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Geometric reasoning: burgers and cameras
Aug. 5th, 2017 05:37 pmI was arranging beefburgers of two different kinds on a baking tray and I chose to use a sort of nested arrow shape to make it easy to determine which burgers were of which kind even after rotating the tray. Our oven heats unevenly so it helps if the information encoded in the cunning arrangement of the food survives reorientations.
It occurred to me that there must be some mathematical theory that applies well to arranging beefburgers on baking trays. Relevant factors would include how many burgers wide and deep the tray is, or perhaps the theory generalizes to non-rectangular trays with different degrees of rotational symmetry that lend themselves to different patterns of arranging the burgers: for instance, three kinds of burgers on a pentagonal tray. Maybe there is some standard library of patterns of burgers comprising a toolkit that can be employed for every situation.
Of course, bakers of pies may avail themselves of the more straightforward approach of scoring the pastry lids differently according to kind of pie. I am not aware of an easy comparable approach with burgers, especially if they begin frozen and may be flipped once or twice during baking.
I once developed a sensor fusion system that, given inputs from radars and cameras, could bootstrap from some knowns to determine where in space the remaining targets and sensors were. The algorithm could make use of detected features for disambiguation; perhaps sometimes one cannot tell which kind of burger is which but can tell which burgers are of the same kind. Anyhow, the sensor fusion code would not be a great solution but it occurred to me that it could be used to verify arrangements of burgers: only a good arrangement should leave the system able to confidently determine the orientation of a camera that has the burgers in view.
Beyond my beefburger placement problem, which may already be well-known by some entirely different name, the sensor fusion system offered two interesting mathematical digressions. One is how the geometry of each situation affects the number of solutions for how a camera is located and oriented according to what it sees of targets whose location has now been determined by other sensors. The other is the heuristic of guessing which targets may correspond to which as viewed by different sensors: there may be intractably many combinations but, for example, a simple trick may be to guess that targets close together in one sensor's view are also close together in another's. More certainly, an object in between two others from one perspective will also be so from another.
It occurred to me that there must be some mathematical theory that applies well to arranging beefburgers on baking trays. Relevant factors would include how many burgers wide and deep the tray is, or perhaps the theory generalizes to non-rectangular trays with different degrees of rotational symmetry that lend themselves to different patterns of arranging the burgers: for instance, three kinds of burgers on a pentagonal tray. Maybe there is some standard library of patterns of burgers comprising a toolkit that can be employed for every situation.
Of course, bakers of pies may avail themselves of the more straightforward approach of scoring the pastry lids differently according to kind of pie. I am not aware of an easy comparable approach with burgers, especially if they begin frozen and may be flipped once or twice during baking.
I once developed a sensor fusion system that, given inputs from radars and cameras, could bootstrap from some knowns to determine where in space the remaining targets and sensors were. The algorithm could make use of detected features for disambiguation; perhaps sometimes one cannot tell which kind of burger is which but can tell which burgers are of the same kind. Anyhow, the sensor fusion code would not be a great solution but it occurred to me that it could be used to verify arrangements of burgers: only a good arrangement should leave the system able to confidently determine the orientation of a camera that has the burgers in view.
Beyond my beefburger placement problem, which may already be well-known by some entirely different name, the sensor fusion system offered two interesting mathematical digressions. One is how the geometry of each situation affects the number of solutions for how a camera is located and oriented according to what it sees of targets whose location has now been determined by other sensors. The other is the heuristic of guessing which targets may correspond to which as viewed by different sensors: there may be intractably many combinations but, for example, a simple trick may be to guess that targets close together in one sensor's view are also close together in another's. More certainly, an object in between two others from one perspective will also be so from another.
