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Singular spectrum analysisbased image subband decomposition filter banks
EURASIP Journal on Advances in Signal Processing volume 2020, Article number: 29 (2020)
Abstract
This paper presents a novel type of a filter bank for image decomposition that incorporates “singular spectrum analysis” (SSA). SSA is based on “singular value decomposition” (SVD).The presented filter banks provide good directional selectivity. Subband images highlight exposed directional features such as 0^{∘},90^{∘},+ 45^{∘}, and − 45^{∘} of the input image. The proposed method can be seen as a tool to separate an image into layers or to present an image as a sum of its highfrequency directional components and a lowfrequency approximation.
Introduction
Filter banks play an important role in signal processing and are used in many areas. In general, the main purpose of filter banks is to divide an input signal into subbands containing distinct frequency domains. They are known for their applications in the area of image processing, such as image compression and denoising, and many more. Different filter bank designs can be implemented depending on the application. This process is famous under different names: subband decomposition, subband transform, subband filtering, and multiscale transform.
In general, a filter bank consists of an analysis and a synthesis stage. During the analysis stage, the input signal is divided into subbands depending on the requirements of the application. The respective synthesis stage is the inverse transform to reconstruct the input signal from the subbands. Depending on the use case, there is a processing stage between the analysis and the synthesis stage. The processing stage is tailored to one of many possible applications, such as extraction of edges, removal of details, image compression, and compressed sensing.
A comprehensive book [1] of 2018 surveys widely used types of multiscale (subband) transform methods according to the historical development process: fourier analysis and fourier transform, wavelets together with different types of wavelet transform (discrete, continuous, undecimated, biorthogonal, complex, dualtree complex, and quaternion), and “New Multiscale Constructions” (ridgelet, curvelets, contourlet, and shearlet transform). In the field of image processing, filter banks are traditionally based on different types of wavelets, highlighted in many old and worldwide famous books [2–5].
Waveletbased filter banks have a complex structure, require a huge amount of memory, and lack a perfect reconstruction for finite sequences of data. In parallel to the research of waveletbased filter banks, Kakarala and Ogunbona investigated in new approaches of subband decomposition methods based on SVD. In 2001, they published “multiresolution SVD” (MRSVD [6]) to overcome these issues. This laid the base for hybrid methods combining a waveletbased core and SVD, such as “hybrid waveletSVD” (Ashino et al., 2004 [7]) and “SVDwavelet” (Arandiga et al., 2005 [8]). Later, subsequent filter banks that solely base on SVD were proposed by Singh and Kumar (2011 [9]) and Bhatnagar et al., (2014 [10]). Their algorithms show good results in image compression, image encryption, lossy image compression, and face recognition. It was also claimed that SVDbased algorithms are able to outperform waveletbased filter banks.
This work introduces a set of novel filter banks that build on “singular spectrum analysis” (SSA) [11], an algorithm that implements “singular value decomposition” (SVD). These filter banks are referred to as “singular spectrum analysisbased image subband decomposition filter banks” (“SSAiSBD filter banks”).
To adapt onedimensional Basic SSA for a processing of twodimensional image data, the twodimensional image data has to be transformed into a onedimensional vector first. The choice of a vectorization scheme (scanning order) determines the properties of the filter bank. Four such transformation schemes are realized and form the core of four “SSAiSBD 2channel filter banks.” These are used as building blocks to create “SSAiSBD multi (4 and 6)channel filter banks” capable of extracting directional features.
Good directional selectivity is achieved on the basis of “SSAiSBD 6channel filter banks” that decompose an input image into an approximation and highfrequency components. The approximation contains the low frequency components and the latter highlight specific directional features as well as uncaught remains. The distinct highfrequency components contain the horizontal (0^{∘}), vertical (90^{∘}), and diagonal (+ 45^{∘} and − 45^{∘}) directional features, respectively. Directional features regarding the directions + 45^{∘} and − 45^{∘} are clearly separated from each other. This transform is invertible, and the reverse transform is implemented as a trivial addition of the created subband images.
The property of good directional selectivity is useful for image processing applications such as edge and object detection, face and gesture recognition, and feature extraction in machine learning.
To demonstrate the mode of operation of filter banks, the wellknown synthetic “Octagon” image shown in Fig. 1a is used as the input. It was chosen for all experiments. It is a blackandwhite PNG image with 600×600 pixels with a shape providing horizontal, vertical, and diagonal directional features.
Additionally, the natural “Cameraman” image, a black and white PNG image with 256×256 pixels shown on Fig. 1b, is used to demonstrate the performance of the developed algorithms. All experiments were conducted with the MATLAB toolkit.
The remainder of this paper is structured as follows: Section 2 observes the image subband decomposition methods with directional selectivity properties, regarding the directions 0^{∘},90^{∘},+ 45^{∘}, and − 45^{∘}. Section 3.1 provides the algorithm of SSA. Section 3.2 reviews how SSA is applied to vectors (onedimensional data) and to twodimensional image data. Four schemes of how to apply onedimensional Basic SSA to twodimensional image data are introduced in Section 3.3. In Section 3.4, a novel scheme of creating “SSAiSBD 2channel filter banks” is presented. The outputs of the respective filter banks are visualized and discussed in Section 4, where “SSAiSBD multi (4 and 6)channel filter banks” are introduced, offering good directional selectivity. The paper concludes with Section 5.
Related work
Today, the classical “discrete wavelet transform” (DWT) is still a commonly used scheme for subband decomposition of signals (1989, Mallat’s dyadic filter tree [2, 12]). For image processing, 2DDWT is an extension of DWT for twodimensional input signals. Typically used mother wavelets (classes of wavelet bases) are Haar, Daubechies, Biorthogonal, Coiflet, and Symletwavelets [12]. 2DDWT is suited well for image compression in JPEG2000 [12, 13], denoising, and the creation of sparse images for compressed sensing [12, 14]. However, 2DDWT is not good enough for some specific applications such as pattern recognition. The reasons for that are discussed in the following.
In Fig. 2, the “Octagon” input image was decomposed using 2DDWT [15] with “Daubechies D4” as the mother wavelet, showing the first level of 2DDWT decomposition. Every subband image has a size of 300×300 pixels as a result of downsampling in the filter banks (decimation).
The two major drawbacks of 2DDWT in image processing are:

