This paper proposes a geometric approach for comparing tensor-valued images (tensor fields) that is based on the idea of matching intrinsically low-dimensional shapes embedded in a higher-dimensional ambient space. More specifically, instead of regarding the tensor fields as tensor-valued functions defined on a given (image) domain, we consider their image graphs. These tensorial image graphs can naturally be regarded as submanifolds (shapes) in an ambient space that is the cartesian product of their domain and the space of tensors. With this viewpoint, comparisons between tensor fields can naturally be formulated as comparisons between their corresponding shapes, and an intrinsic comparison measure can be developed based on matching these low-dimensional shapes. The proposed approach offers great conceptual clarity and transparency, and thorny issues such as parametric invariance and symmetric registration can be handled effortlessly in this novel framework. Furthermore, we show that the resulting variational framework can be satisfactorily optimized using a gradient descent-based method, and the computed similarities can be used as the affinity measures in a supervised learning framework to yield competitive results on challenging classification problems. In particular, experimental results have shown that the proposed approach is capable of producing impressive results on several classification problems using the OASIS image database, which include classifying the MRI brain images of Alzheimer's disease patients.