Chapter 7 Algebraic Operations Introduction Definitions There are four algebraic operations for images: Addition: Subtraction: Multiplication: Division:

Uses of Algebraic Operations Addition: 1. Averaging multiple images to reduce the effects of additive noise; 2. Superimpose the contents of one image upon another, producing a double-exposure effects. Subtraction: 1. Removal of undesired additive pattern from an image; 2. Motion detection; 3. Compute the gradient. Multiplication: 1. Correct the effects of space variation of a digitizer ; 2. Mask image. Division: Useful in multi-spectral image processing, such as color image processing.

Algebraic operations and the histogram Histogram of sum images Two images is uncorrelated if If at least one image is random and statistically independent to the other, the two image is uncorrelated. If the two images are identical, the above equation is not satisfied.

Histogram of the sum image: Where * means the convolution (卷积) integral. Convolution of two Gaussians produces a third Gaussian that is shifted and broader.

Histogram of difference images Histogram of difference image is similar as that of sum image, i.e., convolution of one image and the negative of another image. When two near-identical images are slightly misaligned, the subtraction produces a partial derivative image If , then

有噪声图象的IOD 1）什么是IOD？ 无噪声理想情况下的定义： 可以得到物体的光密度，用于分析。 对综合光密度的再次考察

有噪声图象的IOD 2）噪声图象的IOD 问题描述：均匀背景上有对比度明显的物体图象，并且被加性噪声污染。请计算物体的IOD。 假设： 其直方图为：

有噪声图象的IOD 为什么???

有噪声图象的IOD

有噪声图象的IOD 问题：在已知输出直方图的情况下，N0是多少？ 假设：噪声直方图左右对称，并且均值为N0。其与HS卷积后，直方图不发生变化。

Applications of algebraic operations Averaging for noise reduction Suppose Define signal-to-noise power ratio as:

When averaging M images yields And the signal-to-noise amplitude ratio is Theorem：Averaging M images contaminated by an additive noise increases the signal-to-noise power ratio of the image by M times.

Proof:

An example of averaging image An example of noise reduction by averaging images The first 5 images are contaminated ones, and the last is the averaged one

Image Subtraction Back ground subtraction Country Road

Subtraction of the back ground

Motion detection

Gradient Magnitude Given a scalar function f(x,y), define its gradient as the following vector function The vector points in the direction of maximum upward slope, the gradient magnitude is equal to the value of the slope

is often approximated by

An image of lung cancer

The gradient image

Multiplication and Division The multiplication is often used for image masking The Division operation may be used to remove effects of a spatially varying digitizer sensitivity function and to generate Ratio images useful in multi-spectral image processing.

Summary of important points The histogram of the sum of two uncorrelated images is given by the convolution of the two input histogram. Averaging images of a stationary scene contaminated by random noise increases the signal-to-noise amplitude ratio by . Subtracting slightly displaced identical images produces a partial derivative image. The convolution of two Gaussian functions produces another , broader Gaussian.

上机实习 在MATLAB软件中，首先对lenna.bmp图象加上随机高斯噪声，然后编制通过求平均值降噪的程序，观察实验效果。