Shift sensitivity: Shift sensitivity or the lack of shift invariance or shift variance causes unpredictable changes in the output coefficients if the input image is shifted slightly [16, 17].

Lack of directionality: A lack of directionality or a poor directional selectivity of a filter bank leads to a nonideal separation of directional features. Two aspects of poor directional selectivity are distinguished:

Horizontal and vertical directional components also contain diagonal features as depicted in Figs. 2b, c. A cleaner extraction of those features is a goal.

The filter bank can not separate diagonal features of − 45^{∘} and + 45^{∘}. Such a compound of both orientations is shown in Fig. 2d. It is preferred to get distinct outputs for − 45^{∘} and + 45^{∘} diagonal features.

These drawbacks motivated scientists and researches to develop extensions of 2DDWT as well as other sophisticated waveletbased filter banks.
Many other types of image subband decomposition schemes were developed, being beneficial in specific applications each. Only a few of them focus on directional selectivity, such as a clear separation of horizontal and vertical directional features, or even diagonal and antidiagonal directional features. Such algorithms are:

One of DWTs extensions, providing shift invariance, referred to as both “undecimated discrete wavelet transform” (UDWT) and “stationary wavelet transform.” Its extension to two dimensions (2DUDWT) is implemented most efficiently by the “algorithme a’trous” [12]. It was introduced by Holschneider et al., (1989 [18]). Fowler [19] provides an overview of different names for this transform. Without downsampling and aliasing, this filter is shift invariant by the cost of redundancy in the output information. The first level of 2DUDWT applied to the “Octagon” image is similar to the result of processing the “Octagon” image with 2DDWT (Fig. 2), with the difference that every subband image created with 2DUDWT has the same size as the decomposed image: 600×600 pixels. 2DUDWT still suffers from poor directional selectivity, but due to its shift invariance, it was successfully implemented in applications such as feature extraction [20] and image classification [21]. Furthermore, 2DUDWT offers a substantial performance improvement in denoising [22, 23].

“Dualtree complex wavelet transform” (DTCWT) is an extension to DWT capable of approaching both shift invariance and good directional selectivity, proposed by Kingsbury (1998 [24–30]). This complex waveletbased method offers a combination of properties that was not available in earlier approaches: a principally perfect reconstruction and good directional selectivity together with near shift invariance. According to Kingsbury [29], DTCWT involves six complex wavelets oriented along the directions ± 15^{∘},± 45^{∘}, and ± 75^{∘}. DTCWT is suited well for numerous applications in image [16] and video [31] processing. A scheme based on DTCWT in combination with a complex spline, developed by Chaundhury and Unser (2010 [32]), is able to detect the directions 0^{∘},90^{∘},− 45^{∘}, and + 45^{∘}. An “Octagon” image decomposed with DTCWT, based on a complex spline, was compared to an “Octagon” image decomposed with 2DDWT [33].

