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Fully Orthogonal 2-D Lattice Structures for Quarter-Plane and Asymmetric Half-Plane Autoregressive Modeling of Random Fields

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Fully Orthogonal 2-D Lattice Structures for Quarter-Plane and Asymmetric Half-Plane Autoregressive Modeling of Random Fields

Ahmet Hamdi Kayran , Senior Member, IEEE, and Erdogan Camcioglu

Abstract—This paper is mainly devoted to the derivation of a new fully orthogonal two-dimensional (2-D) lattice structure for general autoregressive (AR) modeling of random fields. Similar to the 1-D lattice theory, this approach is based on recursive incrementation of the prediction support region by adding a single past observation point at each stage. In addition to developing the basic theory, the presentation includes horizontal and vertical building blocks of the proposed causal 2-D AR lattice filters. The algorithm presented here is useful for high-resolution 2-D spectral analysis applications.

It is shown that the new fully orthogonal 2-D lattice structure can be an efficient tool for high-resolution radar imaging.

Index Terms—2-D signal processing, 2-D lattice filters, linear prediction, autoregressive modeling, 2-D spectral estimation.

I. INTRODUCTION

T

HE great interest in the lattice approach stems from its property of orthogonality [1]. Indeed, the lattice filter trans- forms the sequence of correlated input samples into a sequence of uncorrelated backward prediction errors. This property leads to important applications in the areas of adaptive Wiener filtering [2]. In 1-D signal processing, it is well-known that the succes- sive orthogonalization offered by the lattice structure provides convergence advantages not obtained with tapped-delay-line methods.

Motivated by the success of 1-D lattice methods in the past 40 years, several researchers have attempted to solve the chal- lenging problem of 2-D equivalent lattice algorithms with only partial success [3]–[15]. Since the great majority of the 2-D lattice parameter models in the literature [4]–[8], [11]–[15] are based on simultaneous introduction of many points into the prediction support region (PSR), they can generate a set of mutually orthogonal realization subspaces at each stage. On the other hand, the backward prediction error fields (PEFs) in these subspaces are not mutually orthogonal, in other words, they are correlated. Another fundamental approach to modeling 2-D

Manuscript received December 1, 2018; revised June 27, 2019; accepted July 4, 2019. Date of publication July 23, 2019; date of current version August 8, 2019. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Mark A. Davenport. (Corresponding author: Ahmet Hamdi Kayran.)

A. H. Kayran is with the Faculty of Electrical and Electronics Engineering, Is- tanbul Technical University, Istanbul 34469, Turkey (e-mail: kayran@itu.edu.tr).

E. Camcioglu is with the Faculty of Computer Engineering, Istanbul Rumeli University, Silivri 34570, Turkey (e-mail: erdogan.camcioglu@rumeli.edu.tr).

This paper has supplementary downloadable multimedia material available at http://ieeexplore.ieee.org/http://ieeexplore.ieee.org, provided by the authors.

Digital Object Identifier 10.1109/TSP.2019.2929463

fields is based on the introduction of a single past observation point into the PSR [3], [9], [10]. However, these alternative struc- tures aim in general to produce either a standard quarter-plane (QP) model or an asymmetric half-plane (ASHP) model.

In this paper, we present a new fully orthogonal 2-D lattice structure which can be used to simultaneously obtain all possible QP/ASHP models for general autoregressive (AR) modeling of random fields. Our procedure is conceptually different than all other methods [3]–[15] and increments the PSR by adding only one past observation point at each stage either horizontally or vertically. More importantly, we have succeeded to recursively compute the forward/backward PEFs of all neighboring points around the non symmetric PSR. In our algorithm, there is no assumption about the 2-D data, except the signal is treated as a stationary random field which its statistics are shift-invariant.

No estimation of the autocorrelation function is required in this algorithm.

There are potential applications in areas such as high- resolution radar imaging [16], 2-D joint-process estimation for image restoration and noise cancellation [17], image data com- pression [18] and breast cancer detection and classification [19].

The organization of the paper is as follows: In Section II, we present the main result of this paper. Auxiliary causal half-plane PEFs and ASHP prediction error filters are introduced. The horizontal and vertical recursive order incrementation of the PSR are shown for all possible directions. In Section III, we define vertical and horizontal building blocks of the proposed 2-D lattice structure. The forward/backward 2-D delay matrices and initial delay matrices are given for the horizontal/vertical incrementation. The fully orthogonal 2-D bases vectors are presented in Section IV. In Section V, the computer simulations are carried out to confirm the validity of the proposed theory. The fully orthogonal lattice filters are also applied to high resolution inverse synthetic aperture radar (ISAR) imaging.

