Point operations perform a modification of the pixel values without changing the size, geometry, or local structure of the image.
Homogeneous point operation
The individual new pixel value depend only on the previous value at the same position.
Non-homogeneous point operation
The individual new pixel value depend on the previous value at the same position and including also depend on any other pixel values.
Modifying Image Intensity
Contrast and Brightness
The individual pixel value has a constant contrast value. So if increase contrast by 50%, the new pixel value is multiplied by 1.5. In contrast if decrease contrast by 20%, the new pixel value is multiplied by 0.8.
In the similar way the individual pixel value has a brightness constant value. So if increase brightness by 10 unit, the new pixel value is added by 10. In contrast if decrease brightness by 20 unit, the new pixel value is deleted by 20.
Limiting the Results by Clamping
When the pixel values is performed mathematically, the results of some pixel values may exceed the maximum range of pixel values for a given image type. Clamping is included to avoid this.
After the operation for 8 bit image, if the pixel values that exceed for 255, they are limited any results to the maximum value 255. In similar, they are also limited any results to the minimum value 0 to avoid the negative pixel values.
Inverting Images
It is a converting of the individual pixel value for a given image type. For example in 8 bit image, the pixel values 0 are converted to 255, 1 to 254, 2 to 253, … In the same way the pixel values 255 are also converted to 0, 254 to 1, 253 to 2, … as follow.
Threshold Operation
It is a separating the pixels value in 2 class by determine the threshold value that is usually constant. The threshold function maps all pixels to one of two fixed intensity values
The original distribution (a) is split and merged into two isolated entries at a0 and a1 in the resulting histogram (b) by determine the threshold value ath
The original image (a) and corresponding histogram (c); result after thresholding with ath = 128, a0 = 0, a1 = 255 (b) and corresponding histogram (d).
Contrast & Brightness
Thresholding
Automatic Contrast Adjustment
Automatic contrast adjustment is done by mapping the current lowest and highest pixel values to the minimum and maximum available intensity values, respectively. For 8 bit image, the range of the pixel value low to high is mapped to the new range pixel value min(0) to max(255) as show in figure below.
Auto-contrast operation, The original pixel values a in the range [alow, ahigh] are mapped linearly to the target range [amin, amax].
Effects of auto-contrast and inversion operations on the resulting histograms. Original image (a), result of auto-contrast operation (b), and inversion (c).
As show in figure, the image is brighter than before but histogram of image after the operation is appear gap that cause the low image intensity.
Modified Auto-Contrast
This principle is applied to avoid noises that occurred when auto contrast adjustment. These noises are happened by the yellow part as show in figure below. To avoid these, the modified auto contrast is applied the histogram to determine the noises as quantile (qlow and qhigh) and get rid of them. After that this new alow and ahigh are mapped to the new range again.
Histogram Equalization
Histogram equalization is an adjustment the intensity of image to a uniform distribution.
Histogram equalization on the cumulative histogram. A suitable point operation shifts each histogram line from its original position such that the resulting cumulative histogram Heq is approximately linear.
Linear histogram equalization, The original image (a) and modified image (b), corresponding histograms h, h’ (c, d), and cumulative histograms H, H’ (e, f). The resulting cumulative histogram H (f) approximates a uniformly distributed image. Notice that new peaks are created in the resulting histogram h’ (d) by merging original histogram cells, particularly in the lower and upper intensity ranges.
Histogram Specification
Frequencies and Probabilities
The probability distribution or probability density function (pdf) of a random process, where p(i) is the probability for the occurrence of the pixel value. The resulting function P(i) is (like the cumulative histogram) monotonically increasing. The observed frequency distribution given by the histogram h(i) serves as a (coarse) estimate of the probability distribution p(i) of this random variable.
Principle of histogram specification
Histogram specification modifies the image to match an arbitrary intensity distribution, including the histogram of a given image. This process relies on the alignment of the cumulative histograms by applying a homogeneous point operation.
The goal of histogram specification is to modify a given image by some point operation such that its distribution function matches a reference distribution as closely as possible.
Given is the reference distribution PR (left) and the distribution function for the original image PA (right). The result is the mapping function a → a’ for a point operation, which replaces each pixel a in the original image by a modified value a’. The process has two main steps: (A) For each pixel value a, determine b = PA(a) from the right distribution function. (B) a’ is then found by inverting the left distribution function as a’ = PR^−1(b).
Gamma Correction
Gamma correction compensate for the transfer characteristics of different input and output devices and to map them to a unified intensity space.
“Gamma” originates from analog photography, where the relationship between the light energy and the resulting film density.
Exposure function of photographic film. With respect to the logarithmic light intensity B, the resulting film density D is approximately linear over a wide intensity range. The slope (ΔD/ΔB) of this linear section of the function specifies the “gamma” (γ) value for a particular type of photographic material.
To compensate for the nonlinearities of the receivers, a “gamma correction” was applied to the TV signal once before broadcasting in order to avoid the need for costly correction measures on the receiver side.
Gamma correction is based on the power function. The parameter γ is called the gamma value.
Power function b = fγ(a) = aγ for a ∈ [0, 1] for different gamma values.
Gamma correction in the digital imaging work flow. Images are processed and stored in a “linear” intensity space, where gamma correction is used to compensate for the transfer characteristic of each input and output device.
Reference
Principles of digital image processing, Fundamental techniques, Wilhelm Burger Mark J. Burg
Image Processing 
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