The Gaborlike transform proposed by Chaundhury and Unser (2009 [34]) also provides good directional selectivity. They presented examples of transformed “Octagon” and “Cameraman” images. The Gaborlike transform is based on analytic Bspline wavelets, referred to as Gaborlike wavelets, that are oriented along the four principal directions: 0^{∘},90^{∘},− 45^{∘}, and + 45^{∘}. The added redundancy along the horizontal and vertical directions yields shift invariance along these directions. However, the Gaborlike transform is not invertible. An example of Gaborlike transform implementation in the industry is the identification of damaged elements made of polymer and polymeric composites [35].

“Multiresolution SVD” (MRSVD), presented by Kakarala and Ogunbona (2001 [6]), is an SVDbased method, demonstrating properties of directionality. Subimages contain the mix of 0^{∘},90^{∘},− 45^{∘}, and + 45^{∘} directional features. A comparison of the directional selectivity of MRSVD with that of 2DDWT, incorporating the “Octagon” image, is presented in [7].
Methods
Singular spectrum analysis
“Singular spectrum analysis” (SSA) is a technique initially designed for the analysis of time series. It is a signal processing technique based on “singular value decomposition” (SVD) that decomposes an input signal into several components. These components can be grouped and merged to compose subsequent components. These results can be interpreted as for example a trend, as an oscillation within the input signal, or as noise. SSA can be used for the extraction of trends, for smoothing, for the extraction of periodic components, or for finding structures in time series. It has been applied in physics, signal analysis, and mathematics, but it is also a valuable tool for market research, economics, and meteorology [36].
SSA was initiated by Broomhead and King (1986 [37]), but it was independently introduced by the researchers Danilov and Zhigljavsky in Russia as the “Caterpillar” method [38]. Today, comprehensive descriptions of the theoretical and practical concepts of SSA are available in [11].
Basic SSA (or SSA) is tailored to onedimensional input signals referred to as a 1Darray [39]. Images, however, are twodimensional signals. For such 2Darrays, 2DSSA is available. Basic SSA and 2DSSA are widely used [40]. 2DSSA has been suggested by Danilov and Zhigljavsky(1997 [38]) and was further discussed in [40–43].
In this paper, a preprocessing step is introduced that allows analyzing images with Basic SSA, by transferring the input image into a 1Darray first. This scheme of transforming 2Darrays into 1Darrays and vice versa is discussed later.
Both Basic SSA and 2DSSA rely on the same four processing steps, as described in [39] and [41]. They only differ in step 1 and 4. This difference will also be discussed afterwards.

1.
Compute the trajectory matrix X from the input array

The case of an input 1Darray
1Darray is represented as a vector F of length N:
$$ F = \left(\begin{array}{llll} f_{1} & f_{2} & \cdots & f_{\mathrm{N}} \\ \end{array}\right)^{\mathrm{T}} $$(1)Here, the trajectory matrix X^{1D} is a Hankel matrix constructed from the input 1Darray F. The dimensions of this Hankel matrix are defined only by the window length L that must confirm to 1≤L≤N. Let K=(N−L+1) to get a Hankel matrix of dimensions L×K. The elements of the input 1Darray F are arranged into the Hankel matrix X^{1D} according to the following scheme [39]:
$$ X^{\mathrm{1D}} = \left(\begin{array}{cccc} f_{1} & f_{2} & \cdots & f_{K} \\ f_{2} & f_{3} & \cdots & f_{{K}+1} \\ \vdots & \vdots & \ddots & \vdots \\ f_{L} & f_{{L}+1} & \cdots & f_{\mathrm{N}} \end{array}\right) $$(2) 
The case of an input 2Darray
Computing the trajectory matrix regarding a 2Darray, we have a matrix I of size N_{x}×N_{y}:
$$ I = \left(\begin{array}{cccc} i_{1,1} & i_{1,2} & \cdots & i_{1,\mathrm{N_{y}}} \\ i_{2,1} & i_{2,2} & \cdots & i_{2,\mathrm{N_{y}}} \\ \vdots & \vdots & \ddots & \vdots \\ i_{\mathrm{N_{x}},1} & i_{\mathrm{N_{x}},2} & \cdots & i_{\mathrm{N_{x}},\mathrm{N_{y}}} \end{array}\right) $$(3)The trajectory matrix is the “HankelblockHankel” (HbH) matrix X^{2D} constructed from the input 2Darray I. The dimensions of the HbH matrix are defined by two window sizes L_{x} and L_{y}, chosen that 1≤L_{x}≤N_{x},1≤L_{y}≤N_{y}, and 1<L_{x}L_{y}<N_{x}N_{y}. Let again be K_{x}=(N_{x}−L_{x}+1) and K_{y}=(N_{y}−L_{y}+1) for convenience.
The input 2Darray I is arranged into the HbH matrix X^{2D} of size (L_{x}L_{y}×K_{x}K_{y}) according to the following rule [41]:
$$ X^{\mathrm{2D}} = \left(\begin{array}{cccc} H_{1} & H_{2} & \cdots & H_{\mathrm{K_{y}}} \\ H_{2} & H_{3} & \cdots & H_{\mathrm{K_{y}} + 1} \\ \vdots & \vdots & \ddots & \vdots \\ H_{\mathrm{L_{y}}} & H_{\mathrm{L_{y}} + 1} & \cdots & H_{\mathrm{N_{y}}} \end{array}\right) $$(4)Each H_{j} is a Hankel matrix of size L_{x}×K_{x} consisting of elements from the input 2Darray I:
$$ H_{j} = \left(\begin{array}{cccc} i_{1,j} & i_{2,j} & \cdots & i_{\mathrm{K_{x}},j} \\ i_{2,j} & i_{3,j} & \cdots & i_{\mathrm{K_{x}}+1,j} \\ \vdots & \vdots & \ddots & \vdots \\ i_{\mathrm{L_{x}},j} & i_{\mathrm{L_{x}}+1,j} & \cdots & i_{\mathrm{N_{x}},j} \end{array}\right) $$(5)


