<p>We propose a new image compression method and algorithm using a closed-form shrinkage function based on Cauchy distribution and a logarithmic transform in wavelet domain. We use a logarithmic transform in image transform stage to solve the arithmetic precision problem and to improve the image quality after decompression. To improve the noise robustness of ultrasonic medical image compression, we also use a closed-form shrinkage function based on Cauchy distribution for the thresholding of wavelet coefficients in image transform stage. Next, we evaluate the image compression performance of the proposed method compared with the classical image compression methods based on Discrete Wavelet Transform (DWT) such as Embedded Zerotree Wavelet (EZW), Set Partitioning In Hierarchical Trees (SPIHT), Spatial Orientation Tree Wavelet Compression (STW), Wavelet Difference Reduction (WDR) and Discrete Wavelet Transform-Vector Quantization (DWT-VQ). Results show that CR and PSNR of the proposed method are highest, MSE and BPP are smallest, and SSIM and CC are closer to one than other existing methods for noisy reference image at different speckle levels. Moreover, the rate-distortion performance is improved. CR and ENL of the proposed method are highest and BPP, EN and SD are smallest of other existing methods for noisy real ultrasonic medical image.</p>