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To replace compromised biometric templates, cancelable biometrics has recently been introduced. The concept is to transform a biometric signal or feature into a new one for enrollment and matching. For making cancelable fingerprint templates, previous approaches used either the relative position of a minutia to a core point or the absolute position of a minutia in a given fingerprint image. Thus, a query fingerprint is required to be accurately aligned to the enrolled fingerprint in order to obtain identically transformed minutiae. In this paper, we propose a new method for making cancelable fingerprint templates that do not require alignment. For each minutia, a rotation and translation invariant value is computed from the orientation information of neighboring local regions around the minutia. The invariant value is used as the input to two changing functions that output two values for the translational and rotational movements of the original minutia, respectively, in the cancelable template. When a template is compromised, it is replaced by a new one generated by different changing functions. Our approach preserves the original geometric relationships (translation and rotation) between the enrolled and query templates after they are transformed. Therefore, the transformed templates can be used to verify a person without requiring alignment of the input fingerprint images. In our experiments, we evaluated the proposed method in terms of two criteria: performance and changeability. When evaluating the performance, we examined how verification accuracy varied as the transformed templates were used for matching. When evaluating the changeability, we measured the dissimilarities between the original and transformed templates, and between two differently transformed templates, which were obtained from the same original fingerprint. The experimental results show that the two criteria mutually affect each other and can be controlled by varying the control parameters of - - the changing functions.