients that are too small, i.e., have no relevant contribution, are set to zero; see Appendix B.1 for the details.

Original: Consistency To have consistent parameterizations for all slices, we ensure that u is always traversed anti clock-wise (using opencv ).
Revised: Consistency To have consistent parameterizations for all slices, we ensure that u is always traversed anti clock-wise (using opencv ).

Original: Furthermore, since the parameterization is only determined up to a translation in time, we need to pick out a speciﬁc one.
Revised: Furthermore, since the parameterization is only determined up to a translation in time, we need to pick out a speciﬁc one.

Original: We choose the unique parameterization such that the contour starts at angle zero at time zero relative to the midpoint c
Revised: We choose the unique parameterization such that the contour starts at angle zero at time zero relative to the midpoint c

Original: = ( c 1 , c 2 ) ∈ Rof R .
Revised: = ( c 1 , c 2 ) ∈ Rof R .

Original: The implementation details are provided in Appendix B.2.
Revised: The implementation details are provided in Appendix B.2.

Original: Approximation and detail coefﬁcients
Revised: Approximation and detail coefﬁcients

Original: Altogether, the above steps yield a contour γ with Fourier coefﬁcients ( γ m )
Revised: Altogether, the above steps yield a contour γ with Fourier coefﬁcients ( γ m )

Original: N − 1 m =1 − N and period l .
Revised: N − 1 m =1 − N and period l .

Original: We reparameterize γ to have period 1 , as explained in Section 2.3, and “center” the contour using the average midpoint computed over the training-set.
Revised: We reparameterize γ to have period 1 , as explained in Section 2.3, and “center” the contour using the average midpoint computed over the training-set.

Original: The initial approximation coefﬁcients at level j 2 are then computed using Lemma A.1.
Revised: The initial approximation coefﬁcients at level j 2 are then computed using Lemma A.1.

Original: Next, the Pyramid
Revised: Next, the Pyramid

Original: Algorithm is used to compute approximation and detail coefﬁcients at levels j 0 ≤ j ≤ j 2 − 1 .
Revised: Algorithm is used to compute approximation and detail coefﬁcients at levels j 0 ≤ j ≤ j 2 − 1 .

Original: We set the detail coefﬁcients which are in absolute value below ε = 5 · 10 − 3 to zero to reduce noise.
Revised: We set the detail coefﬁcients which are in absolute value below ε =· 10 − 3 to zero to reduce noise.

Original: Subsequently, we reconstruct the approximation coefﬁcients at levels j 0 + 1 ≤ j ≤ jusing the thresholded detail coefﬁcients.
Revised: Subsequently, we reconstruct the approximation coefﬁcients at levels j 0 + 1 ≤ j ≤ j 1 using the thresholded detail coefﬁcients.

Original: No detail coefﬁcients are used to compute approximation coefﬁcients at levels
Revised: No detail coefﬁcients are used to compute approximation coefﬁcients at levels

Original: j 1 + 1 ≤ j ≤ j 2 .
Revised: j 1 + 1 ≤ j ≤ j 2 .

Original: The ﬁnal approximation and (thresholded) detail coefﬁcients a∈ (cid:76) j 2 j = j 0 (cid:98) V j × (cid:98) V j and d∈ (cid:76) j 1 − 1 j = j 0 (cid:99) W j × (cid:99) W j , respectively, are used as ground-truth.
Revised: The ﬁnal approximation and (thresholded) detail coefﬁcients a and d , respectively, are used as ground-truth.

Original: The resulting dataset D thus consists of tuples ( x, a, d ) .
Revised: The resulting dataset D thus consists of tuples ( x, a, d ) .

Original: The training and validation set D train and
Revised: The training and validation set D train and

Original: D val , respectively, are obtained by randomly omitting subjects from the the full dataset D .
Revised: D val , respectively, are obtained by randomly omitting subjects from the the full dataset D .

Original: For the toy-example, spleen and prostate we have ( |D train | , |D val | ) =
Revised: For the toy-example, spleen and prostate we have ( |D train | , |D val | ) =

Original: (1650 , 250) , ( |D train | , |D val | )
Revised: (1650 , 250) , ( |D train | , |D val | )

Original: = (527 , 75) , and ( |D train | , |D val | ) =
Revised: = (527 , 75) , and ( |D train | , |D val | ) =

Original: (3148 , 502) , respectively.
Revised: (3148 , 502) , respectively.

Original: Before feeding the images x into the model, we linearly rescale the image intensities at each instance to [0 , 1] .
Revised: Before feeding the images x into the model, we linearly rescale the image intensities at each instance to [0 , 1] .

Original: Furthermore, for the spleen and prostate, we use extensive data augmentation: we use random shifts, random rotations, random scaling, elastic deformations, horizontal shearing and random cropping.
Revised: Furthermore, for the spleen and prostate, we use extensive data augmentation: we use random shifts, random rotations, random scaling, elastic deformations, horizontal shearing and random cropping.

Original: 4. M ODEL TRAINING
Revised: 4. M ODEL TRAINING

Original: We use the Adam optimizer Kingma & Ba (2015) to train our network for 150 epochs.
Revised: We use the Adam optimizer Kingma & Ba (2015) to train our network for 150 epochs.

Original: In the ﬁrst ﬁve epochs, we use a linearly increasing learning rate from 5 · 10 − 4 to 10 − 3 , which is subsequently decayed by a factor of 0 . 5 each time the loss does not signiﬁcantly decrease for 10 subsequent epochs (for the remaining 145 epochs).
Revised: In the ﬁrst ﬁve epochs, we use a linearly increasing learning rate from 5 · 10 − 4 to 10 − 3 , which is subsequently decayed by a factor of 0 . 5 each time the loss does not signiﬁcantly decrease for 10 subsequent epochs (for the remaining 145 epochs).

Original: A batch size of 16 samples is used in each descent step.
Revised: A batch size of 16 samples is used in each descent step.

Original: Finally, the model with the lowest loss is selected.
Revised: Finally, the model with the lowest loss is selected.

Original: The computations were performed in P Y T ORCH on a Geforce RTX 2080 Ti.
Revised: The computations were performed in P Y T ORCH on a Geforce RTX 2080 Ti.

Original:  R ESULTS
Revised:  R ESULTS

Original: In this section we examine the performance of our method.
Revised: In this section we examine the performance of our method and its dependence on the choice of basis.

Original: We investigate the dependence on the choice of basis and smoothness; we consider the Debauches wavelets db p with p ∈ { 1 , 2 , 4 , 8 , 16 } vanishing moments.
Revised: We ﬁx all other hyper-parameters (as much as possible). We consider the Debauches wavelets db p with p ∈ { 1 , 2 , 4 , 8 , 16 } vanishing moments.

