<< >> Up Title Contents Index

Statistics

In image processing it is quite common to use simple statistical descriptions of images and sub-images. The notion of a statistic is intimately connected to the concept of a probability distribution, generally the distribution of signal amplitudes. For a given region--which could conceivably be an entire image--we can define the probability distribution function of the brightnesses in that region and the probability density function of the brightnesses in that region. We will assume in the discussion that follows that we are dealing with a digitized image a[m,n].

Probability distribution function of the brightnesses

The probability distribution function, P(a), is the probability that a brightness chosen from the region is less than or equal to a given brightness value a. As a increases from - to +, P(a) increases from 0 to 1. P(a) is monotonic, non-decreasing in a and thus dP/da >= 0.

Probability density function of the brightnesses

The probability that a brightness in a region falls between a and a+a, given the probability distribution function P(a), can be expressed as p(a)a where p(a) is the probability density function:


T.1 Rectangle





picture 1

picture 2




T.2 Pyramid





picture 1

picture 2




T.3 Pill Box





picture 1

picture 2




T.4 Cone





picture 1

picture 2





T.5 Airy PSF





picture 1

picture 2




T.6 Gaussian





picture 1

picture 2




T.7 Peak





picture 1

picture 2




T.8 Exponential

Decay





picture 1

picture 2

Table 4: 2D Images and their Fourier Transforms

Because of the monotonic, non-decreasing character of P(a) we have that:

For an image with quantized (integer) brightness amplitudes, the interpretation of a is the width of a brightness interval. We assume constant width intervals. The brightness probability density function is frequently estimated by counting the number of times that each brightness occurs in the region to generate a histogram, h[a]. The histogram can then be normalized so that the total area under the histogram is 1 (eq. ). Said another way, the p[a] for a region is the normalized count of the number of pixels, , in a region that have quantized brightness a:

The brightness probability distribution function for the image shown in Figure 4a is shown in Figure 6a. The (unnormalized) brightness histogram of Figure 4a which is proportional to the estimated brightness probability density function is shown in Figure 6b. The height in this histogram corresponds to the number of pixels with a given brightness.

(a) (b)

Figure 6: (a) Brightness distribution function of Figure 4a with minimum, median, and maximum indicated. See text for explanation. (b) Brightness histogram of Figure 4a.

Both the distribution function and the histogram as measured from a region are a statistical description of that region. It must be emphasized that both P[a] and p[a] should be viewed as estimates of true distributions when they are computed from a specific region. That is, we view an image and a specific region as one realization of the various random processes involved in the formation of that image and that region. In the same context, the statistics defined below must be viewed as estimates of the underlying parameters.

Average

The average brightness of a region is defined as the sample mean of the pixel brightnesses within that region. The average, ma, of the brightnesses over the pixels within a region () is given by:

Alternatively, we can use a formulation based upon the (unnormalized) brightness histogram, h(a) = *p(a), with discrete brightness values a. This gives:

The average brightness, ma, is an estimate of the mean brightness, ua, of the underlying brightness probability distribution.

Standard deviation

The unbiased estimate of the standard deviation, sa, of the brightnesses within a region () with pixels is called the sample standard deviation and is given by:

Using the histogram formulation gives:

The standard deviation, sa, is an estimate of a of the underlying brightness probability distribution.

Coefficient-of-variation

The dimensionless coefficient-of-variation, CV, is defined as:

Percentiles

The percentile, p%, of an unquantized brightness distribution is defined as that value of the brightness a such that:

P(a) = p%

or equivalently

Three special cases are frequently used in digital image processing.

* 0% the minimum value in the region

* 50% the median value in the region

* 100% the maximum value in the region

All three of these values can be determined from Figure 6a.

Mode

The mode of the distribution is the most frequent brightness value. There is no guarantee that a mode exists or that it is unique.

SignaltoNoise ratio

The signal-to-noise ratio, SNR, can have several definitions. The noise is characterized by its standard deviation, sn. The characterization of the signal can differ. If the signal is known to lie between two boundaries, amin <= a <= amax, then the SNR is defined as:

Bounded signal -

If the signal is not bounded but has a statistical distribution then two other definitions are known:

Stochastic signal - S & N inter-dependent

S & N independent

where ma and sa are defined above.

The various statistics are given in Table 5 for the image and the region shown in Figure 7.

Figure 7 Table 5 Region is the interior of the circle. Statistics from Figure 7

A SNR calculation for the entire image based on eq. is not directly available. The variations in the image brightnesses that lead to the large value of s (=49.5) are not, in general, due to noise but to the variation in local information. With the help of the region there is a way to estimate the SNR. We can use the s (=4.0) and the dynamic range, amax - amin, for the image (=241-56) to calculate a global SNR (=33.3 dB). The underlying assumptions are that 1) the signal is approximately constant in that region and the variation in the region is therefore due to noise, and, 2) that the noise is the same over the entire image with a standard deviation given by sn = s.

<< >> Up Title Contents Index