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An adaptive image-coding algorithm for compression of medical ultrasound (US) images in the wavelet domain is presented. First, it is shown that the histograms of wavelet coefficients of the subbands in the US images are heavy-tailed and can be better modelled by using the generalised Student's t-distribution. Then, by exploiting these statistics, an adaptive image coder named JTQVS-WV is designed, which unifies the two approaches to image-adaptive coding: rate-distortion (R-D) optimised quantiser selection and R-D optimal thresholding, and is based on the varying-slope quantisation strategy. The use of varying-slope quantisation strategy (instead of fixed R-D slope) allows coding of the wavelet coefficients across various scales according to their importance for the quality of reconstructed image. The experimental results show that the varying-slope quantisation strategy leads to a significant improvement in the compression performance of the JTQVS-WV over the best state-of-the-art image coder, SPIHT, JPEG2000 and the fixed-slope variant of JTQVS-WV named JTQ-WV. For example, the coding of US images at 0.5 bpp yields a peak signal-to-noise ratio gain of >0.6, 3.86 and 0.3 dB over the benchmark, SPIHT, JPEG2000 and JTQ-WV, respectively.