Original: Roughly speaking, db p corresponds to the unique MRA for which the mother wavelet has minimal support and p vanishing moments.
Revised: Roughly speaking, db p corresponds to the unique MRA for which the mother wavelet has minimal support and p vanishing moments.

Original: For each basis, we ﬁx appropriate resolution levels j 0 , j 1 and j 2 as follows.
Revised: For each basis, we ﬁx appropriate resolution levels j 0 , j 1 and j 2 as follows.

Original: We choose j 0 as the smallest  level for which  2 j 0 − 1 is larger or equal to the order of the low-pass-ﬁlter.
Revised: We ﬁx j 0 as the smallest possible resolution level for which the length of our signal ( 2 j 0 ) still exceeds the length of the low-pass ﬁlter.

Original: Sincethe low-pass ﬁlter associated to db p has order 2 p , we set j 0 ( p ) =
Revised: Since db p has a low-pass ﬁlter of length 2 p , this forces j 0 ( p ) =

Original: p .
Revised: p .

Revised: The choice for j 1 ≥ j 0 is based on the observations that the norm of d j decreases as j increases and small detail coefﬁcients have no signiﬁcant impact anymore (see Figure 11).

Original: We choose j 1 > j 0 as the smallest level for which the detail coefﬁcients are below ε = 5 · 10 − 3 for all samples in the training set.
Revised: We ﬁx j 1 ≥ j 0 as the smallest level for which | [ d j 1 k ] s | < 5 · 10 − 3 for all samples in the training set (also see the discussion in Section 4.2).

Revised: The ﬁnal resolution level j 2 and the support of the mother wavelet determine how much of the contour is covered, see Figure 3.

Original: We choose j 2 ≥ j 1 as the smallest resolution level for which the distance between the end-points of the curves are within 1 pixel distance for all samples in the training set.
Revised: We therefore ﬁx j 2 ≥ j 1 as the smallest resolution level for which the distance between the end-points of the curves are within 1 pixel distance for all samples in the training set.

Original: Finally, we use 64, 32 and 64 Fourier coefﬁcients for the the toy-problem, spleen and prostate, respectively, to initialize theapproximation coefﬁcients.
Revised: We use 64 Fourier coefﬁcients for the the toy-problem, spleen and prostate, respectively to initialize the

Original: Baseline For the prostate and spleen we use theoriginal two-dimensional U-Net in Ronneberger et al.
Revised: Baseline For the prostate and spleen we use the U-Net in Ronneberger et al.

Original: (2015) as baseline with  the following differences: we use 32 ﬁlters in the ﬁrst layer instead of 64 and use group normalization with four groups.
Revised: (2015) and a recently developed variant in Nikolov et al. (2018) for comparison. We have modiﬁed the network architectures to match the parameter count with our networks (approximately), while maintaining the structure and idea’s presented in the original papers as much as possible, see Appendix C. We stress, however, that our objective, i.e., parameterizing contours, is different from the U-Net’s objective. The binary ground-truth matched by a U-Net is a fundamentally different (often easier) object than the continuous representation of a curve matched by our networks.

Revised: Subtle curvature and geometry may be accurately presented using our ground-truth curves, e.g., by using a sufﬁciently large number of Fourier coefﬁcients to compute approximation coefﬁcients.

Revised: Binary ground-truth masks, however, cannot capture such subtle geometry due to their discrete nature.

Original: Performance measures We evaluate accuracy using two-dimensional quantities only, since our models are 2 d. We compute the component-wise L 2 -errors between observed and predicted curves on the highest resolution level j 2 by taking the (cid:96) 2 -norm of the approximation coefﬁcients.
Revised: Performance measures We evaluate accuracy using two-dimensional quantities only, since our models are 2 d. We compute the component-wise L 2 -errors between observed and predicted curves on the highest resolution level j 2 by taking the (cid:96) 2 -norm of the approximation coefﬁcients.

Original: Furthermore, we compute the dice score and Haussdorf distance between curves using the implementation in SHAPELY .
Revised: Furthermore, we compute the dice score and Haussdorf distance between curves using the implementation in SHAPELY .

Original: This requires a polygonal approximation of the contour, which is easily obtained using the approximation coefﬁcients at level j 2 .
Revised: This requires a polygonal approximation of the contour, which is easily obtained using the approximation coefﬁcients at level j 2 .

Original: Note that the Hausdorff distance between subsets of a general metric space(baseline) may differ from the Hausdorff distance between the associated boundaries (wavelet models), but coincide for (compact) simply connected subsets of R n .
Revised: Note that the Hausdorff distance between subsets of a general metric space may differ from the Hausdorff distance between the associated boundaries (wavelet models), but coincide for (compact) simply connected subsets of R n .

Original: The results are shown in Table 1.
Revised: The results are shown in Table 1.

Original: The predictions for the best performing wavelet models with respect to the Hausdorff-score are shown in Figure 1.
Revised: The predictions for the best performing wavelet models with respect to the Hausdorffscore are shown in Figure 5.

Original: Predictions for other bases, corresponding wavelet decompositions, and detailed visualization of statistics (violin plots) can be found in Appendix C.
Revised: Predictions for other bases, corresponding wavelet decompositions, and detailed visualization of statistics (violin plots) can be found in Appendix D.

Original: Toy problem We observe that all models perform well and are capable of accurately parameterizing piecewise smooth curves.
Revised: Toy problem We observe that all models perform well and are capable of accurately parameterizing piecewise smooth curves.

Original: For db 1 , however, we observe relatively large gaps between the end-points, since its support is relatively small.
Revised: For db 1 , however, we observe relatively large gaps between the end-points, since its support is relatively small.

Original: In addition, db 1 has difﬁculty with accurately predicting small “densely” sampled cycloids.
Revised: In addition, db 1 has difﬁculty with accurately predicting small “densely” sampled cycloids.

Original: In general, the predictions associated to the less regular wavelets db 1 and db 2 sometimes exhibit “small” oscillatory behavior.
Revised: In general, the predictions associated to the less regular wavelets db 1 and db 2 sometimes exhibit “small” oscillatory behavior.

Original: We found that the latter two issues were caused by a too large resolution level j 2 .
Revised: We found that the latter two issues were caused by a too large resolution level j 2 .

Original: To see why, note that the features extracted from the images only contain information up to a certain resolution level.
Revised: To see why, note that the features extracted from the images only contain information up to a certain resolution level.

