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The infiltrative nature of lesions is a significant feature that implies a malignant breast lesion in ultrasound images. Characterizing the infiltrative nature of lesions with computationally inexpensive and highly efficacious features is crucial for the realization of a computer-aided diagnosis system. In this study, the infiltrative nature of lesions is regarded as an energy that produces irregular and considerably local variances in a 1-D signal. The local variances can be characterized by a few high octave energies (i.e., the channel energies close to low frequency bands) in 1-D discrete periodized wavelet transform (DPWT). To reduce computation cost, high octave decomposition is performed by a reversible round-off 1-D nonrecursive DPWT (1-D RRO-NRDPWT). A test dataset of breast sonograms with the lesion contour delineated by an experienced physician and three datasets of breast sonograms with the lesion contour delineated by a Java-based image processing program, ImageJ, are built for feature efficacy evaluation. Evaluation with the receiver operating characteristic (ROC) parameters, the area under ROC curve Az, accuracy Ac, sensitivity Se, specificity (Sp), and positive (ppv) and negative predictive values (npv), shows that the proposed feature has an individual performance of (Az, Ac, Se, Sp, ppv, npv) = (0.991, 0.951, 0.985, 0.933, 0.973, 0.992) and (0.934, 0.844, 0.933, 0.795, 0.714, 0.956) for manual and ImageJ-generated datasets, respectively. The performance differences in the three ImageJ-generated datasets derived by variant setting parameters are not significant. Experimental results also reveal that the proposed feature is suitable for combination with some morphometric parameters for performance improvement.