II. AUXILIARYCAUSALHALF-PLANEPEFS ANDASHP PREDICTIONERRORFILTERS

Similar to the 1-D lattice signal modeling, at our proposed 2-D lattice structure the PSR order will be recursively incre- mented by adding a single past observation point at each stage either horizontally or vertically. However unlike the 1-D case, at vertical and horizontal directions, there are two possible ways of incrementation of the PSR. Due to this natural diversity, with the

1053-587X © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.

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Fig. 1. Definition of the auxiliary horizontal and vertical forward/backward PEFs for the rectangular PSRS(p,q)k,l .

choice of the initial augmentation point, our method will be able to provide all possible asymmetric half-plane models as well as the quarter-plane ones.

For the(p, q)th order stage, the rectangular PSR can be defined as

Sk,l(p,q)= span{y(k − i, l − j); 1 ≤ i ≤ p, 1 ≤ j ≤ q}. (1) The prediction of the neighboring p points on the right of the PSR, namely, y(k − 1, l), . . . , y(k − p, l) will give the auxiliary vertical forward PEFs, v(p−1,q)1 (k − 1, l), . . . , vp(p−1,q)(k − 1, l), respectively. In similar fashion, the prediction of the neighboring q points at the bottom at the PSR, namely, y(k, l − 1), . . . , y(k, l − q) will also give the auxiliary hori- zontal forward PEFs, h(p,q−1)1 (k, l − 1), . . . , h(p,q−1)q (k, l − 1), respectively. In addition to these PEFs the prediction of the neighboring 2 points at bottom corners of the PSR, namely, y(k, l) and y(k, l − q − 1) will be named as the auxiliary diago- nal forward PEFs, d(p,q)1 (k, l) and d(p,q)2 (k, l − 1), respectively.

Similarly, the prediction of neighboring p+ q + 2 past points of the PSR, S(p,q)k,l , can be made as follows: The prediction of p points on its left, namely, y(k − p, l − q − 1), . . . , y(k − 1, l − q − 1), and q points at its top, namely, y(k − p − 1, l − q), . . . , y(k − p − 1, l), and 2 points at its top corners, namely, y(k − p − 1, l − q − 1) and y(k − p − 1, l) will be defined as the auxiliary vertical backward PEFs, v˜1(p−1,q)(k − 1, l − 1), . . . , ˜v(p−1,q)p (k − 1, l − 1) and the auxiliary horizontal back- ward PEFs, ˜h(p,q−1)1 (k − 1, l − 1), . . . , ˜h(p,q−1)p (k − 1, l − 1) and the auxiliary diagonal backward PEFs, ˜d(p,q)1 (k − 1, l − 1) and ˜d(p,q)2 (k, l − 1), respectively.

Remark 2.1: The rectangular PSR given in (1) are used to de- fine all possible neighboring auxiliary forward/backward PEFs as shown in Fig. 1. However, after the horizontal or vertical incrementation of the past observations, the resulting PSR will have a non symmetric shape. Hence we will have two different ASHP shaped PSRs. Indeed, they are mirror images of each

Fig. 2. Horizontal recursive order incrementation from left-to-right and defi- nition of forward/backward PEFs.

other with respect to the vertical and horizontal axes. This issue will be clarified in the following subsections.

It is important to notice that the definition of S(p,q)k,l in (1) assumes that the samples from the 2-D field are arranged in the matrix form, with k being the row index, and l the column index. In other words, our description is the90rotated version of the conventional(x, y) arrangement of sample points, in which k corresponds to the y− coordinate, and l corresponds to the x− coordinate.

A. Horizontal Recursive Order Incrementation

Adding only one past observation point at each stage from left-to-right or from right-to-left along the horizontal direction, a single row of q new observations can be added to the top of the PSR as shown in Figs. 2 and 3. These type PSRs will be enable to define a set of four ASHP predictions error filters whose output computed recursively row by row.

Remark 2.2: In order to avoid confusion and ensure conve- nience, when presenting the error propagation equations of our new orthogonal 2-D lattice model in this work, α, β, and γ will be used to define the reflection coefficients or the lattice parameters for the auxiliary vertical, horizontal, and diagonal PEFs. Similarly, δ will be defined as the reflection coefficient for the ASHP AR PEFs. Moreover, in order to calculate the reflections at each stage, the sum of the forward and backward PEFs should be minimized in the mean-square sense.

1) Incrementation From Left-To-Right: As a result of this horizontal recursive order incrementation from left-to-right, a set of new auxiliary vertical forward and backward PEFs, namely, v(p,q)1,l

j (k, l) and ˜v1,l(p,q)

j (k, l) are introduced, where the subscript lj denotes the number points added to the top of the PSR from left-to-right. At the same time, the existing auxiliary vertical

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Fig. 3. Horizontal recursive order incrementation from right-to-left and defi- nition of forward/backward PEFs.

forward and backward PEFs of Sk,l(p,q)in Fig. 1, are renumbered by increasing one by one. This arrangement is shown in Fig. 2.

