Subpixel object segmentation using wavelets and multiresolution analysis

1 INTRODUCTION . Semantic image segmentation is a core component of many medical imaging related tasks . Both as part of a pipeline to find a region of interest , or a task by itself , e.g. , for measuring tumor volume . Nowadays , almost all segmentation algorithms in medical imaging are replaced by U-Net-like architectures Ronneberger et al . ( 2015 ) combining an encoder and decoder . 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 . 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 . 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 . Typically , the pixels are assumed to be independent so that the joint-likelihood is tractable . 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 . 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 . 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 . A fundamental component in the setup of our framework is the decision on how to represent ( closed ) curves . 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 . For this reason , we have chosen to represent contours using wavelets and Multi Resolution Analysis ( MRA ) instead . 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 . This Under review as a conference paper at ICLR 2022 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 . The filters in the decoder are not learned but uniquely determined by the chosen wavelet basis . Any wavelet basis induced by a MRA can be used . This flexibility allows for incorporation of priors on the smoothness of curves . Related work Previous work by Chen et al . ( 2019 ) ; Marcos et al . ( 2018 ) ; Hatamizadeh et al . ( 2020 ) also proposed models to predict contours by combining Active Contour Models ( ACM ) with a CNN into an end-to-end model . In these papers the representation of curves is ultimately still based on pixel-based computations . For instance , in Hatamizadeh et al . ( 2020 ) curves are modelled as level sets of distance maps defined on a discretization of the domain of the image . In Chen et al . ( 2019 ) a similar approach is followed , but a smoothed approximation of an indicator function is used instead of a distance map . The work in Marcos et al . ( 2018 ) is perhaps most closely related to ours ; they directly construct polygonal approximations of curves and represent them using ( pixel ) coordinates of the nodes . 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 . These properties are implicitly encoded in an objective function ( energy-functional ) defined on a space of distance maps Hatamizadeh et al . ( 2020 ) , suitable approximations of indicator functions Chen et al . ( 2019 ) , or family of polygons Marcos et al . ( 2018 ) . However , in contrast to our approach , the above methods all provide pixel-based output . 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 . This paper is organized as follows . In Section 2 we review the mathematical background needed to construct our model . We explain how contours can be decomposed and reconstructed on different resolution levels . The reconstruction algorithm , the Pyramid Algorithm , forms a core component of our network architecture . In Section 3 we set up the model architecture and loss . Subsequently the datasets , training method and performance measures are described in Section 4 . We end the paper with results and a discussion in Section 5 . Further mathematical details are provided in the appendix . 2 BACKGROUND AND MATHEMATICAL SETUP . 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. We assume that each image contains a ( uniquely identifiable ) simply connected region R ( x ) ⊂ R2 , 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 ) . We will develop a deep learning framework for computing such parameterizations γ ( x ) using Multi Resolution Analysis ( MRA ) . Since wavelets play a seminal role in our set up , we first review the necessary theory in Sections 2.1 and 2.2 as a subject in its own right in the context of general scalar-valued signals γ . 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 . 2.1 MULTI RESOLUTION ANALYSIS . In this section we briefly review wavelet theory using the framework of a multi resolution analysis ( MRA ) . We closely follow the exposition in Pereyra & Ward ( 2012 ) and Montefusco & Puccio ( 2014 ) . Throughout this section we denote the space of square integrable functions on R , equipped with standard inner product , by L2 ( R ) . The uncertainty principle in Fourier analysis dictates that a signal γ ∈ L2 ( R ) can not be simultaneously localized in the time and frequency domain , see Pereyra & Ward ( 2012 ) . Multi resolution analysis aims to address this shortcoming by decomposing a signal on different discrete resolution levels . The idea is to construct subspaces Vj ⊂ L2 ( R ) , associated to various resolution levels j ∈ Z , spanned by integer shifts of a localized mapping ϕj . The level of localization associated to Vj is determined by taking an appropriate dilation of a prescribed map ϕ ; the so-called scaling function . In the MRA framework the dilation factors are chosen to be powers of two . 