Original: The subpixel information needed to “ﬁll in the blanks”, so to speak, is in part provided by the ground-truth data and in part by the chosen wavelet basis.
Revised: The subpixel information needed to “ﬁll in the blanks”, so to speak, is in part provided by the ground-truth data and in part by the chosen wavelet basis.

Original: The regularity and support of the wavelet determines to which extent, i.e., up to which resolution level, subpixel information can be “ﬁlled in”.
Revised: The regularity and support of the wavelet determines to which extent, i.e., up to which resolution level, subpixel information can be “ﬁlled in”.

Original: As the regularity (and support) of the wavelet decreases, the maximal achievable resolution level decreases as well.
Revised: As the regularity (and support) of the wavelet decreases, the maximal achievable resolution level decreases as well.

Original: Spleen The predictions of our models are accurate and on par with the baseline U-Net.
Revised: Spleen The predictions of our models are accurate and on par with the baseline U-Nets.

Original: In terms of the dice score, the U-Net performs slightly better.
Revised: The more advanced U-Net in Nikolov et al.

Original: This is mostly due to “edge” cases where the boundary of the spleen is small and about to disappear from our two-dimensional sliced-view.
Revised: (2018) performs slightly better; this is mostly due to “edge” cases where the boundary of the spleen is small and about to disappear from our two-dimensional slicedview.

Original: In such cases it may sometimes be ambiguous to deﬁne an accurate ground-truth contour.
Revised: In such cases it may sometimes be ambiguous to deﬁne an accurate ground-truth contour,

Original: On the other hand, our models perform slightly better in terms of Hausdorff-distance. This is reﬂected in
Revised: [Table] [Table]

Original: [Table]
Revised: [Table]

Original: smaller averages, but more noticeably our models follow the boundaries more faithfully resulting in a smaller variation around the mean.
Revised: resulting in curves with subtle spurious curvature. Such geometry is not (and cannot be) present in the binary ground-truth mask due its discrete nature. Prostate central gland

Original: Prostate The wavelet models produce accurate predictions and are on par with the U-Net.
Revised: The wavelet models produce accurate predictions and are on par with the U-Nets.

Original: Our models perform slightly better in terms of Hausdorff distance.
Revised: Our models perform slightly better in terms of Hausdorff distance.

Original: While our models produce accurate predictions for most examples, there are instances where both our model and the baseline U-Net fail to produce accurate predictions; see the last two columns of Figure 1.
Revised: While all models produce accurate predictions for most examples, there are instances where both the wavelet models and the baseline U-Nets fail to produce accurate predictions; see the last two columns of Figure 5.

Original: In these examples, the detail coefﬁcients associated to parts of the curve with high curvature are too small in magnitude to be accurately predicted.
Revised: In these examples, the detail coefﬁcients associated to parts of the curve with high curvature are too small in magnitude to be accurately predicted.

Original: While detail coefﬁcients of such small magnitude were less relevant in the latter two examples, they are important for the prostate.
Revised: While detail coefﬁcients of such small magnitude were less relevant in the latter two examples, they are important for the prostate.

Original: Ablation study (no detail coefﬁcients) To demonstrate the importance of the detail coefﬁcients, we have trained models without them (and skip-connections).
Revised: Ablation study (no detail coefﬁcients) To demonstrate the importance of the detail coefﬁcients, we have trained models without them, i.e., without the skip-connections.

Original: In this set up, we do not supervise the predictions on the lowest resolution level during training.
Revised: In this set up, we do not supervise the predictions on the lowest resolution level during training.

Original: The results demonstrate, as expected, that the lower-order wavelets db 1 , db 2 perform signiﬁcantly worse without detail coefﬁcients.
Revised: The results demonstrate, as expected, that the lower-order wavelets db 1 , db 2 perform signiﬁcantly worse without detail coefﬁcients.

Original: In fact, for db 1 our model failed to produce any sensible approximations that can be evaluated and were therefore omitted.
Revised: In fact, for db 1 our model failed to produce any sensible approximations that can be evaluated and were therefore omitted.

Original: A small drop in performance is observed for db 4 .
Revised: A small drop in performance is observed for db 4 .

Original: In general, for db 16 there is not much gain, with respect to accuracy, memory-footprint and inference time, to explicitly model the detail coefﬁcients.
Revised: In general, for dbthere is not much gain, with respect to accuracy, memory-footprint and inference time, to explicitly model the detail coefﬁcients.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: We have introduced a novel method to model boundaries of two dimensional simply connected domains using wavelets and MRAs.
Revised: We have introduced a novel method to model boundaries of two dimensional simply connected domains using wavelets and MRAs.

Original: In effect this allows for subpixel segmentations.
Revised: In effect this allows for subpixel segmentations.

Original: The efﬁcacy of the method has been demonstrated by modeling the boundaries of hypercloids (toy-example), spleen and prostate,and demonstrating that the results are on par with a traditional U-Net yielding up to ﬁve times faster inference speed.
Revised: The efﬁcacy of the method has been demonstrated by modeling the boundaries of hypercloids (toy-example), spleen and prostate, demonstrating that the results are on par with typical U-Nets, yielding up to ﬁve times faster inference speed.

Original: A T HEORETICAL RESULTS
Revised: A T HEORETICAL RESULTS

Original: In this section we prove theoretical results regarding the computation of wavelet decompositions.
Revised: In this section we prove theoretical results regarding the computation of wavelet decompositions.

Original: In addition, we provide all the implementation details.
Revised: In addition we provide all the implementation details

Original: A. E XPLICIT EXPRESSION FOR a jk
Revised: A. E XPLICIT EXPRESSION FOR a jk

Original: To address the issue that periodic signals are not contained in L 2 ( R ), we consider the cut-off ˜ γ :
Revised: To address the issue that periodic signals are not contained in L 2 ( R ) we consider the cut-off ˜ γ :

Original: =
Revised: =

Original: γ 1 [ − l,l ] .
Revised: γ 1 [ − l,l ] .

Original: In this section we derive an explicit formula for (cid:104) ˜ γ, ϕ jk (cid:105) by exploiting the periodicity of γ .
Revised: In this section we derive an explicit formula for (cid:104) ˜ γ, ϕ jk (cid:105) by exploiting the periodicity of γ .

Original: In particular, we will quantify the claim that (scaled) sample values of γ may be used as approximation coefﬁcients.
Revised: In particular, we will quantify the claim that (scaled) sample values of γ may be used as approximation coefﬁcients.

Original: For this purpose, we ﬁrst review some facts about the scaling equation.
Revised: For this purpose, we ﬁrst review some facts about the scaling equation.