At this stage, in addition to these new vertical PEFs, a set of symmetric half-plane forward and backward PEFs are also presented. For j= 0, the horizontal ASHP model PEFs are initialized by the auxiliary horizontal forward/backward PEFs of the rectangular PSR, Sk,l(p,q), as follows: For i= 1, . . . , q,

ri,l(p,q−1)0 (k, l) = h(p,q−1)i (k, l)

˜

ri,l(p,q−1)0 (k, l) = ˜h(p,q−1)i (k − 1, l). (2) The number of these ASHP PEFs decreases as the incrementa- tion index j of the new points increases. For the incrementation with j points, the ASHP model PEFs, can be obtained by a linear combination of the(j − 1) points ones as follows: For j = 1, . . . , q and i = 1, . . . , q − j,

ri,l(p,q−1)j (k, l) = ri+1,l(p,q−1)j−1(k, l) − δi,l(p,q−1)j r1,l(p,q−1)j−1 (k, l)

˜ ri,l(p,q−1)

j (k, l) = ˜ri+1,l(p,q−1)

j−1(k, l) − δ(p,q−1)i,lj r˜(p,q−1)1,l

j−1 (k, l). (3) The first horizontal auxiliary forward/backward ASHP PEFs, namely, ri,l(p,q−1)j (k, l) and ˜ri,l(p,q−1)j (k, l) for j = 1, . . . , q − 2, can be used to generate2(q − 2) different ASHP model causal 2-D lattice filters. The outputs of their synthesis models can be computed row by row from left to right or vice versa. This row by row computation property is the result of horizontal recursive order incrementation.

Moreover, it is interesting to note that r(p,q−1)i,l

q−1 (k, l) and

˜ r(p,q−1)i,l

q−1 (k, l) for j = q − 1 become the horizontal second- and fourth-quadrant QP AR models PEFs, namely, r(p,q−1)QP2 (k, l)

and˜rQP(p,q−1)2 (k, l), respectively, and they can be written as, r(p,q−1)QP

2 (k, l) = r(p,q−1)2,lq−2 (k, l) − δ1,l(p,q−1)q−1 r(p,q−1)1,l

q−2 (k, l)

˜

rQP(p,q−1)2 (k, l) = ˜r(p,q−1)2,lq−2 (k, l) − δ1,l(p,q−1)q−1 ˜r1,l(p,q−1)q−2 (k, l). (4) The outputs of their synthesis model can be computed row by row or column by column unlike the ASHP models. However, for both the ASHP and the QP AR models, the computational procedure starts from the NE corner or from SW corner of the given data set. Indeed, this distinguishing computational feature of the proposed lattice model is the result of recursive order incrementation from left-to-right as shown in Fig. 2.

For the horizontal incrementation from left-to-right the auxil- iary first diagonal PEFs, namely, d(p,q)1 (k, l) and ˜d(p,q)1 (k, l) are used to initiate a set of new auxiliary vertical PEFs, v(p,q)1,l0 (k, l) and ˜v1,l(p,q)

0 (k, l), respectively. On the other hand, the existing auxiliary vertical forward PEFs of Sk,l(p,q), namely, v(p−1,q)i (k, l), for i= 1, . . . , q, will be shifted horizontally as shown in Fig. 2.

Hence for j = 0 the initial auxiliary vertical forward/backward PEFs are given by the following equations: For i= 1, . . . , p + 1,

v(p,q)i,l

0 (k, l) =

⎧⎨

d(p,q)1 (k, l), for i= 1

v(p−1,q)i−1 (k − 1, l), for i= 2, . . . , p + 1, (5) and

˜

vi,l(p,q)0 (k, l) =

⎧⎨

d˜(p,q)1 (k, l), for i= 1

˜

vi−1(p−1,q)(k, l), for i= 2, . . . , p + 1.

