1. An ultrasonic image processing apparatus for processing a plurality of signals obtained by scanning an object to be inspected with an ultrasonic beam and respectively representing pixel values of a plurality of pixels within an ultrasonic image, said apparatus comprising: a morphology processing unit for processing the plurality of signals to reduce speckles appearing in the ultrasonic image; and a digital scan converter for converting a scan format of the plurality of signals processed by said morphology processing unit; wherein said morphology processing unit performs first signal processing on the plurality of signals while moving a mask area determined based on a structure element in morphology processing, said first signal processing including multiplying pixel values of a plurality of pixels within the mask area by a first group of weighting factors set depending on positions within the mask area, respectively, and converting a pixel value at a center of the mask area into a maximum value of the pixel values multiplied by the first group of weighting factors, and performs second signal processing on the plurality of signals subjected to the first signal processing while moving a mask area determined based on a structure element in morphology processing, said second signal processing including multiplying pixel values of a plurality of pixels within the mask area by a second group of weighting factors set depending on positions within the mask area, respectively, and converting a pixel value at a center of the mask area into a minimum value of the pixel values multiplied by the second group of weighting factors; and wherein said first and second groups of weighting factors are set such that a value obtained by multiplying corresponding factors within the mask area in said first and second groups together becomes 1.