Combine 3/4 cup water, cranberries, and 1/2 cup sugar in a small saucepan; stir well.
Bring cranberry mixture to a boil over medium heat; cook 8 minutes or until cranberries pop, stirring frequently.
Remove from heat, and let cool.
Position knife blade in food processor bowl.
Add cranberry mixture; process until smooth.
Strain puree through a sieve; discard pulp.
Combine cranberry puree, remaining 3 3/4 cups water, remaining 3/4 cup sugar, pineapple juice, and orange juice in a large bowl; stir well.
Pour cranberry mixture into freezer can of a 1-gallon hand-turned or electric freezer.
Freeze according to manufacturer's instructions.
Pack freezer with additional ice and rock salt; let stand 1 hour, if desired.
Scoop mixture into individual dessert bowls.
Garnish with mint sprigs, if desired.