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After the above initialization, the error propagation equations for the p+ 1 auxiliary vertical forward/backward PEFs can be written for j = 1, . . . , q and i = 1, . . . , p + 1 as follows,

vi,l(p,q)

j (k, l) = v(p,q)i,lj−1(k, l) − αi,l(p,q)j r1,l(p,q−1)

j−1 (k, l − 1)

˜

vi,l(p,q)j (k, l) = ˜v(p,q)i,lj−1(k, l) − αi,l(p,q)j r˜(p,q−1)1,lj−1 (k, l). (7) In a similar way, the error propagation equations for q horizontal forward/backward PEFs, can be written as, for j = 1, . . . , q and i = 1, . . . , q,

h(p+1,q−1)i,lj (k, l) = h(p+1,q−1)i,lj−1 (k, l)

− βi,l(p+1,q−1)j ˜r1,l(p,q−1)

j−1 (k − 1, l)

˜h(p+1,q−1)i,l

j (k, l) = ˜h(p+1,q−1)i,lj−1 (k, l)

− βi,l(p+1,q−1)j r(p,q−1)1,lj−1 (k, l) (8) where for j= 0 and i = 1, . . . , q,

h(p+1,q−1)i,l0 (k, l) = h(p,q−1)i (k, l)

˜h(p+1,q−1)i,l

0 (k, l) = ˜h(p,q−1)i (k, l). (9) The first-diagonal forward/backward PEFs, d(p,q)1 (k, l) and d˜(p,q)1 (k, l) are used to initiate the new vertical PEFs that arise

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as a result of the incrementation left to right as indicated in (5) and (6). Their error propagation equations can be given as, for j = 1, . . . , q,

d(p+1,q)1,l

j (k, l) = d(p+1,q)1,lj−1 (k, l)−γ1,l(p+1,q)j r˜(p,q−1)1,l

j−1 (k − 1, l − 1) d˜(p+1,q)1,lj (k, l) = ˜d(p+1,q)1,lj−1 (k, l) − γ1,l(p+1,q)j r1,l(p,q−1)j−1 (k, l) (10) where for j= 0,

d(p+1,q)1,l

0 (k, l) = d(p,q)1 (k, l)

d˜(p+1,q)1,l0 (k, l) = ˜d(p,q)1 (k − 1, l). (11) 2) Incrementation From Right-To-Left: Horizontally recur- sive order incrementation from right to left generates a set of new auxiliary vertical forward and backward PEFs, namely, vp+1,r(p,q)j(k, l) and ˜v(p,q)p+1,rj(k, l) as shown in Fig. 3, where the subscript rjdenotes the number of past points added to the top of the PSR from right to left.

Similar to the left to right incrementation case, a set of aux- iliary ASHP forward/backward PEFs are also presented. These PEFs are initiated for j= 0 and i = 1, . . . , q as follows:

ri,r(p,q−1)0 (k, l) = h(p,q−1)q−i+1 (k, l)

˜

ri,r(p,q−1)0 (k, l) = ˜h(p,q−1)q−i+1 (k − 1, l). (12) It is interesting to note that above initiation is made at the reversed order so that for example, the first ASHP PEF is equal to the last horizontal one, r1,r(p,q−1)0 (k, l) = h(p,q−1)q (k, l), and so on.

As the incremented points increase right to left, the decreasing number of ASHP PEFs can be given as; for j= 1, . . . , q and i = 1, . . . , q − j,

ri,r(p,q−1)j (k, l) = ri+1,r(p,q−1)j−1(k, l) − δi,r(p,q−1)j r(p,q−1)1,rj−1 (k, l)

˜

ri,r(p,q−1)j (k, l) = ˜ri+1,r(p,q−1)j−1(k, l) − δi,r(p,q−1)j r˜1,r(p,q−1)j−1 (k, l). (13) Analogously the left to right case, the first ASHP PEFs, r(p,q−1)1,rj (k, l) and ˜r(p,q−1)1,rj (k, l) for j = 1, . . . , q − 2 can be used to generate 2(q − 2) ASHP AR model causal 2-D lat- tice filters as well as for j= q − 1 the QP AR model ones, namely, rQ1(p,q−1)(k, l) = r1,r(p,q−1)q−1 (k, l) and ˜r(p,q−1)Q1 (k, l) =

˜

r(p,q−1)1,rq−1 (k, l).

Although the output of these ASHP models is computed in the same way, namely, row by row, their computational procedure start from the NW corner and the SE corner of the input data for the forward and the backward PEFs, respectively.

The error propagation equations for the p+ 1 auxiliary ver- tical forward/backward PEFs can be given for j= 1, . . . , q and i = 1, . . . , p + 1 as follows:

vi,r(p,q)j (k, l) = v(p,q)i,rj−1(k, l) − αi,r(p,q)j r˜1,r(p,q−1)j−1 (k, l − 1)

˜

v(p,q)i,rj (k, l) = ˜v(p,q)i,rj−1(k, l) − α(p,q)i,rj r(p,q−1)1,rj−1 (k, l). (14)

where for j= 0 and i = 1, . . . , p + 1,

vi,r(p,q)0 (k, l) =

⎧⎨

vi(p−1,q)(k, l), for i= 1, . . . , p d˜(p,q)2 (k, l), for i= p + 1,

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and

˜

v(p,q)i,r0 (k, l) =

⎧⎨

˜

vi(p−1,q)(k − 1, l), for i= 1, . . . , p d(p,q)2 (k, l), for i= p + 1.