2.
Singular value decomposition (SVD) of the trajectory matrix X.
Denote \(\lambda _{1}, \dots,\lambda _{L}\) as eigenvalues of XX^{T} taken in the decreasing order \((\lambda _{1}\geq \ \dots \geq \ \lambda _{L} \geq \ 0)\) and \(U_{1}, \dots,U_{L}\) denote as corresponding eigenvectors. Set d=rank(X), for simplicity is considered d=L.
$$ X_{i}= \sqrt{\lambda_{i}} U_{i} V_{i}^{\mathrm{T}} $$(6)where V_{i} is defined as
$$ V_{i}=X^{\mathrm{T}}U_{i}/\sqrt{\lambda_{i}}.\ $$(7)SVD of the trajectory matrix X is represented by
$$ \ X=X_{1}+ + X_{i}+ \dots+X_{d}. $$(8)\(\sqrt {\lambda _{i}}\) – is the ith singular value of X; U_{i} and V_{i} – are the associated left and right singular vectors of X, for \(i=1, \dots, d\).
The collection (\(\sqrt {\lambda _{i}}, U_{i}, V_{i} \)) is defined as the ith eigentriple of the SVD.

3.
Eigentriple grouping
The grouping procedure splits the set of indices \(\{ 1, \dots,d \}\) into m subsets \(I_{1}, \dots,I_{m} \). Let \(I = \{i_{1}, \dots,i_{p} \} \). Then, corresponding to group I, the resultant matrix X_{I} can be written as \( X_{I}=X_{i{_{1}}}+ \dots +X_{i{_{p}}} \). The resultant matrices are calculated for each group \(I_{1}, \dots,I_{m} \) and the expansion(8) provides the decomposition
$$ \ X=X_{I{_{1}}}+ \dots +X_{I{_{m}}} $$(9)The choosing procedure of the sets \(I_{1}, \dots,I_{m}\) is defined as eigentriple grouping.

The case of an input 1Darray: \( X^{\mathrm {1D}}=X^{\mathrm {1D}}_{I{_{1}}}+ \dots +X^{\mathrm {1D}}_{I{_{m}}} \)

The case of an input 2Darray: \( X^{\mathrm {2D}}=X^{\mathrm {2D}}_{I{_{1}}}+ \dots +X^{\mathrm {2D}}_{I{_{m}}} \)


4.
Reconstruction of the signal, regarding each group.

The case of an input 1Darray
Opposed to the trajectory matrix X^{1D}, the approximated matrix \(X^{\mathrm {1D}}_{I{_{m}}} \) is not necessarily a Hankel matrix anymore. To “hankelize” it, all elements of each antidiagonal are replaced with their respective averaged value. Then, the reconstructed 1Darray \(\tilde {F}\) can be assembled by simply arranging elements from this created Hankel matrix. Finally, the original 1Darray F can be reconstructed by:
$$ \ F=\tilde{F}_{1}+\tilde{F}_{2}+ \dots +\tilde{F}_{M} $$(10) 
The case of an input 2Darray
The matrices \(X^{\mathrm {2D}}_{I{_{m}}}\) is not necessarily a HbH matrix anymore. To allow reprojection, which is again a rearrangement of elements to create the reconstructed 2Darray \(\tilde {I}, X^{\mathrm {2D}}_{I{_{m}}}\) must be transferred to a HbH matrix first. This is performed in two steps. First, each submatrix of \(X^{\mathrm {2D}}_{I{_{m}}}\) (see Eq. (4)) is “hankelized” individually by averaging the elements of their antidiagonals. At second, all hankelized submatrices with the same index are averaged elementwise. \(\tilde {I}_{M}\) is the 2Darray projected from group \(X^{\mathrm {2D}}_{I{_{m}}}\).
$$ \ I=\tilde{I}_{1}+\tilde{I}_{2}+ \dots +\tilde{I}_{M} $$(11)

