## 1 Introduction  
Medical B-mode ultrasound [1, 2] has long been imposed as the first examination modality for the diagnosis of many pediatric diseases. The other modalities (X-ray and magnetic resonance imaging scanners) are associated with various, often restrictive, inconveniences (irradiation, sedation, anesthesia of patient, and parent-child accessibility). The interest in B-mode ultrasound lies in associated operational advantages--it is painless, readily available at the bedside and not expensive--favoring regular use, and thus multiplication of diagnostic and monitoring procedures without risk. The significant use of B-mode ultrasound in medical imaging is due to the main characteristic of soft biological tissues. Indeed, the acoustic impedance contrasts between tissues are relatively small, making them ideal media for implementing linear methods of ultrasonic wave propagation. The ultrasonic wave (compression wave only) propagates in an almost homogeneous and isotropic medium, the acoustic characteristics of which are very similar to those of water. Drawbacks limit B-mode ultrasound being used as a standardized and comprehensive modality:  
* B-mode ultrasound displays the acoustic impedance contrast as a grayscale map that does not quantitatively match the acoustic impedance value, or any other acoustic parameter. * The frequencies of the transmitters used in medical practice are quite high in the ultrasonic frequency range, between 2 MHz and 20 MHz, sometimes as high as 50 MHz [3]. The higher the frequency, the stronger the absorption. The propagation can be limited in depth to display deep structures, structures obscured by bones, or in obese patients. * If the acoustic impedance changes abruptly, as when passing an interface between hypoechogenic soft tissues and hyperechogenic bone structures, the wave/medium interaction phenomena are no longer linear. The linear theories and approximations used become limited. Taking into account the physical phenomena of wave propagation, wave refraction or beam deflection, the unattenuated shear waves, or the inhomogeneity induced by high echogenicity, modifies the acquisition protocols or image reconstruction strategies requiring the resolution of nonlinear inverse problems. B-mode ultrasound is thus poorly adapted to these complex areas, raising resolution problems in the interdiaphyseal areas of joined bones (tibia/fibula, for example), leading to a lack of distinct information between the overlying tissue (_periosteum_) and the underlying tissue (the deep face of the cortical bone and the medullary canal). Ultrasound computed tomography (USCT) seems to be an alternative or complementary modality to B-mode ultrasound, including mathematical modeling more adapted to the physical phenomena induced. USCT provides access to significant acoustic parameters of biological tissues. The signal acquisition geometry is no longer linear or sectoral as in B-mode ultrasound, but circular in the orthogonal plane. USCT makes it possible to combine morphometric imaging related to an interface resolution problem and quantitative imaging related to the identification of physical parameters. In diffraction mode, USCT exploits the information resulting from the diffraction of an incident acoustic field by an organ, and more particularly the mutual information from several transducers or several elements of an ultrasonic array [4; 5; 6; 7]. Diffraction-mode USCT is effective for imaging the soft tissues of an organ, for example a breast, for which a fluid-like model is workable, and provides promising benefits [7]. However, USCT can be hampered by the acquisition of a large volume of recorded data, long processing times, or high computation costs, especially for iterative nonlinear approaches [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18].