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Altogether , this yields an increasing sequence of closed subspaces Vj ⊂ Vj+1 ⊂ L2 ( R ) dense in L2 ( R ) , where Vj+1 is the next level up in resolution after Vj . The representation of γ at resolution level j , denoted by γj : = Pjγ , is its orthogonal projection onto Vj . Here Pj denotes the orthogonal projection onto Vj . The coefficients of γj with respect to the basis for Vj , denoted by aj ( γ ) = ( ajk ( γ ) ) k∈Z , are called the approximation coefficients . The associated subspaces Vj are referred to as the approximation subspaces ( see Figure 1b ) . To study the information that is lost when a signal in Vj+1 is projected onto Vj , we consider the operator Qj : = Pj+1 − Pj . The range of Qj , denoted by Wj , is referred to as the detail subspace at level j ; it is the orthogonal complement of Vj in Vj+1 . The detail subspaces ( Wj ) j∈Z are mutually disjoint and orthogonal by construction . A fundamental result , known as Mallat ’ s Theorem , states that the subspaces Wj can too be spanned by dilating and shifting a single map . More precisely , there exists a map ψ ∈ W0 , the so-called mother wavelet , such that ( ψjk ) k∈Z is an orthonormal basis for Wj , see Pereyra & Ward ( 2012 ) . The coefficients of Qjγ with respect to this basis , denoted by dj ( γ ) : = ( djk ( γ ) ) k∈Z , are referred to as the detail coefficients of γ at resolution level j . The detail coefficients store the information needed to go back one level up in resolution , since Pj+1 = Pj+Qj by construction . We often write aj ( γ ) = aj and dj ( γ ) = dj for brevity . In practice , we only approximate a finite number of approximation and detail coefficients , see Section 2.3 . 2.2 THE DISCRETE WAVELET TRANSFORM . In this section we describe how to compute the approximation and detail coefficients given a prescribed scaling function ϕ . Many fundamental aspects of MRA ’ s , both theoretical and computational , can be traced back to the following key observation . Since V0 ⊂ V1 , there must exist coefficients h = ( hk ) k∈Z such that ϕ = ∑ k∈Z hkϕ1k . This equation is referred to as the scaling equation ; one of the fundamental properties of a scaling function . The sequence h , the so-called lowpass filter , completely characterizes the scaling function . Similarly , since ψ ∈W0 ⊂ V1 , there exist coefficients g = ( gk ) k∈Z , the so-called high-pass filter associated to h , such that ψ = ∑ k∈Z gkϕ1k . For Mallat ’ s mother wavelet , we have gk = ( −1 ) k−1h1−k . 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 aj+1 . Conversely , the orthogonal decomposition Vj+1 = Vj ⊕Wj can be used to reconstruct aj+1 given the approximation and detail coefficients aj and dj , respectively , at resolution level j . The computations are summarized in Figure 2 . The reconstruction and decomposition formulae together form the well-known Pyramid Algorithm Mallat ( 2008 ) . In practice , our signals are periodic and do not directly fit into the MRA framework , since non-zero periodic signals are not elements in L2 ( R ) . We will address this issue in the next section by using an appropriate cut-off . Here we only remark that the periodicity needs to be carefully taken into account in the decomposition and reconstruction formulae , see Appendix A.2 . 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 < /latexit > In this section we explain how a scalar-valued periodic signal γ with period l > 0 can be approximated using a MRA . Let j1 ∈ N be a desired resolution-level . 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 . Next , to address the issue that periodic signals are not contained in L2 ( R ) , we restrict the re-parameterized curve to [ −1 , 1 ] , i.e. , set γ∗ ( t ) : = γ ( lt ) 1 [ −1,1 ] ( t ) . In general , this will introduce discontinuities at the boundary points−1 and 1 . This is , however , not an issue , since we only need information about γ∗ on a strict subset [ I0 , I1 ] ⊂ [ −1 , 1 ] of length 1 . It is shown in Lemma A.1 how ( and which ) approximation coefficients can be related to the sample values of γ . In particular , if j1 is sufficiently large , the approximation coefficients needed to ( approximately ) cover [ −1 , 1 ] are ( aj1k ( γ∗ ) ) b2 j1−βc k=−2j1 . Here β > 0 is the support of the underlying wavelet . These coefficients will be close to the ( scaled ) sample values of γ on { k2−j1 : −2j1 ≤ k ≤ b2j1 − βc } . Motivated by this observation , and the fact that we only need γ on [ − 12 , 12 ] , we use the coefficients ( aj1k ( γ ∗ ) ) 2 j1−1−1 k=−2j1−1 only , which cover [ − 12 , 1−2 1−j1 2 ] approximately , see Figure 3 . To ensure that 2j1−1 − 1 < b2j1 − βc , we require that j1 ≥ ⌈ log ( β−1 ) log ( 2 ) + 1 ⌉ . This quantity is well-defined for all bases considered in this paper , except the Haar-basis .

Summary: The paper proposes a method for boundary extraction from images using a U-net type network architecture. The boundaries are considered as closed curves represented by multiresolution analysis (wavelets). The down-sampling part of the network consists of a 2D encoder and the up-sampling part is a 1D decoder that reconstructs the boundaries.