Grouping of eigentriples
In Section 3.1, the step of grouping the eigentriples was not specified yet. The algorithm proposed in this paper separates the set of eigentriples into two groups. Each of both groups is used to reconstruct an output signal, resulting in two reconstructed output signals for each given input signal.
The first group of eigentriples contains only the first eigentriple related to the most significant singular value \(\sqrt {\lambda _{i}}\ \) of the trajectory matrix X. The reconstructed signal is referred to as main vector\(\tilde {F}_{\text {main}}\) for the 1Darray and main component\(\tilde {I}_{\text {main}}\) for the 2Darray, respectively.
The second group of eigentriples contains all eigentriples except the first one. The reconstructed signal is referred to as residual vector\(\tilde {F}_{\text {res}}\) (1Darray) and residual component\(\tilde {I}_{\text {res}}\) (2Darray).
An interesting feature of this grouping scheme is that \(F = \tilde {F}_{\text {main}} + \tilde {F}_{\text {res}}\) and \(I = \tilde {I}_{\text {main}} + \tilde {I}_{\text {res}}\). In consequence, this scheme implements a twochannel filter bank with a simple reverse transform. This property can be exploited to lower the computational efforts: if only \(\tilde {F}_{\text {main}}\) or \(\tilde {I}_{\text {main}}\) is calculated, then \(\tilde {F}_{\text {res}}\) or \(\tilde {I}_{\text {res}}\) can be calculated through a substraction: \(\tilde {F}_{\text {res}} = F  \tilde {F}_{\text {main}}\) and \(\tilde {I}_{\text {res}} = I  \tilde {I}_{\text {main}}\).
Processing images with Basic SSA
An input image I, being a 2Darray, can directly be filtered using 2DSSA. Thus, it can be presented as a sum of a main and a residual component. In this section, four schemes are proposed that allow processing 2Darrays with Basic SSA.
To achieve that, the 2Darray I is transformed to a 1Darray F first to be able to perform Basic SSA. Then, after grouping the eigentriples into two groups as described in Section 3.2, the resulting 1Darrays \(\tilde {F}_{\text {main}}\) and \(\tilde {F}_{\text {res}}\) are rearranged elementwise into the 2Darrays \(\tilde {I}_{\text {main}}\) and \(\tilde {I}_{\text {res}}\), respectively. Thus, Basic SSA is used to represent an input image I as a sum of a main (\(\tilde {I}_{\text {main}}\)) and a residual component (\(\tilde {I}_{\text {res}}\)).
As there is no unique scheme defining how to convert a 2Darray into a 1Darray and viceversa, four possible ways of creating F from I are introduced now. They allow processing an input 2Darray with Basic SSA. They also offer the respective reverse transform to derive the output images from the resulting 1Darrays:

1.
f_{i} are the elements of all columns of I, transposed and arranged one after the other as depicted in Fig. 3a. This modification of SSA is referred to as “VerticalSSA.”

2.
f_{i} are the elements of all rows of I, one after the other, as demonstrated in Fig. 3b. This modification of SSA is referred to as “HorizontalSSA.”

3.
f_{i} is the ith element of I regarding the path shown on Fig. 3c. This modification of SSA is referred to as “DiagonalSSA.”

4.
f_{i} is the ith element of I regarding the path shown on Fig. 3d. This modification of SSA is referred to as “AntidiagonalSSA.”
Presentation of “SSAiSBD 2channel filter bank”
As mentioned in Section 3.2, the proposed SSAbased algorithms decompose an image into a main and a residual component, configurable by selecting one of the introduced eigentriple grouping schemes. The decomposition is considered a “SSAiSBD 2channel filter bank as depicted in Fig. 4. Here, and in all following diagrams of this type, the main and residual component are reflected by a dashed and a solid line, respectively. Important to mention, the theory of SSA does not propose an optimal value for the window size/length. Instead, the window size/length depends on the application, and all window sizes/lengths used in this paper were chosen empirically to achieve a good visual presentation.
To demonstrate, the “Octagon” input image was processed with the “SSAiSBD 2channel filter bank,” where 2DSSA is used as “chosenSSA,” which can be also called “2DSSAiSBD 2channel filter bank,” with the window sizes (L_{x},L_{y})= (10, 10) and (30, 30). The resulting main and residual component are shown in Fig. 5.
Note that the smoothness of main components and the thickness of the lines highlighting the edges of the “Octagon” in the residual components relate to the window size. The “2DSSAiSBD 2channel filter bank” operates as a filter bank that separates the input image into a lowfrequency and a highfrequency part. The cutoff frequency relates to the chosen window size.
To present alternatives to incorporating 2DSSA as the “chosenSSA” building block, four alternatives are presented here to create a “SSAiSBD 2channel filter bank.” These are “VerticalSSA,” “HorizontalSSA,” “DiagonalSSA,” and “AntidiagonalSSA.” The resulting main and residual components of the “Octagon” input image are shown on Fig. 6. The window length was L = 30 in all cases. A comparison of only the residual components highlights the differences of these filter banks with the bank incorporating 2DSSA, Fig. 5d:

