Figures of merit for image quality are derived on the basis of the performance of mathematical observers on specific detection and estimation tasks. The tasks include detection of a known signal superimposed on a known background, detection of a known signal on a random background, estimation of Fourier coefficients of the object, and estimation of the integral of the object over a specified region of interest. The chosen observer for the detection tasks is the ideal linear discriminant, which we call the Hotelling observer. The figures of merit are based on the Fisher information matrix relevant to estimation of the Fourier coefficients and the closely related Fourier crosstalk matrix introduced earlier by Barrett and Gifford [Phys. Med. Biol. 39, 451 (1994)]. A finite submatrix of the infinite Fisher information matrix is used to set Cramer-Rao lower bounds on the variances of the estimates of the first N Fourier coefficients. The figures of merit for detection tasks are shown to be closely related to the concepts of noise-equivalent quanta (NEQ) and generalized NEQ, originally derived for linear, shift-invariant imaging systems and stationary noise. Application of these results to the design of imaging systems is discussed.
We make the following statement precise under fairly weak conditions: in an experiment, if we summarize n statistically independent observations (xi,. .. ,x.) in n m < n real numbers (yi,. . .,ym), where y = z Jfj(xi) and the fj are given funci=1 tions, and if we assume we have lost no information by the summary, then the family of probabilities associated with the experiment must be an exponential family.Let (X,21, IPt:t & T}) be fixed, where T is a set, 2 is a sigma-algebra of subsets of X, and {P1} is a family of probabilities, which satisfy P,(A) = 0 if and only if P,'(A) = 0 for (t,t',A) C T X T X 21. We say that {Pi} is an exponential family if for a fixed to & T there are p + 1 real-valued functions c; on T and p real-valued Borel functions pj on X, [spj-1(B) C 21 when B c R is a Borel set], so that Pt(A) = co(t) exp ( cj(t)
For independent X and Y in the inequality P(X ≤ Y + μ), we give sharp lower bounds for unimodal distributions having finite variance, and sharp upper bounds assuming symmetric densities bounded by a finite constant. The lower bounds depend on a result of Dubins about extreme points and the upper bounds depend on a symmetric rearrangement theorem of F. Riesz. The inequality was motivated by medical imaging: find bounds on the area under the Receiver Operating Characteristic curve (ROC).
AMS 2000 subject classificationsPrimary 62G32; 60E15; secondary 92C55
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