Original: If we take the Fourier transform of both sides of the scaling equation, we obtain the relation ˆ ϕ ( ξ ) =
Revised: If we take the Fourier transform of both sides of the scaling equation, we obtain the relation ˆ ϕ ( ξ ) =

Original: H ( ξ 2 ) ˆ ϕ ( ξ 2 ) for ξ ∈ R , where H ( ξ ) = 1 √ 2 (cid:80)
Revised: H ( ξ 2 ) ˆ ϕ ( ξ 2 ) for ξ ∈ R , where H ( ξ ) = 1 √ 2 (cid:80)

Original: k
Revised: k

Original: ∈
Revised: ∈

Original: Z h k e − i 2 πξk .
Revised: Z h k e − i 2 πξk .

Original: The map H is referred to as the reﬁnement mask associated to h .
Revised: The map H is referred to as the reﬁnement mask associated to h .

Original: Throughout this paper we assume that h is ﬁnite, in which case ϕ has compact support and H is a trigonometric polynomial of period 1 .
Revised: Throughout this paper we assume that h is ﬁnite, in which case ϕ has compact support and H is a trigonometric polynomial of period 1 .

Original: It is beyond the scope of this paper to discuss the properties of H in detail and refer the reader to Pereyra & Ward (2012);
Revised: It is beyond the scope of this paper to discuss the properties of H in detail and refer the reader to Pereyra & Ward (2012);

Original: Montefusco & Puccio (2014).
Revised: Montefusco & Puccio (2014).

Original: Here we only need the following result.
Revised: Here we only need the following result.

Original: Under suitable conditions, one may iterate the equation for ˆ ϕ and show that ˆ ϕ ( ξ )
Revised: Under suitable conditions, one may iterate the equation for ˆ ϕ and show that ˆ ϕ ( ξ )

Original: =
Revised: =

Original: (cid:81) ∞
Revised: (cid:81) ∞

Original: k =1 H ( ξ 2 k ) for all ξ ∈ R .
Revised: k =1 H ( ξ 2 k ) for all ξ ∈ R .

Original: Lemma A.1 (Initialization approximation coefﬁcients) .
Revised: Lemma A.1 (Initialization approximation coefﬁcients) .

Original: Let h ∈ (cid:96) 

Revised: Let h ∈ (cid:96)

Original: ( Z ) be a low-pass-ﬁlter deﬁning a MRA on L 2 ( R ) and H the associated reﬁnement mask.
Revised: ( Z ) be a low-pass-ﬁlter deﬁning a MRA on L 2 ( R ) and H the associated reﬁnement mask.

Original: Assume h is non-zero for only a ﬁnite number of coefﬁcients, supp ( ϕ ) ⊂
Revised: Assume h is non-zero for only a ﬁnite number of coefﬁcients, supp ( ϕ ) ⊂

Original: [0 , β ] for some β > 0  and ϕ is piecewise continuous.
Revised: [0 , β ] for some β > 0 , and ϕ is bounded.

Original: Furthermore, suppose that ˆ ϕ ( ξ ) =
Revised: Furthermore, suppose that ˆ ϕ ( ξ ) =

Original: (cid:81) ∞ n =1 H ( ξ 2 n ) for all ξ ∈ R .
Revised: (cid:81) ∞ n =1 H ( ξ 2 n ) for all ξ ∈ R .

Original: If γ ∈ C 2 per ([0 , l ]) is a l -periodic map with Fourier coefﬁcients ( γ m ) m ∈ Z , then
Revised: If γ ∈ C 2 per ([0 , l ]) is a l -periodic map with Fourier coefﬁcients ( γ m ) m ∈ Z , then

Original: [Equation]
Revised: [Equation]

Original: ) where ω ( l ) := 2 πl is the angular frequency of γ , for any j ∈ Z and k ∈ {(cid:100)− 2 j l (cid:101) , . . .
Revised: where ω ( l ) := 2 πl is the angular frequency of γ , for any j ∈ Z and k ∈ {(cid:100)− 2 j l (cid:101) , . . .

Original: , (cid:98) 

Revised: , (cid:98)

Original: j l − β (cid:99)} .
Revised: j l − β (cid:99)} .

Original: Proof.
Revised: Proof.

Original: Let j ∈ Z and k ∈ {(cid:100)− 2 j l (cid:101) , . .
Revised: Let j ∈ Z and k ∈ {(cid:100)− 2 j l (cid:101) , . .

Original: , (cid:98) 

Revised: , (cid:98)

Original: j l − β (cid:99)} be arbitrary.
Revised: j l − β (cid:99)} be arbitrary.

Original: A change of variables shows that
Revised: A change of variables shows that

Original: [Equation]
Revised: [Equation]

Original: since supp ( ϕ ) ⊂ [0 , β ] .
Revised: since supp ( ϕ ) ⊂ [0 , β ] .

Original: In particular, note that 2 − j ( t + k ) ∈
Revised: In particular, note that 2 − j ( t + k ) ∈

Original: [ − l, l ] for t ∈ [0 , β ] , since k ∈ {(cid:100)− 2 j l (cid:101) , . . .
Revised: [ − l, l ] for t ∈ [0 , β ] , since k ∈ {(cid:100)− 2 j l (cid:101) , . . .

Original: , (cid:98) 

Revised: , (cid:98)

Original: j l − β (cid:99)} .
Revised: j l − β (cid:99)} .

Original: Therefore, we may plug in the Fourier expansion for γ and compute
Revised: Therefore, we may plug in the Fourier expansion for γ and compute

Original: [Equation]
Revised: [Equation]

Original: Next, note that that series inside the integral converges pointwise to γ (cid:0) 2 − j ( t + k ) (cid:1) ϕ
Revised: Next, note that that series inside the integral converges pointwise to γ (cid:0) 2 − j ( t + k ) (cid:1) ϕ

Original: ( t ) on [0 , β ] .
Revised: ( t ) on [0 , β ] .

Original: Furthermore, the partial sums can be bounded from above on [0 , β ] by a constant, since γ ∈ C 2 per ([0 , l ]) and ϕ is bounded.
Revised: Furthermore, the partial sums can be bounded from above on [0 , β ] by a constant, since γ ∈ C 2 per ([0 , l ]) and ϕ is bounded.

Original: Therefore, we may interchange the order of summation and integration by the Dominated Convergence Theorem:
Revised: Therefore, we may interchange the order of summation and integration by the Dominated Convergence Theorem:

Original: [Equation]
Revised: [Equation]

Original: Finally, changing the domain of integration to R again, we see that
Revised: Finally, changing the domain of integration to R again, we see that

Original: [Equation]
Revised: [Equation]

Original: The stated result now follows from the assumption that ˆ ϕ ( ξ )
Revised: The stated result now follows from the assumption that ˆ ϕ ( ξ )

Original: =
Revised: =

Original: (cid:81) ∞
Revised: (cid:81) ∞

Original: k =1 H ( ξ 2 k ) for any ξ ∈ R .
Revised: k =1 H ( ξ 2 k ) for any ξ ∈ R .

