II.D. Registration Procedure

The overall algorithm for the proposed method is shown in ^{Algorithm. 1}. In the beginning, we initialized an untrained ConvNet (f_θ) for a given pair of moving and fixed images, I_m and I_f. First, the untrained f_θ produces an initial deformation field, ϕ. Second, we deform the moving image with ϕ (i.e., I_m ◦ ϕ). Then, the registration loss is computed as:

where ℒ_sim represents the similarity measure between I_d and I_f, ℛ represents the value of the regularizer applied to the deformation field, and λ is a user-defined weighting parameter to control the effectiveness of ℛ. The loss is back-propagated to update the parameters in f_θ. The above procedure is repeated for a pre-specified number of iterations.

Since no information other than the given image pair is needed, the proposed method requires no prior training and is thus fully and truly unsupervised. The ConvNet is capable of learning an ”optimal” deformation from a single pair of images. In the next section, we discuss a series of experiments that were performed to study the effectiveness of the proposed method.