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Similarly the q auxiliary horizontal forward/backward PEFs in Fig. 3 can be written as, for j= 1, . . . , q and i = 1, . . . , q,

h(p+1,q−1)i,r

j (k, l) = h(p+1,q−1)i,rj−1 (k, l)

− βi,r(p+1,q−1)j r˜(p,q−1)1,rj−1 (k − 1, l)

˜h(p+1,q−1)i,rj (k, l) = ˜h(p+1,q−1)i,rj−1 (k, l)

− βi,r(p+1,q−1)j r1,r(p,q−1)j−1 (k, l) (17) where for j= 0 and i = 1, . . . , q,

h(p+1,q−1)i,r0 (k, l) = h(p,q−1)i (k, l)

˜h(p+1,q−1)i,r0 (k, l) = ˜h(p,q−1)i (k − 1, l). (18) The auxiliary second diagonal forward/backward PEFs, d(p,q)2 (k, l) and ˜d(p,q)2 (k, l) are used to initiate the new verti- cal PEFs in (15), (16), namely,v˜p+1,r(p,q)0(k, l) and vp+1,r(p,q)0(k, l), respectively. Their error propagation equations are expressed for j = 1, . . . , q, as follows:

d2,r(p+1,q)j (k, l) = d(p+1,q)2,rj−1 (k, l) − γ2,r(p+1,q)j ˜r1,r(p,q−1)j−1 (k − 1, l) d˜(p+1,q)2,rj (k, l) = ˜d(p+1,q)2,rj−1 (k, l) − γ2,r(p+1,q)j r(p,q−1)1,rj−1 (k, l) (19) where for j= 0,

d(p+1,q)2,r0 (k, l) = d(p,q)2 (k, l)

d˜(p+1,q)2,r0 (k, l) = ˜d(p,q)2 (k − 1, l). (20)

B. Vertical Recursive Order Incrementation

Similar to the horizontal order incrementation, adding only one past observation point at each stage from top to bottom or from bottom to top along the vertical direction, a single column of p new observations can be added to the left of the PSR as shown in Figs. 4 and 5. the vertical ASHP PEFs, namely, e(p−1,q)1,tj (k, l), e(p−1,q)1,bj (k, l) and the backward ones, for j = 1, . . . , p − 2, can be used to generate 4(p − 2) different ASHP casual 2-D lattice filters. Moreover, the output of their synthesis models can be computed column by column from top to bottom or vice versa.

1) Incrementation From Top-To-Bottom: Fig. 4 shows the vertical incrementation from top to bottom. For this case a set of new auxiliary horizontal forward and backward PEFs, namely, h(p,q)1,tj (k, l) and ˜h(p,q)1,tj (k, l) for j = 1, . . . , p, are introduced, where the subscript tjdenotes the number of points added to the left of the PSR from top to bottom. Furthermore, a set of vertical

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Fig. 4. Vertical recursive order incrementation from top-to-bottom and defi- nition of forward/backward PEFs.

Fig. 5. Vertical recursive order incrementation from bottom-to-top and defi- nition of forward/backward PEFs.

ASHP forward/backward PEFs are presented additionally. They are initialized as follows: For j= 0 and i = 1, . . . , p,

e(p−1,q)i,t0 (k, l) = v(p−1,q)i (k, l)

˜

e(p−1,q)i,t0 (k, l) = ˜v(p−1,q)i (k, l). (21) As expected, the number of these vertical ASHP PEFs will decrease as the number of added new points increases. Indeed, for an incrementation with j points, we have only p− j PEFs as shown in Fig. 4. For j= 1, . . . , p and i = 1, . . . , p − j their error propagation are given as

ei,t(p−1,q)j (k, l) = e(p−1,q)i+1,tj−1(k, l) − δi,t(p−1,q)j e(p−1,q)1,tj−1 (k, l)

˜

e(p−1,q)i,tj (k, l) = ˜e(p−1,q)i+1,tj−1(k, l) − δi,t(p−1,q)j ˜e(p−1,q)1,tj−1 (k, l). (22) For j= 1, . . . , p − 2 we get 2(p − 2) vertical ASHP PEFs, e(p−1,q)1,tj (k, l) and ˜e(p−1,q)1,tj (k, l) and for j = p − 1, e(p−1,q)1,tp−1 (k, l) ande˜(p−1,q)1,tp−1 (k, l) become the vertical first- and third-quadrant QP AR models, respectively. e(p−1,q)QP

1 (k, l) and ˜e(p−1,q)QP

1 (k, l) can be written as in (4).

