In the Kalman filter, an observation model h(x) is used to transform the predicted state variables x from the state space to the measurement space: z = h(x)
If the state space is identical to measurement space, then h(x)=x and the predicted state can simply be represented as: z = x
The state covariance P is transformed using H=dh/dx into the measurement space as HPH^T, where H is the partial derivative of the observation model h(x) with respect to the state variables. The problem is that H is usually nonlinear and tricky to calculate.
I am wondering whether the Kalman filter process would still work if the observation model and all subsequent steps are "flipped" so that the transformation goes from measurement space to state space, instead of state space to measurement space, through the flipped observation model h* where: x = h*(z).
In the flipped process, the innovation would be computed in state space after the measurement matrix is transformed into state space, and the Kalman gain would be computed in state space after the measurement covariance is transformed into the state space.
A similar question was asked about whether an "inverse observation model" would work, which implied that the observation model h needs to be an invertible matrix, but it doesn't sound like this was intended. To avoid this confusion, I'm calling it the "flipped" observation model. https://stats.stackexchange.com/questions/71886/inverse-observation-model-and-kalman-filtering/71895#71895
To take it another step further, regardless of the complexity of the transformation between the state variables and the actual measurement variables, can't the measurement space be defined to be identical to the state space (we'll call this the faux measurement space) for the sole purpose of simplifying calculation of h and H, and just make sure to transform the actual measurement variables into the faux measurement space prior to the update step of the Kalman filter?
Thanks in advance.
You propose 2 questions summarized as
Question 1:
Not really. If you flip the equations in the KF, you would essentially be estimating the measurement in contrast to estimating the states (althouhg it is somewhat unclear by what flipping "all" the equations means). You could rewrite you KF defining your state to be the measurements obtained by the sensors, but I'm not sure the benefit of doing this because the states are what you are interested in estimating.
Question 2:
Yes, but the implementation is the same. This is exactly what you are doing in a KF. If you notice in the KF, you multiple the covariance matrix by the Jacobian of the observation model. This is transforming the covariance from the state space to the observation space. If you do this transformation beforehand, you will still need to transform your covariance matrix, and you will just end up with the exact same equations.