1. A teat detection method comprising: obtaining, by a processor, a three-dimensional (3D) image of a rearview of a dairy livestock in a stall, wherein: the dairy livestock is oriented in the 3D image with respect to: an x-axis corresponding with a horizontal dimension of the 3D image, a y-axis corresponding with a vertical dimension of the 3D image, and a z-axis corresponding with a depth dimension into the 3D image; and each pixel of the 3D image is associated with a depth value along the z-axis; identifying, by the processor, one or more regions within the 3D image comprising depth values greater than a depth value threshold; applying, by the processor, a thigh gap detection rule set to the one or more regions to identify a thigh gap region among the one or more regions, wherein the thigh gap region comprises an area between hind legs of the dairy livestock; demarcating, by the processor, an access region within the thigh gap region, wherein the access region is defined by: a first vertical edge, a second vertical edge, a first upper edge spanning between the first vertical edge and the second vertical edge, and a first lower edge spanning between the first vertical edge and the second vertical edge; demarcating, by the processor, a teat detection region, wherein the teat detection region is defined by: a third vertical edge extending vertically from the first vertical edge of the access region, a fourth vertical edge extending vertically from the second vertical edge of the access region, a second upper edge spanning between the third vertical edge and the fourth vertical edge, and a second lower edge spanning between the third vertical edge and the fourth vertical edge; partitioning, by the processor, the 3D image within the teat detection region along the z-axis to generate a plurality of image depth planes; examining, by the processor, each of the plurality of image depth planes, wherein examining each of the image depth planes comprises: identifying one or more teat candidates within the image depth plane; and applying a teat detection rule set to the one or more teat candidates to identify one or more teats; and determining, by the processor, position information for the one or more teats.