feat: Add LeastMeanSquareVelocityFilter for advanced tracking velocity estimation

- Introduced a new `LeastMeanSquareVelocityFilter` class to enhance tracking velocity estimation using historical detections.
- Implemented methods for updating measurements and predicting future states, laying the groundwork for advanced tracking capabilities.
- Improved import organization and added necessary dependencies for the new filter functionality.
- Updated class documentation to reflect the new filter's purpose and methods.

app/tracking/__init__.py

@@ -19,8 +19,8 @@ from beartype import beartype
 from beartype.typing import Mapping, Sequence
 from jax import Array
 from jaxtyping import Array, Float, Int, jaxtyped
-from pyrsistent import PVector
+from pyrsistent import PVector, v
-
+from itertools import chain
 from app.camera import Detection
 
 
@@ -80,7 +80,7 @@ class GenericVelocityFilter(Protocol):
 
 class LastDifferenceVelocityFilter(GenericVelocityFilter):
     """
-    a velocity filter that uses the last difference of keypoints
+    a naive velocity filter that uses the last difference of keypoints
     """
 
     _last_timestamp: datetime
@@ -128,6 +128,54 @@ class LastDifferenceVelocityFilter(GenericVelocityFilter):
         self._last_velocity = None
 
 
+class LeastMeanSquareVelocityFilter(GenericVelocityFilter):
+    _get_historical_detections: Callable[[], Sequence[Detection]]
+    """
+    get the current historical detections, assuming the detections are sorted by
+    timestamp incrementally (i.e. index 0 is the oldest detection, index -1 is
+    the newest detection)
+    """
+    _velocity: Optional[Float[Array, "J 3"]] = None
+
+    def __init__(self, get_historical_detections: Callable[[], Sequence[Detection]]):
+        self._get_historical_detections = get_historical_detections
+
+    def predict(self, timestamp: datetime) -> TrackingPrediction:
+        raise NotImplementedError
+
+    @jaxtyped(typechecker=beartype)
+    def _update(
+        self,
+        keypoints: Float[Array, "N J 3"],
+        timestamps: Float[Array, "N"],
+    ) -> None:
+        """
+        update measurements with least mean square method
+        """
+        raise NotImplementedError
+
+    def update(self, keypoints: Float[Array, "J 3"], timestamp: datetime) -> None:
+        historical_detections = self._get_historical_detections()
+        t_0 = min(d.timestamp for d in historical_detections)
+        detections = jnp.array(
+            chain((d.keypoints for d in historical_detections), (keypoints,))
+        )
+        # timestamps relative to t_0 (the oldest detection timestamp)
+        timestamps = jnp.array(
+            chain(
+                ((d.timestamp - t_0).total_seconds() for d in historical_detections),
+                ((timestamp - t_0).total_seconds(),),
+            )
+        )
+        raise NotImplementedError
+
+    def get(self) -> TrackingPrediction:
+        raise NotImplementedError
+
+    def reset(self, keypoints: Float[Array, "J 3"], timestamp: datetime) -> None:
+        raise NotImplementedError
+
+
 @jaxtyped(typechecker=beartype)
 @dataclass(frozen=True)
 class Tracking: