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# Simultaneous stitching

Suppose the measurements of two neighboring subapertures are triplets W1 = {w = (uy,1, Vj,1, фуд), j = 1,2,..., N1} and W2 = {W2 = (uy,2, v,2, фу,2), j = 1, 2, ..., N2}, respectively, where ф is the measured height on pixel (u, v). N1 and N2 are the total number of measuring points in subapertures 1 and 2. The two data sets are described in different coordinate systems since each subaperture is individually aligned. A global coordinate system is built as a reference frame, in which the two data sets can be transformed and described according to Eq. (24)

In this global frame, the subapertures should be consistent in the overlapping region, i.e., the height in subaperture 1 should ideally equal that in subaperture 2 at the same point. While measurement error always exists practically, the overlapping region should be consistent in the LS sense. Denote the data in the overlapping region by = {w^Km^^^^), jo = 1,2,..., No} and W2o = {w2o= (wjo,2, vjo,2, 9jo,2)}, respectively, where No is the number of overlapping point pairs and the subscript “o” indicates the overlap. Based on the LS principle, the mathematical model of two-subaperture stitching is formulated as follows:

No

min F = ^ (zjo,1 - Zj0,2 )2

jo=1

Let m = [ah bb cb pi, tb 01, аг, b2, ег, pi, h, 02]T. b = (ф,ъ,2 - Ф^д) is a column

vector of No elements. The column j of coefficient matrix A is defined as

Therefore, Eq. (28) is rewritten as a system of linear equations Am = b for LS solutions. The stitching problem can be solved by solving the linear equations to get the optimal misalignment coefficients, which are then substituted into Eq. (27) to transform the subaperture measurements into the global frame.

Generally, more than two subapertures are required in a subaperture stitching test. Based on the above model, the sequential stitching mode stitches two neighboring subapertures sequentially in each step. It obviously suffers the problem of error stack-up, and the stitching order affects the accuracy. In contrast, we prefer the simultaneous stitching mode where all subaperture overlapping regions are considered and made consistent at the same time by correcting the misalignment.

Suppose there are a total of s subapertures. The measurements of subapertures i and k are triplets W = {ж=(мл^,ф,д), j = 1,2,... ,N} and Wk = {w*=(wj,*,Vj,*,9j,*),

j = 1,2,... ,Nk}. The data in the overlapping region between these two subapertures are Wto = {wio=(lkUjo,t, 'j, lktyjo,i), jo = 1,2, ..,lkNo] and Wko = [wko=(lkUjo,2, lkVj0,2,ф/ъ,2)}. lkNo is the number of overlapping point pairs. The left superscript “ik" indicates the overlap between subapertures i and k. Based on the LS principle, the simultaneous stitching is formulated as the following LS model

s— 1 s lkNo

min F = (ikZj0,i — ikZj0,k)2

i=1 k=i+1 jo=1

Let m = [«1, b1, C1, p1, h, 01, .., as, bs, cs, ds, ps, ts, 0S]T. b = (lk9jo,k — lktyjo,t) is a

column vector of No elements where

is the total number of overlapping point pairs. The matrix A is a No x (6s) sparse matrix. Therefore, Eq. (31) is still rewritten as a system of linear equations Am = b for LS solutions. By solving the linear equations we get the optimal misalignment coefficients, which are then used to transform the subaperture measurements into the global frame.

The basic problem for stitching is hence solving a system of linear equations. Mature algorithms such as SVD and QR decomposition64 are available and succeed in extensive applications in engineering computations.

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