“VerticalSSA” (Fig. 6b): all vertical directional features are removed,

“HorizontalSSA” (Fig. 6d): all horizontal directional features are removed,

“DiagonalSSA” (Fig. 6f): all diagonal directional features are removed, and

“AntidiagonalSSA” (Fig. 6h): all antidiagonal directional features are removed.
Note that the “SSAiSBD 2channel filter bank” provides a “redundant form” of image decomposition: every component (main and residual) contains the same amount of pixels as the input image (N=N_{x}N_{y}). Thus, N input pixels are transformed to a total of 2N output pixels. Independent from the type of SSA referred to by “chosenSSA,” the input image can always be reconstructed by just adding the main and the residual component.
Discussion and results
In Section 3.4, “SSAiSBD 2channel filter banks” were presented. It was shown that in the case of using “VerticalSSA,” “HorizontalSSA,” “DiagonalSSA,” or “AntidiagonalSSA,” the residual component lacks one of the directional features. As such a removal is not the goal, this section introduces that a chaining of multiple “SSAiSBD 2channel filter banks” perform an extraction of directional features. These filter banks implement a multichannel decomposition that deconstruct a given input image into a sum of multiple subimages.
“SSAiSBD 4channel filter bank”
As a first step to realize such filter banks, the proposed SSA types are organized into orthogonal pairs. These pairs correspond to the properties of the residual components they create, whereupon the removed directional features are orthogonal to each other. The following orthogonal pairs are introduced:

“VerticalSSA” and “HorizontalSSA” with the same window length L and

“DiagonalSSA” and “AntidiagonalSSA” with the same window length L.
The first SSA type of such an orthogonal pair is referred to as “chosenSSA” and the remaining is referred to as “complementSSA.” Both members of the pair are equivalent, so each can be selected to become the “chosenSSA.”
Figure 7 shows the “SSAiSBD 4channel filter bank.” It performs a 2stage decomposition that decomposes an input image into a sum of four subimages. Each output subimage is the result of having the input image passing two consecutive “SSAiSBD 2channel filter banks” implementing an orthogonal pair.
The first filter bank implements “VerticalSSA” and “HorizontalSSA” as its orthogonal pair. This filter bank is referred to as “SSAiSBD 4channel filter bank,” type “vertical/horizontal,” with “VerticalSSA” picked as the “chosenSSA.” Figure 8 shows the four generated subimages, whereas a window length of L = 30 was used.

Subimage 1 is the approximation, containing the lowfrequency components (Fig. 8a),

Subimage 2 provides all vertical (90^{∘}) directional features (Fig. 8b),

Subimage 3 contains all horizontal (0^{∘}) directional features with orientation (Fig. 8c), and

Subimage 4 consists of the remains. They contain all diagonal (+ 45^{∘} and − 45^{∘}) directional features (Fig. 8d).
Subimage 2 in Fig. 8b and subimage 3 in Fig. 8c contain the extracted 0^{∘} and 90^{∘} directional features, respectively. Noticeably, subimage 1 in Fig. 8a contains only lowfrequency components. This demonstrates that both “VerticalSSA” (stage 1) and “HorizontalSSA” (stage 2) extract not only the desired directional features, but they also let the lowfrequency components pass. Furthermore, subimage 4 in Fig. 8d contains all diagonal directional components. This highlights that the directional selectivity of all stages of the filter bank are good: if the lowfrequency components, the vertical, and the horizontal directional features are removed, then the “remains” must contain the omitted diagonal directional features.
The second filter bank incorporates “DiagonalSSA” and “AntidiagonalSSA” as its orthogonal pair, with “DiagonalSSA” as the “chosenSSA.” The resulting filter bank, referred to as “SSAiSBD 4channel filter bank,” type “diagonal/antidiagonal,” extracts + 45^{∘} and − 45^{∘} directional features. The four resulting subimages created with this filter bank are as shown in Fig. 9.

