▁elli ps es ▁are ▁orient ed ▁along ▁principal ▁axes , ▁the ▁parameter ▁space ▁is ▁four - dimensional . ▁ ▁( x , y ) ▁defines ▁the ▁center ▁of ▁the ▁el lipse , ▁and ▁a ▁and ▁b ▁denote ▁the ▁two ▁rad ii . ▁ ▁Allow ing ▁the ▁center ▁to ▁be ▁anywhere ▁in ▁the ▁image , ▁adds ▁the ▁constraint ▁ 0 < x < 8 0 0 ▁and ▁ 0 < y < 6 0 0 . ▁If ▁the ▁rad ii ▁are ▁given ▁the ▁same ▁values ▁as ▁constraints , ▁what ▁is ▁left ▁is ▁a ▁sp ars ely ▁filled ▁accum ulator ▁array ▁of ▁more ▁than ▁ 2 3 0 ▁billion ▁values . ▁ ▁A ▁program ▁thus ▁conce ived ▁is ▁unlikely ▁to ▁be ▁allowed ▁to ▁allocate ▁sufficient ▁memory . ▁ ▁This ▁doesn ' t ▁mean ▁that ▁the ▁problem ▁can ' t ▁be ▁solved , ▁but ▁only ▁that ▁new ▁ways ▁to ▁const rain ▁the ▁size ▁of ▁the ▁accum ulator ▁array ▁are ▁to ▁be ▁found , ▁which ▁makes ▁it ▁feas ible . ▁ ▁For ▁instance : ▁▁ ▁If ▁it ▁is ▁reasonable ▁to ▁assume ▁that ▁the ▁elli ps es ▁are ▁each ▁contained ▁entirely ▁within ▁the ▁image , ▁the ▁range ▁of ▁the ▁rad ii ▁can ▁be ▁reduced . ▁▁▁ ▁The ▁largest ▁the ▁rad ii ▁can ▁be ▁is ▁if ▁the ▁center ▁of ▁the ▁el lipse ▁is ▁in ▁the ▁center ▁of ▁the ▁image , ▁allowing ▁the ▁edges ▁of ▁the ▁el lipse ▁to ▁stretch ▁to ▁the ▁edges . ▁ ▁In ▁this ▁extreme ▁case , ▁the ▁rad ii ▁can ▁only ▁each ▁be ▁half ▁the ▁magnitude