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: In this paper, the authors propose a novel deep learning framework for the fast prediction of boundaries. The boundary is modeled as smooth closed curves with wavelets and multi-resolution analysis. A U-shape model is trained to predict for coefficients of the curve. The proposed method is evaluated on medical images and achieves similar performances with up to 5x faster inference speed. The main contribution is that it is the first to use wavelet analysis to construct pixel-independent representations of curves.