The error propagation equations for the q+ 1 auxiliary hor- izontal forward/backward PEFs are given for j= 1, . . . , p and i = 1, . . . , q + 1, as follows:

hi,t(p,q)j (k, l) = h(p,q)i,tj−1(k, l) − βi,t(p,q)j e(p−1,q)1,tj−1 (k − 1, l)

˜hi,t(p,q)j (k, l) = ˜h(p,q)i,tj−1(k, l) − βi,t(p,q)j ˜e(p−1,q)1,tj−1 (k, l) (23) where for j= 0 and i = 1, . . . , q + 1,

h(p,q)i,t

0 (k, l) =

⎧⎨

d(p,q)1 (k, l), for i= 1

h(p,q−1)i−1 (k, l − 1), for i= 2, . . . , q + 1, (24) and

˜h(p,q)i,t0 (k, l) =

⎧⎨

d˜(p,q)1 (k, l), for i= 1

˜h(p,q−1)i−1 (k, l), for i= 2, . . . , q + 1.

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Likewise the error propagation equations for the p vertical PEFs can be written as, for j= 1, . . . , p and i = 1, . . . , p,

vi,t(p−1,q+1)j (k, l) = v(p−1,q+1)i,tj−1 (k, l) − αi,t(p−1,q+1)j e˜(p−1,q)1,tj−1

× (k, l − 1)

˜

v(p−1,q+1)i,t

j (k, l) = ˜v(p−1,q+1)i,t

j−1 (k, l) − α(p−1,q+1)i,tj e(p−1,q)1,tj−1 (k, l) (26) where for j= 0 and j = 1, . . . , p,

v(p−1,q+1)i,t0 (k, l) = v(p−1,q)i (k, l)

˜

v(p−1,q+1)i,t0 (k, l) = ˜v(p−1,q)i (k, l − 1). (27) The first diagonal auxiliary PEFs, d(p,q)1 (k, l) and ˜d(p,q)1 (k, l) are also employed to initiate the new horizontal PEFs at this case as shown in (24), (25). Their error propagation equations similarly are defined for j= 1, . . . , p, as follows:

d1,t(p,q+1)j (k, l) = d(p,q+1)1,tj−1 (k, l) − γ1,t(p,q+1)j e˜(p−1,q)1,tj−1

× (k − 1, l − 1)

d˜(p,q+1)1,tj (k, l) = ˜d(p,q+1)1,tj−1 (k, l) − γ1,t(p,q+1)j e(p−1,q)1,tj−1 (k, l) (28) where for j= 0,

d(p,q+1)1,t0 (k, l) = d(p,q)1 (k, l)

d˜(p,q+1)1,t0 (k, l) = ˜d(p,q)1 (k, l − 1). (29) 2) Incrementation From Bottom-To-Top: Vertically recur- sive order incrementation from bottom to top generate a set new auxiliary horizontal forward and backward PEFs, namely, h(p,q)q+1,b

j(k, l) and ˜h(p,q)q+1,bj(k, l) as shown in Fig. 5, where the subscript bjdenotes the number of past points added to the right side of the PSR from bottom to top. However, unlike the top to bottom case which is discussed in the above Subsection II-B1, this is accomplished by the use of the auxiliary PEFs, namely, d(p,q)2 (k, l) and ˜d(p,q)2 (k, l) instead of the first diagonal ones.

(6)

Furthermore, similar to the right to left case in (12), the auxiliary vertical ASHP forward/backward PEFs are initiated at the reversed order for j= 0 and i = 1, . . . , p as follows:

e(p−1,q)i,b

0 (k, l) = vp+1−i(p−1,q)(k, l)

˜

e(p−1,q)i,b0 (k, l) = ˜vp+1−i(p−1,q)(k, l). (30) As the incremented points increase bottom to top, the decreasing number of ASHP vertical PEFs can be written as follows: For j = 1, . . . , p and i = 1, . . . , p − j,

e(p−1,q)i,b

j (k, l) = e(p−1,q)i+1,bj−1(k, l) − δi,b(p−1,q)j e(p−1,q)1,b

j−1 (k, l)

˜

e(p−1,q)i,bj (k, l) = ˜e(p−1,q)i+1,bj−1(k, l) − δi,b(p−1,q)j e˜(p−1,q)1,bj−1 (k, l). (31) For j= 1, . . . , p − 2 we can also get a different set of 2(p − 2) vertical ASHP PEFs, e(p−1,q)1,b

j (k, l) and ˜e(p−1,q)1,b

j (k, l), from the bottom to top case. Similarly for j= p − 1, e(p−1,q)1,bj−1 (k, l) and

˜ e(p−1,q)1,b

j−1 (k, l) are the first- and third-quadrant QP AR models, respectively.

