On minimax rates of convergence in image models under sequential design
Consider a binary image (``\(0\)'' and ``\(1\)'') in \(C=[0,1]\times[0,1]\) such that the set of ``\(1\)'' looks like \[ G=\{(x,y) |0\leq x\leq 1,\;0\leq y\leq g(x)\}, \] where \(g:[0,1]\rightarrow[0,1]\) is the edge function. An estimator \(g_{n}^{\ast}\) of \(g\) is defined based on a set \(\Lambda _{n}=\{X_{1},...,X_{n}\}\) of \(C\) (the design) and observations \(Y_{1},...,Y_{n}\) defined by \(Y_{i}=1_{G}(X_{i})\) with probability \(p>1/2\) and \(Y_{i}=1-1_{G}(X_{i})\). The design is sequential, i.e. each consecutive design point starting from \(X_{2}\) is a function of all the previous data: \(X_{2}=X_{2}(X_{1},Y_{1}),...,X_{i+1}=X_{i+1}(X_{1},Y_{1},...,X_{i},Y_{i})\), \(1\leq i<n\). It is proved that in the case of the \(L_{s}\)-norm loss (\(1\leq s<\infty\)) the estimator \(g_{n}^{\ast}\) is an approximation of \(g\;\)whose accuracy is \(O(1/n)\) as \(n\rightarrow\infty\). The case of high-dimensional images is studied. It is a nice and recommendable paper for all statisticians interested in the estimation of edges of images. The proofs are easy to understand.
scientific article; zbMATH DE number 1395195