v2010.10.26 - Convex Optimization

420 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.4.2.3.6 Proof (sketch). Only the sensor location x 1 is unknown.The objective function together with the equality constraints make a linearsystem of equations in Gram matrix variable GtrG = ‖x 1 ‖ 2 + ‖ˇx 2 ‖ 2 + ‖ˇx 3 ‖ 2 + ‖ˇx 4 ‖ 2tr(GΦ i1 = ďi1 , i = 2, 3, 4tr ( ) (941)Ge i e T i = ‖ˇxi ‖ 2 , i = 2, 3, 4tr(G(e i e T j + e j e T i )/2) = ˇx T i ˇx j , 2 ≤ i < j = 3, 4which is invertible:⎡⎤svec(I) Tsvec(Φ 21 ) Tsvec(Φ 31 ) Tsvec(Φ 41 ) Tsvec(e 2 e T 2 ) Tsvec G =svec(e 3 e T 3 ) Tsvec(e 4 e T 4 ) Tsvec ( (e⎢ 2 e T 3 + e 3 e T 2 )/2 ) T⎣ svec ( (e 2 e T 4 + e 4 e T 2 )/2 ) T ⎥svec ( (e 3 e T 4 + e 4 e T 3 )/2 ) ⎦T−1⎡‖x 1 ‖ 2 + ‖ˇx 2 ‖ 2 + ‖ˇx 3 ‖ 2 + ‖ˇx 4 ‖ 2 ⎤ď 21ď 31ď 41‖ˇx 2 ‖ 2‖ˇx 3 ‖ 2‖ˇx 4 ‖ 2⎢ ˇx T 2ˇx 3⎥⎣ ˇx T 2ˇx 4⎦ˇx T 3ˇx 4(942)That line in the ambient space S 4 of G , claimed on page 419, is traced by‖x 1 ‖ 2 ∈ R on the right-hand side, as it turns out. One must show this line tobe tangential (2.1.7.1.2) to S 4 + in order to prove uniqueness. Tangency ispossible for affine dimension 1 or 2 while its occurrence depends completelyon the known measurement data.But as soon as significant noise is introduced or whenever distance data isincomplete, such problems can remain convex although the set of all optimalsolutions generally becomes a convex set bigger than a single point (and stillcontaining the noiseless solution).5.4.2.3.7 Definition. Isometric reconstruction. (confer5.5.3)Isometric reconstruction from an EDM means building a list X correct towithin a rotation, reflection, and translation; in other terms, reconstructionof relative position, correct to within an isometry, correct to within a rigidtransformation.△