From Fig. 5, the q+ 1 auxiliary horizontal PEFs are defined as for j = 1, . . . , p and i = 1, . . . , q + 1,

hi,b(p,q)j (k, l) = h(p,q)i,bj−1(k, l) − βi,b(p,q)j e˜(p−1,q)1,bj−1 (k − 1, l)

˜h(p,q)i,b

j (k, l) = ˜h(p,q)i,bj−1(k, l) − βi,b(p,q)j e(p−1,q)1,b

j−1 (k, l) (32) for j = 0 and i = 1, . . . , q + 1,

h(p,q)i,b0 (k, l) =

⎧⎨

h(p,q−1)i (k, l), for i= 1, . . . , q d(p,q)2 (k, l), for i= q + 1,

(33)

and

˜h(p,q)i,b

0 (k, l) =

⎧⎨

˜h(p,q−1)i (k, l − 1), for i= 1, . . . , q d˜(p,q)2 (k, l), for i= q + 1.

(34)

The error propagation equations for the p vertical PEFs can be written as for i, j= 1, . . . , p,

vi,b(p−1,q+1)

j (k, l) = vi,b(p−1,q+1)j−1 (k, l) − α(p−1,q+1)i,bj e˜(p−1,q)1,b

j−1

× (k, l − 1)

˜

vi,b(p−1,q+1)

j (k, l) = ˜vi,b(p−1,q+1)

j−1 (k, l) − α(p−1,q+1)i,bj e(p−1,q)1,b

j−1 (k, l) (35) where for j= 0 and j = 1, . . . , p,

vi,b(p−1,q+1)

0 (k, l) = vi(p−1,q)(k, l)

˜

vi,b(p−1,q+1)

0 (k, l) = ˜vi(p−1,q)(k, l − 1). (36) In this case, d(p,q)2 (k, l) and ˜d(p,q)2 (k, l) are used to initiate the new horizontal PEFs as depicted in (33), (34). The second diagonal PEFs are given, for j= 1, . . . , p, as follows:

d2,b(p,q+1)j (k, l) = d(p,q+1)2,bj−1 (k, l) − γ2,b(p,q+1)j e(p−1,q)1,bj−1 (k − 1, l) d˜(p,q+1)2,b

j (k, l) = ˜d(p,q+1)b,b

j−1 (k, l) − γ2,b(p,q+1)j e˜(p−1,q)1,b

j−1 (k, l − 1) (37)

for j= 0,

d(p,q+1)2,b

0 (k, l) = d(p,q)2 (k, l − 1)

d˜(p,q+1)2,b0 (k, l) = ˜d(p,q)2 (k, l). (38) Remark 2.3: In this subsection, we have introduced 12 PEFs (i.e., e,e, r, ˜˜ r, v, ˜v, h, ˜h, d1, ˜d1, d2, ˜d2), and 5 lattice coefficients (namely, α, β, γ1, γ2, δ). Those are related through 8 recursions for the horizontal recursive order incrementations (i.e., (3), (7), (8), (10), (13), (15), (17), (19)) and 8 recursions for the vertical ones (i.e., (22), (23), (26), (28), (31), (32), (35), (37)). The general recursion pair can be written as,

xj(k, l) = xj−1(k, l) − Kjyj−1(k, l)

˜

xj(k, l) = ˜xj−1(k, l) − Kjy˜j−1(k, l) (39) In order to obtain the lattice coefficients, Burg’s method was used as in the 1-D case [1], [2]. Therefore, the parameter Kj in (39) is computed by minimizing the sum of the forward and backward PEFs,

Pj = E

|xj(k, l)|2 + E

|˜xj(k, l)|2

. (40)

Substituting (39) in (40) and determining the complex-valued gradient of Pjwith respect to Kj, and setting this gradient equal to zero, we can find optimum value of the reflection coefficient as follows:

Kj =E

yj−1(k, l)xj−1(k, l) + E

˜

xj−1(k, l)˜yj−1(k, l) E [|yj−1(k, l)|2+ |˜yj−1(k, l)|2]

(41) where E[·] denotes the expected value over the PEFs. When working with measurements, the probabilistics correlations are approximated by sample correlations.

C. First Order Stage

At the initial zeroth order stage the PSR, Sk,l(0,0)is an empty set. Therefore, the first order auxiliary vertical, horizontal and diagonal forward/backward PEFs are initiated by the given 2-D observation data v1(0,0)(k, l) = h(0,0)1 (k, l) = d(0,0)i (k, l) = y(k, l) and ˜v1(0,0)(k, l) = ˜h(0,0)1 (k, l) = ˜d(0,0)i (k, l) = y(k, l), for i= 1, 2, respectively.

