Place a cheesecloth over a sieve. Put the sieve over a bowl, pour the yogurt in the prepared sieve and refrigerate for about 3 to 4 hours to drain.
Place the blackberries and 1/4 cup of the honey in a saucepan over medium-low heat. Cook, stirring occasionally for about 7 minutes. Add vanilla extract, remove the pan from the heat, and crush the berries gently keeping some of them intact. Set aside and let the sauce chill at room temperature.
Put the strained yogurt (about 2 cups) in the bowl of the blender, add 1/4 cup honey, vanilla and almond extracts if used, and blend until smooth. Pour the yogurt mixture into a covered storage container and place in the freezer. After half an hour stir again to break up the ice crystals. Stir every 30 minutes for two or three times more or until it reaches the desired consistency.
Meanwhile strain the blackberry mixture through a fine-mesh strainer set over a medium bowl. Keep the juice for another use. Set the strained berries aside.
When the yogurt is halfway through the freezing process fold in the strained berries with a spatula creating the marble effect, being careful not to overmix.
Freeze for about 2 hours before serving or let it overnight. Allow the frozen yogurt to sit at room temperature for 15 minutes before serving if frozen solid.
Serve in bowl or waffle cones and sprinkle with fresh blackberries.