Original: Remark A.2.
Revised: Remark A.2.

Original: The bounds (cid:100)− 2 j l (cid:101) and (cid:98) 

Revised: The bounds (cid:100)− 2 j l (cid:101) and (cid:98)

Original: j l − β (cid:99) are the smallest and largest integer, respectively, for which 2 − j ( t + k ) ∈ [ − l, l ] for all t ∈ [0 , β ] .
Revised: j l − β (cid:99) are the smallest and largest integer, respectively, for which− j ( t + k ) ∈ [ − l, l ] for all t ∈ [0 , β ] .

Original: The bounds on k are somewhat artiﬁcial, however, since the argument may be repeated for any truncation of γ on [ − nl, nl ] , where n ∈ N .
Revised: The bounds on k are somewhat artiﬁcial, however, since the argument may be repeated for any truncation of γ on [ − nl, nl ] , where n ∈ N .

Original: This observation is reﬂected in the righthand-side of (1) , which is well-deﬁned for all k ∈ Z (but not squared-summable).
Revised: This observation is reﬂected in the righthand-side of (1) , which is well-deﬁned for all k ∈ Z (but not squared-summable).

Original: The reason for choosing this particular truncation is to identify a minimal number of approximation coefﬁcients needed to cover the full signal γ .
Revised: The reason for choosing this particular truncation is to identify a minimal number of approximation coefﬁcients needed to cover the full signal γ .

Original: The requirement that ˆ ϕ ( ξ ) = (cid:81) ∞ k =1 H ( ξ 2 k ) for ξ ∈ R is satisﬁed for all MRA’s considered in this paper.
Revised: The requirement that ˆ ϕ ( ξ ) = (cid:81) ∞ k =1 H ( ξ 2 k ) for ξ ∈ R is satisﬁed for all MRA’s considered in this paper.

Original: In particular, we remark that ˆ ϕ is continuous at ξ = 0 and ˆ ϕ (0)
Revised: In particular, we remark that ˆ ϕ is continuous at ξ = 0 and ˆ ϕ (0)

Original: = 1 .
Revised: = 1 .

Original: It follows from these observations and Lemma A.1 that a jk (˜ γ )
Revised: It follows from these observations and Lemma A.1 that a jk (˜ γ )

Original: = a jk ( γ ) ≈ 2 − j 2 γ ( k 2 − j ) for j sufﬁciently large and k constrained to {(cid:100)− 2 j l (cid:101) , . . .
Revised: = a jk ( γ ) ≈ 2 − j 2 γ ( k 2 − j ) for j sufﬁciently large and k constrained to {(cid:100)− 2 j l (cid:101) , . . .

Original: , (cid:98) 

Revised: , (cid:98)

Original: j l − β (cid:99)} .
Revised: j l − β (cid:99)} .

Original: A. P YRAMID A LGORITHM - IMPLEMENTATION DETAILS
Revised: A. P YRAMID A LGORITHM - IMPLEMENTATION DETAILS

Original: In this section we provide the computational details for how to compute approximation and detail coefﬁcients using the Discrete Fourier Transform (DFT).
Revised: In this section we provide the computational details for how to compute approximation and detail coefﬁcients using the Discrete Fourier Transform (DFT).

Original: In the following arguments all sequences are assumed to be two-sided.
Revised: In the following arguments all sequences are assumed to be two-sided.

Original: We will frequently abuse notation and write that a ﬁnite sequence is an element in C n or R n .
Revised: We will frequently abuse notation and write that a ﬁnite sequence is an element in C n or R n .

Original: What we actually mean by this, is that we have a two-sided sequence of length n which can be embedded in (cid:96) 2 ( Z ) by appropriately padding with zeros.
Revised: What we actually mean by this, is that we have a two-sided sequence of length n which can be embedded in (cid:96) 2 ( Z ) by appropriately padding with zeros.

Original: In such a situation we will explicitly specify the indices of the sequence so that the intended ordering is clear.
Revised: In such a situation we will explicitly specify the indices of the sequence so that the intended ordering is clear.

Original: Similarly, all operators in this section are implicitly assumed to be deﬁned on the associated (two-sided) sequence spaces, even though we may write that they are deﬁned on C n or R n .
Revised: Similarly, all operators in this section are implicitly assumed to be deﬁned on the associated (two-sided) sequence spaces, even though we may write that they are deﬁned on C n or R n .

Original: Let a =
Revised: Let a =

Original: ( a k ) N − 1 k = −
Revised: ( a k ) N − 1 k = −

Original: N ∈ R 2 N and g =
Revised: N ∈ R 2 N and g =

Original: ( g k ) M − 1 k =1 − M ∈ R
Revised: ( g k ) M − 1 k =1 − M ∈ R

Original: M − 1 be arbitrary.
Revised: M − 1 be arbitrary.

Original: Here g may be interpreted as a high-pass-ﬁlter of length M with N ≥ M ≥ 2 .
Revised: Here g may be interpreted as a high-pass-ﬁlter of length M with N ≥ M ≥ 2 .

Original: Similarly, we may think of a as the approximation coefﬁcients of a 1 -periodic signal at a speciﬁc resolution level j , with N = 2 j − 1 , as explained in Section 2.3.
Revised: Similarly, we may think of a as the approximation coefﬁcients of a 1 -periodic signal at a speciﬁc resolution level j , with N = 2 j − 1 , as explained in Section 2.3.

Original: In this section, however, we will not emphasize these interpretations, e.g., write a jk instead of a k , to avoid clutter in the notation.
Revised: In this section, however, we will not emphasize these interpretations, e.g., write a jk instead of a k , to avoid clutter in the notation.

Original: Observe that the righthand-side of (1) is a 2 j -periodic sequence for 1 -periodic signals.
Revised: Observe that the righthand-side of (1) is a 2 j -periodic sequence for 1 -periodic signals.

Original: Therefore, to properly deal with “boundary terms”, we will use the 2 N -periodic extension ˜ a ∈ R Z of a to evaluate discrete convolutions.
Revised: Therefore, to properly deal with “boundary terms”, we will use the 2 N -periodic extension ˜ a ∈ R Z of a to evaluate discrete convolutions.