Since the proposed 2-D lattice model require only one of the diagonal prediction error fields, d(0,0)1 (k, l) or d(0,0)2 (k, l) can be chosen at this initial stage. Fig. 6 shows two possible initial stages for the vertical/horizontal order incrementation.

As discussed in the previous subsections, the horizontal in- crementation from left-to-right and the vertical incrementation from top-to-bottom require the employment of d(p,q)1 (k, l) and d˜(p,q)1 (k, l) in order to initiate the necessary new vertical and hor- izontal PEFs as shown in Figs. 2 and 4. Moreover the ASHP AR models in Figs. 2 and 4 lead only the second- and fourth-quadrant QP AR models PEFs, namely, r(p,q−1)QP2 (k, l), e(p−1,q)QP2 (k, l) and

˜ rQP(p,q−1)

2 (k, l), ˜e(p−1,q)QP

2 (k, l), respectively.

(7)

Fig. 6. Definition of the initial auxiliary horizontal and vertical for- ward/backward PEFs forSk,l(1,1).

However, the horizontal incrementation from right-to-left and the vertical incrementation from bottom-to-top call for the uti- lization of d(p,q)2 (k, l) and ˜d(p,q)2 (k, l) for the initiation of the new PEFs as shown in Figs. 3 and 5. Unlike the previous ones, all of these ASHP AR models direct only the first- and third-quadrant QP AR models PEFs, namely, rQP(p,q−1)1 (k, l), e(p−1,q)QP1 (k, l) and

˜

r(p,q−1)QP1 (k, l), ˜e(p−1,q)QP1 (k, l), respectively.

The initial first order 2-D lattice sections can be written in the well-known form:

v1(0,1)(k, l)

˜

v1(0,1)(k, l)



=

 1 −α(0,1)1

−α(0,1)1 1

  y(k, l) y(k, l − 1)

 (42)

h(1,0)1 (k, l)

˜h(1,0)1 (k, l)



=

 1 −β1(1,0)

−β1(1,0) 1

  y(k, l) y(k − 1, l)

 (43) and

d(1,1)1 (k, l) d˜(1,1)1 (k, l)



=

 1 −γ1(1,1)

−γ1(1,1) 1

  y(k, l) y(k − 1, l − 1)



(44) or

d(1,1)2 (k, l) d˜(1,1)2 (k, l)



=

 1 ] −γ2(1,1)

−γ2(1,1)] 1

 y(k, l − 1) y(k − 1, l)

 . (45) The first order forward/backward vertical and horizontal PEFs in (42) and (43), namely, v(0,1)1 (k, l), ˜v1(0,1)(k, l), h(1,0)1 (k, l) and ˜h(1,0)1 (k, l), along with one of the diagonal PEFs given in (44) and (45), namely, d(1,1)1 (k, l), ˜d(1,1)1 (k, l) or d(1,1)2 (k, l), d˜(1,1)2 (k, l) are then combined to increment the PSR vertically and horizontally by adding only one more past observation point at each successive stage. Fig. 7 shows the generation at the first- and second-quadrant QP AR 2-D filters. Table I gives the algorithm of the proposed 2-D lattice structıure when d1 is chosen for the incrementation.

III. VERTICALANDHORIZONTALBUILDINGBLOCKS OF THE

PROPOSED2-D LATTICEFILTERS

The 2-D z-transform of the prediction error propagation equations for the horizontal recursive incrementation can be rewritten in more compact forms which will lead the horizontal building blocks, namely,H(p+1,q)xj (z1, z2) for x = l or r and

Fig. 7. The basic recursive order incrementation types for the first order initial stage and the generation of the first- and second-quadrant QP AR 2-D filters.

TABLE I

ALGORITHM OF THEPROPOSED2-D LATTICESTRUCTURE BYUSINGd1

j = 1, 2, . . . , q. Indeed, from (7), (8) and (10) for ”left-to-right”

(l) incrementation and from (14), (17) and (19) for ”right-to-left”

(r) incrementation, we can write the error propagation of the auxiliary PEFs in the vector form as follows: For j= 1, . . . , p and x= l or r,

F(p+1,q)xj (z1, z2) F˜(p+1,q)xj (z1, z2)

⎦ =

F(p+1,q)xj−1 (z1, z2) F˜(p+1,q)xj−1 (z1, z2)

K(p+1,q)xj Δ(p+1,q)x K(p+1,q)xj Δ˜(p+1,q)x

R(p,q−1)1,xj−1 (z1, z2) R˜(p,q−1)1,xj−1 (z1, z2)

 (46)

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