Sampling is time limited. The finite sampling time results in artifacts caused by what is known as "the Gibbs Phenomenon". The representation in the image domain near a discontinuity, for example, includes an "oscillatory overshoot" which is approximately 9% of the magnitude of the signal at the discontinuity. An artifact due to the Gibbs phenomen appears as "ringing" in the image. The ringing is often referred to as "Gibbs artifact". See the book entitled "The Fourier Transform and its Applictions" by R. Bracewell, published by McGraw-Hill Book Co. (1965) pp. 209 et seq.
If more sampling points are taken, the amplitude remains 9%, but the overshoot is compressed towards the edge of the discontinuity which reduces the artifact and accordingly improves the spatial resolution. While taking more sampling points improves spatial resoltion, it requires time which reduces throughput. Also, as is well known, the signal-to-noise ratio (SNR) is proportional to the inverse of the square root of N where N is the number of sampling points. Accordingly, to obtain enough sampling points to effectively reduce the Gibbs artifact not only reduces throughput, but also reduces the SNR of the image to the point where the improved resolution may be obscured by noise.
A multiplicative filter in the time domain can effectively reduce the overshoot and increase the SNR. However, such a filter reduces the resolution of the image. The reduction in resolution occurs because a time domain filter that reduces the overshoot also increases the transition width of the function. The spatial resolution; i.e. the smallest size detectable, is proportional to the transition width; so that, an increased transition width means the smallest size detectable increases.
In magnetic resonance imaging there are many instances, for example, for thoracic images when an image of 256.times.256 is not required. In fact many times an image providing less resolution, but improved SNR and acquisition time would be preferred. Until now the lower resolution images have not been used because of the Gibbs artifact, which obscures the image and especially an image with less resolution. Therefore, what is required is a reduction of the Gibbs artifact without adversely effecting the resolution, the SNR or the scan time.
Accordingly, it is an object of the present invention to effectively and substantially reduce the ringing artifact in the display image while substantially maintaining a given resolution and the signal-to-noise ratio of the final display image without increasing the scan time.