Subimage 1 is the approximation containing the lowfrequency components (Fig. 9a),

Subimage 2 provides all diagonal (+ 45^{∘}) directional features (Fig. 9b),

Subimage 3 contains all antidiagonal (− 45^{∘}) directional features (Fig. 9c), and

Subimage 4 consists of the remains, in this case vertical (0^{∘}) and horizontal (90^{∘}) directional features (Fig. 9d).
Noticeably, the subimages 2 and subimages 3 created with both exemplary filter banks highlight not only the desired directional components, but they also provide slightly visible traces of the directional components contained in subimages 4.
Regarding each of the “SSAiSBD 4channel filter banks,” N input pixels are transformed to a total of 4N pixels. The sum of all four subimages is again equal to the input image.
“SSAiSBD 6channel filter bank”
The next target is a clear separation 0^{∘},90^{∘},+ 45^{∘}, and − 45^{∘} directional features with a single multistage filter bank. The “SSAiSBD 4channel filter bank,” type “vertical/horizontal,” ispresented in Section 4.1, is already able to separate 0^{∘} and 90^{∘} directional components and to provide a compound of all diagonal directional components (see Fig. 8). However, there are still two issues: the directional features with + 45^{∘} and − 45^{∘} are still not separated from each other, and the respective subimage 4 does not explicitly contain the diagonal features, but the remains. Thus, if there were subsequent omitted components, they also would have been mixed into the remains. However, with subsequent filtering steps, it is possible to extract and to separate all diagonal directional features into distinct subimages.
Here, “SSAiSBD 6channel filter banks” are created as an extension to a “SSAiSBD 4channel filter bank.” The filter bank structure presented on the Fig. 10 shows such a filter bank that is based on a “SSAiSBD 4channel filter bank,” type “vertical/horizontal.” The six resulting subimages presented in Fig. 11 provide the following “Octagon” decomposition:

Subimage 1 is the approximation containing the lowfrequency components (Fig. 11a),

Subimage 2 provides all vertical (90^{∘}) directional features (Fig. 11b),

Subimage 3 contains all horizontal (0^{∘}) directional features (Fig. 11c),

Subimage 4 highlights all diagonal (+ 45^{∘}) directional features (Fig. 11d),

Subimage 5 shows all antidiagonal (− 45^{∘}) directional features (Fig. 11e), and