Original: Convolution and multiplication of trigonometric polynomials To compute the detail and approximation coefﬁcients, we need to evaluate expressions of the form
Revised: Convolution and multiplication of trigonometric polynomials To compute the detail and approximation coefﬁcients, we need to evaluate expressions of the form

Original: [Equation]
Revised: [Equation]

Original: for − N ≤ k ≤ N − 1 .
Revised: for − N ≤ k ≤ N − 1 .

Original: Note that although ˜ a is an inﬁnite 2 N -periodic sequence, the series in (2) contains only a ﬁnite number of nonzero-terms, since g k 2 = 0 for | k 2 | ≥ M .
Revised: Note that although ˜ a is an inﬁnite 2 N -periodic sequence, the series in (2) contains only a ﬁnite number of nonzero-terms, since g k 2 = 0 for | k 2 | ≥ M .

Original: Furthermore, for − N ≤ k ≤ N − 1 , we do not need the full periodic extension ˜ a , but only a partial (ﬁnite) extension
Revised: Furthermore, for − N ≤ k ≤ N − 1 , we do not need the full periodic extension ˜ a , but only a partial (ﬁnite) extension

Original: P K ( a ) , where P K : C 2 N → C K is deﬁned by
Revised: P K ( a ) , where P K : C 2 N → C K is deﬁned by

Original: [Equation]
Revised: [Equation]

Original: and K := 2( N + M − 1) .
Revised: 2 N N ≤ k ≤ N + M −and K := 2( N + M − 1) .

Original: That is, (˜ a ∗ g ) k
Revised: That is, (˜ a ∗ g ) k

Original: = ( P K ( a ) ∗ g ) k for − N ≤ k ≤ N − 1 .
Revised: = ( P K ( a ) ∗ g ) k for − N ≤ k ≤ N − 1 .

Original: We use standard arguments to compute P K ( a ) ∗ g .
Revised: We use standard arguments to compute P K ( a ) ∗ g .

Original: Namely, we interpret P K ( a ) ∗ g as the Fourier coefﬁcients of uv , where u, v : R → C are the trigonometric polynomials deﬁned by
Revised: Namely, we interpret P K ( a ) ∗ g as the Fourier coefﬁcients of uv , where u, v : R → C are the trigonometric polynomials deﬁned by

Original: [Equation]
Revised: [Equation]

Original: The product uv is a trigonometric polynomial with ˜ K :
Revised: The product uv is a trigonometric polynomial with ˜ K :

Original: = K + 2( M − 1) non-zero coefﬁcients corresponding to terms of order − ˜ K 2 ≤ k ≤ ˜ K 2 − 1 .
Revised: = K + 2( M − 1) non-zero coefﬁcients corresponding to terms of order − ˜ K 2 ≤ k ≤ ˜ K 2 − 1 .

Original: The coefﬁcients of uv can be characterized by evaluating it on ˜
Revised: The coefﬁcients of uv can be characterized by evaluating it on ˜

Original: K distinct points in C .
Revised: K distinct points in C .

Original: After ﬁxing ˜ K such points, we may go back and forth between value and coefﬁcient representations of u , v and uv using the isomorphism deﬁned by the evaluation operator.
Revised: After ﬁxing ˜ K such points, we may go back and forth between value and coefﬁcient representations of u , v and uv using the isomorphism deﬁned by the evaluation operator.

Original: Evaluation at the roots of unity We evaluate u and v , in the complex variable z = e iθ , at the ˜
Revised: Evaluation at the roots of unity We evaluate u and v , in the complex variable z = e iθ , at the ˜

Original: Kth roots of unity.
Revised: Kth roots of unity.

Original: To do this, we ﬁrst extend P K ( a ) and g to sequences in C ˜ K by padding with zeros.
Revised: To do this, we ﬁrst extend P K ( a ) and g to sequences in C ˜ K by padding with zeros.

Original: More precisely, deﬁne Z even ˜ K :
Revised: More precisely, deﬁne Z even ˜ K :

Original: C K → C ˜ K by ( Z even ˜ K ( b ))
Revised: C K → C ˜ K by ( Z even ˜ K ( b ))

Original: k
Revised: k

Original: = b k for − K 2 ≤ k ≤ K 2 − 1 and zero for − ˜ K 2 ≤ k < − K
Revised: = b k for − K 2 ≤ k ≤ K 2 − 1 and zero for − ˜ K 2 ≤ k < − K

Original: and K 2 ≤ k < ˜ K 2 .
Revised: and K 2 ≤ k < ˜ K 2 .

Original: Similarly, deﬁne Z odd M : C 2 M − 1 → C ˜ K by ( Z odd M ( b ))
Revised: Similarly, deﬁne Z odd M : C 2 M − 1 → C ˜ K by ( Z odd M ( b ))

Original: k
Revised: k

Original: = b k for | k | < M and zero for − ˜ K 2 ≤ k ≤ − M and M ≤ k < ˜ K 2 .
Revised: = b k for | k | < M and zero for − ˜ K 2 ≤ k ≤ − M and M ≤ k < ˜ K 2 .

Original: We can now evaluate u and v at the ˜ K -th roots of unity by computing
Revised: We can now evaluate u and v at the ˜ K -th roots of unity by computing

Original: DFT ˜ K ◦ S ˜ K ◦ Z even ˜ K ◦ P K ( a ) and ˆG
Revised: DFT ˜ K ◦ S ˜ K ◦ Z even ˜ K ◦ P K ( a ) and ˆG

Original: ˜ K :=
Revised: ˜ K :=

Original: DFT ˜ K ◦ S ˜ K ◦ Z odd M ( g ) , respectively, where S ˜ K : C ˜ K
Revised: DFT ˜ K ◦ S ˜ K ◦ Z odd M ( g ) , respectively, where S ˜ K : C ˜ K

Original: → C ˜ K is deﬁned by
Revised: → C ˜ K is deﬁned by

Original: [Equation]
Revised: [Equation]

Original: Consequently, uv can be evaluated at the ˜ K -th roots of unity by taking the element-wise product of the latter two vectors.
Revised: Consequently, uv can be evaluated at the ˜ K -th roots of unity by taking the element-wise product of the latter two vectors.

Original: Finally, the desired coefﬁcients ( P K ( a ) ∗ g ) N − 1 k = − N are obtained by going back to coefﬁcient space using the inverse DFT , i.e.,
Revised: Finally, the desired coefﬁcients ( P K ( a ) ∗ g ) N − 1 k = − N are obtained by going back to coefﬁcient space using the inverse DFT , i.e.,

Original: [Equation]
Revised: [Equation]

Original: Here (cid:12) denotes the Hadamard-product and Π N : C ˜ K
Revised: Here (cid:12) denotes the Hadamard-product and Π N : C ˜ K

Original: → C 2 N is the truncation operator deﬁned by Π N ( b ) :
Revised: → C 2 N is the truncation operator deﬁned by Π N ( b ) :

Original: = ( b k ) N − 1 k = − N .
Revised: = ( b k ) N − 1 k = − N .

