4.3 Non Adaptive Histogram Modification

Fig 4.9 gives another example of non adaptive histogram equalization. In the figure,

H_F (j) ,for j = 1,2,…J represents the fractional number of pixels in an input image whose amplitude is quantized to the jth reconstruction level.

Histogram Equalization seeks to produce an output image field G by point rescaling such that the normalized gray level histogram H_G (k) = 1/K, for k=1,2,..,K. In the example of Fig 4.9, the number of output levels is set to one half of the number of input levels.

Fig 4.9 Gray Level Histogram Equalization

The scaling algorithm is developed as follows. The average value of the histogram is computed. Then starting at the lowest gray level of the original the pixels in the quantization band are combined until the sum is closest to the average. All of these pixels are then rescaled to the new first reconstruction level at the midpoint of the enhanced image first quantization band. The process is repeated for higher value gray levels. If the number of reconstruction levels of the original image is large, it is possible to rescale the gray levels so that the enhanced image histogram is almost constant.

It should be noted that the number of reconstruction levels of the enhanced image must be less than the number of levels of the original image to provide proper gray scale redistribution if all the pixels in each quantization level are to be treated similarly. This process results in a somewhat large quantization error. It is possible to perform the gray scale histogram equalization process with the same number of gray levels for the original and enhanced images, and still achieve a constant histogram of the enhanced image, by randomly redistributing pixels from input to output quantization bands. The histogram modification process can be considered to be a monotonic point transformation g_k = T{f_j } for which the input amplitude variable f₁ <= f_j <= f _J is mapped into an output variable g₁ <= g_j <= g _J such that the output probability distribution P_R{g_k=b_k} follows some desired form for a given input probability distribution P_R {f_j = a_j } where aj and bj are reconstruction values for the jth and kth levels. Clearly, the input and output probability distributions must each sum to unity. Thus,

S P_R {f_j = a_j } = 1

S P_R{g_k = b_k } = 1

Furthermore, the cumulative distributions must equate for any input index j. That is, the probability that pixels in the input image have an amplitude less than or equal to a_j must be equal to the probability that pixels in the output image have amplitude less than or equal to b_k where b_k=T{a_j} because the transformation is monotonic. Hence,

S P_R { g_n= b_n } = S P_R {f_m = a_m }

Histogram equalization usually performs best on images with detail hidden in dark regions. Good quality originals are often degraded by histogram equalization. Some common transformation functions used in nonadaptive histogram modification are shown in Fig 4.10.

Fig 4.10 Histogram Modification Transfer functions

Prev Section Contents Next Section

Back To Home Page