Subimage 6 consists of the remains (Fig. 11f).
Obviously, other similar structures of “SSAiSBD 6channel filter banks” can be created, for example, a filter bank that is based on a “SSAiSBD 4channel filter bank,” type “diagonal/antidiagonal.” Such a filter bank will differ in the order of extraction of the directional features.
The presented “SSAiSBD 6channel filter bank” decomposes an image into a sum of six subimages and provides a clear separation of 0^{∘},90^{∘},− 45^{∘}, and + 45^{∘} directional features. To compare, a related transform consisting of a combination of DTCWT and a complex spline [32] is also able to detect the directions 0^{∘},90^{∘},− 45^{∘}, and + 45^{∘}. DTCWT is famous for its good directional selectivity. Using that transform, a decomposed “Octagon” image also provides clearly separated 0^{∘} and 90^{∘} directional components with slightly visible traces of the − 45^{∘} and + 45^{∘} directional features [33]. Thus, the “SSAiSBD 6channel filter banks” presented in this paper shares this property of providing good directional selectivity.
As in the previous examples, all window lengths are L = 30. N input pixels are transformed to a total of 6N pixels. For the inverse transform to get a reconstruction, all six subimages have to be added.
Demonstration of “SSAiSBD multichannel filter banks” using a natural image
So far, all provided examples involved the synthetic “Octagon” test image. In this section, the performance of the “SSAiSBD multichannel filter bank” is rated using the natural “Cameraman” image (see Fig. 1b).
At first, Fig. 12 presents the results of processing the “Cameraman” image with the “SSAiSBD 4channel filter bank,” type “vertical/horizontal” (see Fig. 7).
Figure 13 presents the results of processing the “Cameraman” image with the “SSAiSBD 4channel filter bank,” type “diagonal/antidiagonal.”
In the next experiment, the “Cameraman” image was decomposed with the “SSAiSBD 6channel filter bank.” Figure 14 presents the calculated subimages 1–6 according to the filter bank structure shown on Fig. 10 with a window length L = 12.
As mentioned before, the window length L has to be determined empirically depending on the application. Here, the larger window length of L = 12 was selected to demonstrate its influence on the extraction of directional features.
A sidebyside comparison of Figs. 12a–c and 14a–c shows the influence of the chosen window size L on the cutoff frequency. The larger the value of L is, the smaller becomes the part of the spectrum contained in the lowfrequency approximation.
All subimages of Figs. 12, 13, and 14 have been contraststretched. The value of each pixel fits into the range from 0 to 255, calculated in the same way as used for the publication of MRSVD [6].
To increase visibility, the same output subimages 1–6 of the “SSAiSBD 6channel filter bank” are presented again on Fig. 15 in its normalized form. To achieve that, the value of each pixel was rescaled as \( min_{{i,j}} \tilde {I}_{{i,j}} \to 0\) and \( max_{{i,j}} \tilde {I}_{{i,j}} \to 1\). They are printed using the standard grayscale colormap. Such presentation is commonly used in related work [40, 44].
A comparable decomposition of the “Cameraman” image was achieved with the first level of DTCWT decomposition [44]. In that publication, Fig. 5 contains subband images that contain separated directional features for ± 15^{∘},± 45^{∘}, and ± 75^{∘} using the same normalization scheme as used here for the images on Fig. 15. The quality of the performance of both filter banks is visually similar: the + 45^{∘} and − 45^{∘} directional features were extracted successfully by both. DTCWT divides an input image into its low and high frequency components, with a cutoff frequency depending on the chosen mother wavelet (type and order). Regarding the “SSAiSBD filter banks,” the window length L provides the means for such adjustments.
Conclusions
The “singular spectrum analysisbased image subband decomposition filter banks” (“SSAiSBD filter banks”) are introduced. Four types of “SSAiSBD 2channel filter banks” based on Basic SSA serve as building blocks for “SSAiSBD multichannel filter banks.”
Due to the subband separation based on SVD, the proposed filter banks are not affected negatively by shift variance. To remind, shift variance is typical for waveletbased image subband decomposition schemes.
Good directional selectivity is provided by the “SSAiSBD 6channel filter bank” that represents an input image as a sum of an approximation containing the lowfrequency components and, separated from each other, the highfrequency components. These highfrequency components refer to distinct horizontal (0^{∘}), vertical (90^{∘}), diagonal (+ 45^{∘}), and antidiagonal (− 45^{∘}) directional features, and the remains. This transform is invertible: a simple addition of all components is sufficient to reproduce the input image. The presented novel filter banks introduce the window length as a parameter that has to be determined empirically depending on the application.
This new method of image subband decomposition is presented for the attention of the scientific community. Mathematicians may be interested in proving or explaining this heuristically developed method while engineers may incorporate this algorithm to improve their specific features extraction, edge detection, or face and gesture recognition tasks.
Availability of data and materials
Please contact the corresponding author for data requests.
Abbreviations
 1D:

Onedimensional
 2D:

Twodimensional
 SSA:

Singular spectrum analysis
 SVD:

Singular value decomposition
 DWT:

Discrete wavelet transform
 2DDWT:

Twodimensionaldiscrete wavelet transform
 MRSVD:

Multiresolution singular value decomposition
 UDWT:

Undecimated discrete wavelet transform
 2DUDWT:

Twodimensional undecimated discrete wavelet transform
 DTCWT:

Dualtree complex wavelet transform
 “SSAiSBD filter bank”:

Singular spectrum analysisbased image subband decomposition filter banks
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Acknowledgements
The corresponding author is deeply grateful for the MSCSP program of the Ilmenau University of Technology and her scholarship of the “German Academic Exchange Service” (DAAD) (Grant No. A/08/77322) that made her masters studies possible. The presented work is based on the knowledge gained during the course AASP (Prof. Dr.Ing. Martin Haardt). We thank the anonymous reviewers for their excellent remarks. They helped to improve this paper.
Funding
The work presented in this paper was solely financed by the research budget of the professorship for Control Technology at the Nuremberg Campus of Technology in Germany (NCT).
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J.E. developed all ideas and algorithms, wrote most of the manuscript together with F.E., and carried out all experiments. F.E. organized focusing discussions and did many corrections according structure of the publication and used terminology. F.G. assisted during the implementation (programming and debugging of some pieces of the code). R.S. supervised and reviewed the work, as a head of research center. The authors read and approved the final manuscript.
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Evers, J., Evers, F., Goppelt, F. et al. Singular spectrum analysisbased image subband decomposition filter banks. EURASIP J. Adv. Signal Process. 2020, 29 (2020). https://doi.org/10.1186/s13634020006854
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Keywords
 Singular spectrum analysis
 Singular value decomposition
 Directional selectivity
 Filter bank
 Wavelet transform
 Subband decomposition