Many current methods for multiple sclerosis (MS) lesion segmentation require radiologist seed points as input, but do not necessarily allow the expert to work in an intuitive or efficient way. Ironically, most methods also assume that the points are placed optimally. This paper examines how seed points can be processed with intuitive heuristics, which provide improved segmentation accuracy while facilitating quick and natural point placement. Using a large set of MRIs from an MS clinical trial, two radiologists are asked to seed the lesions while unaware that the points would be fed into a classifier, based on Parzen windows, that automatically delineates each marked lesion. To evaluate the impact of the new heuristics, an interactive region-growing method is used to provide ground truth and the Dice coefficient (DC) and Spearman’s rank correlation are used as the primary measures of agreement. A stratified analysis is performed to determine the effect on scans with low-, medium-, and high lesion loads. Compared to the unenhanced classifier, the heuristics dramatically improve the DC (+32.91 pt.) and correlation (+0.50) for the scans with low lesion loads, and also improve the DC (+14.55 pt.) and correlation (+0.15) for the scans with medium lesion loads, while having aminimal effect for the scans with high lesion loads, which are already segmented accurately by Parzen windows.With the heuristics, the DC is close to 80% and the correlation is above 0.9 for all three load categories.

The classical Cauchy-Davenport theorem implies the lower bound n+1 for the number of distinct subsums that can be formed from a sequence of n elements of the cyclic group Z p (when p is prime and n < p). We generalize this theorem to a conjecture for the minimum number of distinct subsums that can be formed from elements of a multiset in Z m p ; the conjecture is expected to be valid for multisets that are not "wasteful" by having too many elements in nontrivial subgroups. We prove this conjecture in Z 2 p for multisets of size p + k, when k is not too large in terms of p.Lemma 1.1 (Cauchy-Davenport Theorem). Let A and B be subsets of Z p , and define A + B to be the set of all elements of the formThe lower bound is easily seen to be best possible by taking A and B to be intervals, for example. It is also easy to see that the lower bound of #A + #B − 1 does not hold for general abelian groups G (take A and B to be the same nontrivial subgroup of G). There is, however, a wellknown generalization obtained by Kneser in 1953 [4], which we state in a slightly simplified form that will be quite useful for our purposes (see [7, Theorem 4.1] for an elementary proof): Lemma 1.2 (Kneser's Theorem). Let A and B be subsets of a finite abelian group G, and let m be the largest cardinality of a proper subgroup of G. Then #(A + B) ≥ min{#G, #A + #B − m}.Given a sequence A = (a 1 , . . . , a k ) of (not necessarily distinct) elements of an abelian group G, a related result involves its sumset ΣA, which is the set of all sums of any number of elements chosen from A (not to be confused with A + A, which it contains but usually properly): ΣA = j∈J a j : J ⊆ {1, . . . , k} .(Note that we allow J to be empty, so that the group's identity element is always an element of ΣA.) When G = Z p , one can prove the following result by writing ΣA = {0, a 1 } + · · · + {0, a k } and applying the Cauchy-Davenport theorem inductively: Lemma 1.3. Let A = (a 1 , . . . , a k ) be a sequence of nonzero elements of Z p . Then #ΣA ≥ min{p, k + 1}.

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This article is an expanded version of the talk given by the rst author at the 25th Anniversary Conference of the Centre de R echerches Math ematiques.