Original: B P REPROCESSING
Revised: B P REPROCESSING

Original: In this section we provide the details of our preprocessing steps.
Revised: In this section we provide the details of our preprocessing steps.

Original: B. T RUNCATION F OURIER COEFFICIENTS
Revised: B. T RUNCATION F OURIER COEFFICIENTS

Original: The magnitude of the approximated Fourier coefﬁcients will typically stagnate and stay constant (approximately) beyond some critical order, since all computations are performed in ﬁnite (single) precision.
Revised: The magnitude of the approximated Fourier coefﬁcients will typically stagnate and stay constant (approximately) beyond some critical order, since all computations are performed in ﬁnite (single) precision.

Original: We locate this critical order m ∗ 0 ( s ) ∈ N for each component s ∈ { 1 , 2 } , if present, by iteratively ﬁtting the best line, in the least squares sense, through the points (cid:110)(cid:16) m, (cid:13)(cid:13)(cid:13) ( | [˜ γ ˜ m ] s | ) m˜ m
Revised: We locate this critical order m ∗ 0 ( s ) ∈ N for each component s ∈ { 1 , 2 } , if present, by iteratively ﬁtting the best line, in the least squares sense, through the points (cid:110)(cid:16) m, (cid:13)(cid:13)(cid:13) ( | [˜ γ ˜ m ] [Equation] | ) m

Original:  = 
Revised: [Equation] m = [Equation]

Original: m 0 (cid:13)(cid:13)(cid:13) 1 (cid:17) : m 0 ≤ m ≤ N − 1 (cid:111) for 1 ≤ m 0 ≤ N − 1 .
Revised: 0 (cid:13)(cid:13)(cid:13) 1 (cid:17) : m 0 ≤ m ≤ N − 1 (cid:111) for 1 ≤ m 0 ≤ N − 1 .

Original: We iterate this process until the residual is below a prescribed threshold δ N > 0 .
Revised: We iterate this process until the residual is below a prescribed threshold δ N > 0 .

Original: In practice, we set δ N = 0 .
Revised: In practice, we set δ N = 0 .

Original: The Fourier coefﬁcients with index strictly larger than m ∗ 0 ( s ) are set to zero.
Revised: The Fourier coefﬁcients with index strictly larger than m ∗ 0 ( s ) are set to zero.

Original: B. C ONSISTENT PARAMETERIZATIONS
Revised: B. C ONSISTENT PARAMETERIZATIONS

Original: To have consistent parameterizations we enforce that all contours start at angle zero at time zero relative to the midpoint c
Revised: To have consistent parameterizations we enforce that all contours start at angle zero at time zero relative to the midpoint c

Original: = ( c 1 , c 2 ) ∈ R 2 of the region of interest R .
Revised: = ( c 1 , c 2 ) ∈ R 2 of the region of interest R .

Original: This is accomplished by exploiting the Fourier representation of the curve.
Revised: This is accomplished by exploiting the Fourier representation of the curve.

Original: More precisely, let ˜ γ :=
Revised: More precisely, let ˜ γ :=

Original: t (cid:55)→ (cid:80) | m |≤ N − 1 ˜ γ m e iω ( l ) mt , where ω ( l ) = 

Revised: t (cid:55)→ (cid:80) | m |≤ N − 1 ˜ γ m e iω ( l ) mt , where ω ( l ) =

Original: πl , be the contour with Fourier coefﬁcients Γ :=
Revised: πl , be the contour with Fourier coefﬁcients Γ :=

Original: (˜ γ m ) N − 1 m =1 − N .
Revised: (˜ γ m ) N − 1 m =1 − N .

Original: The midpoint c of the region enclosed by ˜ γ is given by
Revised: The midpoint c of the region enclosed by ˜ γ is given by

Original: [Equation]
Revised: [Equation]

Original: by Green’s Theorem.
Revised: by Green’s Theorem.

Original: Here λ denotes the Lebesgue measure on R 2 , [Γ] s are the Fourier coefﬁcients of [˜ γ ] s , and (Γ (cid:48) ) m :=
Revised: Here λ denotes the Lebesgue measure on R 2 , [Γ] s are the Fourier coefﬁcients of [˜ γ ] s , and (Γ (cid:48) ) m :=

Original: imω ( l ) γ m for | m | ≤ N − 1 .
Revised: imω ( l ) γ m for | m | ≤ N − 1 .

Original: We can now compute the desired parameterization by determining τ ∈ [0 , l ] such that arccos
Revised: We can now compute the desired parameterization by determining τ ∈ [0 , l ] such that arccos

Original: [Equation]
Revised: [Equation]

Original: and deﬁning γ (t )
Revised: and deﬁning γ (

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: ˜ γ ( t − τ )

Original: [Equation]
Revised: [Equation]

Original: While τ can be easily found using Newton’s method, it sufﬁces in practice to simply re-order y from the start, before computing the Fourier coefﬁcients.
Revised: While τ can be easily found using Newton’s method, it sufﬁces in practice to simply re-order y from the start, before computing the Fourier coefﬁcients.

Original: More precisely, we ﬁrst deﬁne a shift ˜ y of y by ˜ y k :=
Revised: More precisely, we ﬁrst deﬁne a shift ˜ y of y by ˜ y k :=

Original: y k + k ∗ mod n p for 0 ≤ k ≤ n p − 1 , where k ∗ :=
Revised: y k + k ∗ mod n p for 0 ≤ k ≤ n p − 1 , where k ∗ :=

Original: argmin
Revised: argmin

Original: [Equation]
Revised: [Equation]

Revised: C U-N ET ARCHITECTURES ( BASELINES )

Revised: In this section we describe our modiﬁcations to the networks presented in Ronneberger et al.

Revised: (2015) and Nikolov et al.

Revised: The goal of the modiﬁcations is to match the parameter count with our networks (approximately), while maintaining the structure and idea’s presented in the original papers as much as possible.

Revised: Our modiﬁcations to the original U-Net in Ronneberger et al.

Revised: (2015) is minimal: we use 32 ﬁlters in the ﬁrst layer instead of 64 and use group normalization with four groups.

Revised: Our modiﬁcations to the network in Nikolov et al.

Revised: (2018) are as follows.

Revised: We use residual blocks of length two (instead of three), we use a bottleneck with 256 channels (instead of 1024 ), and we do not use 1 d convolutions in the “third” direction (to keep our model 2 d).

Revised: Furthermore, we use an encoder and decoder of length six.

Revised: The number of ﬁlters used in the encoder, from top to bottom, is 32 , 32 , 64 , 64 , 128 , and 128 , respectively.

Revised: The number of ﬁlters used in the decoder, from bottom to top, is 128 , 128 , 64 , 64 , 64 and 64 , respectively.

Original: C F IGURES
Revised: D F IGURES

Original: In this section we provide additional visualizations of the statistics, wavelet decompositions and predicted contours
Revised: In this section we provide additional visualizations of the statistics, wavelet decompositions and predicted contours

Original: [Figure]
Revised: [Figure]

Original: C. T OY EXAMPLE
Revised: D. T OY EXAMPLE

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: <latexit 
Revised: <latexit sha1_base64="suyhdlt5/euM6gZqsZd5yW59io0="

Original: [Figure]
Revised: [Figure]

Revised: AAADVXicjVLbShxBED27a4wxF2+PEhiyCnlaZsSQPIq+5FHBVUElzPT2msa5MdOTsCx

Original: [Table]
Revised: [Figure]

Original: +7yysPVR4/X1ntPDqq8Lo0dmzzJy6M4qmziMjv2zif2qChtlMaJPYzPX0v+8KMtK5dn7/yssKdpdJa5qTORJzSexFu7W+/X++Eg1BXcDoZN0Eez9vNeawMnmCCHQY0UFhk84wQRKj7HGCJEQewUc2IlI6d5i89YIbdmlWVFRPSc3zPujhs04140K2UbnpLwLckMsE1OzrqSsZwWaL5WZUH/pT1XTeltxn/caKVEPT4Q/R/vsvK+PPHiMcWuenD0VCgi7kyjUutUpPPgmitPhYKYxBPmS8ZGmZdzDpRTqXeZbaT5n1opqOxNU1vjl3a5TfxEKzK+n3RqqfrIeMqcuak6lX5mRKzuZa6e3Uq+InITD66U756BoaL4F+7VBHnbhjfv1u3gYDQYPh+8eDvq771q7l0Xz7CJHd6tl9jDG+xjzBMcvuArvrW/t393FjpLF6XtVsN5ir9WZ
Revised: g

Original: [Figure]
Revised: [Figure]

Original: 0PGqmq2g
Revised: my0L

Original: [Figure]
Revised: [Figure]

Original: <
Revised: IPkIwVOVURJNSHqYmepTdU7XKTopU1f7MPzW6fZmHs0+nnsy//TZ8xcLi0vLB3XRVMYOTZEW1VES1zZ1uR1651N7VFY2zpLUHibnO5I//GSr2hX5vp+U9jSLz3I3dib2hIajZC1a+7DYDwehruBhELVBH+3aLZY6L3GCEQoYNMhgkcMzThGj5nOMCCFKYqeYEqsYOc1bXGCe3IZVlhUx0XN+z7g7btGce9GslW14Ssq3IjPAOjkF6yrGclqg+UaVBf2b9lQ1pbcJ/0mrlRH1+Ej0X7zbyv/liRePMd6pB0dPpSLizrQqjU5FOg9+ceWpUBKTeMR8xdgo83bOgXJq9S6zjTX [Table] +7yysPVR4/X1ntPDqq8Lo0dmzzJy6M4qmziMjv2zif2qChtlMaJPYzPX0v+8KMtK5dn7/yssKdpdJa5qTORJzSexFu7W+/X++Eg1BXcDoZN0Eez9vNeawMnmCCHQY0UFhk84wQRKj7HGCJEQewUc2IlI6d5i89YIbdmlWVFRPSc3zPujhs04140K2UbnpLwLckMsE1OzrqSsZwWaL5WZUH/pT1XTeltxn/caKVEPT4Q/R/vsvK+PPHiMcWuenD0VCgi7kyjUutUpPPgmitPhYKYxBPmS8ZGmZdzDpRTqXeZbaT5n1opqOxNU1vjl3a5TfxEKzK+n3RqqfrIeMqcuak6lX5mRKzuZa6e3Uq+InITD66U756BoaL4F+7VBHnbhjfv1u3gYDQYPh+8eDvq771q7l0Xz7CJHd6tl9jDG+xjzBMcvuArvrW/t393FjpLF6XtVsN5ir9WZ+0PGqmq2g==</latexit>

Original: [Figure] [Figure] 
Revised: [Figure] [Figure] D.

Revised: S

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Table]
Revised: [Table]

Original: 8KMtK5dn7/
Revised: db <

Original: [Table]
Revised: [Table]

Original: +/
Revised: sha

Original: [Table]
Revised: [Table]

Original: /pT1XTeltxn/

Original: [Table]
Revised: [Table]

Original: /R/

Original: [Table]
Revised: [Table]

Original: +PPHiMcWuenD0VCgi7kyjUutUpPPgmitPhYKYxBPmS8ZGmZdzDpRTqXeZbaT5n1opqOxNU1vjl3a5TfxEKzK

Original: [Table]
Revised: [Table]

Original: n3RqqfrIeMqcuak6lX5mRKzuZa6e3Uq+

Original: [Table] [Table]
Revised: [Table] [Table]

Original: +0PGqmq2g==</latexit> db <latexit sha1_base64="NL+M6hKbXnRwfSFLBzc0mcHfRaI=">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</latexit>
Revised: /apTS9VHxlOmzI3VqfQzIWJ1L3N17FbyFZG7uHer/PAMDBXFv3BvJ8jbFty9W/eD4UY/eN3f3Nvobb9r7t0cXuAV1nm33mAbO9jFgCd8xjm+4aL9vf2zM9OZ/VXabjWcFfyxOt1ry8urEw==</latexit>

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Table]
Revised: [Table]

Original: /apTS9VHxlOmzI3VqfQzIWJ1L3N17FbyFZG7uHer/PAMDBXFv3BvJ8jbFty9W/eD4UY/eN3f3Nvobb9r7t0cXuAV1nm33mAbO9jFgCd8xjm+4aL9vf2zM9OZ/VXabjWcFfyxOt1ry8urEw==</latexit>
Revised: /apTS9VHxlOmzI3VqfQzIWJ1L3N17FbyFZG7uHer/PAMDBXFv3BvJ8jbFty9W/eD4UY/eN3f3Nvobb9r7t0cXuAV1nm33mAbO9jFgCd8xjm+4aL9vf2zM9OZ/VXabjWcFfyxOt1ry8urEw==</latexit>

Original: [Figure]
Revised: [Figure]

Original: The last two columns correspond to “hard” examples.

Original: [Figure]
Revised: [Figure]