## Operations with Multiple Images

This section outlines two enhancement methods that require more than one image of the same scene. In both methods, the images have to be registered and their dynamic ranges have to be comparable to provide a viable outcome.

### 5.1 Noise Suppression by Image Averaging

Noise suppression using image averaging relies on three basic assumptions: (1) that a relatively large number of input images are available, (2) that each input image has been corrupted by the same type of additive noise, and (3) that the additive noise is random with zero mean value and independent of the image. When these assumptions hold, it may be advantageous to acquire multiple images with the specific purpose of using image averaging [1] since with this approach even severely corrupted images can be significantly enhanced. Each of the noisy images a{(m, n) can be represented by a{(m, n) — f (m, n) + di(m, n), where f (m, n) is the underlying noise-free image, and di(m, n) is the additive noise in that image. If a total of Q images are available, the averaged image is

such that

and where E{ • } is the expected value operator, ag is the standard deviation of g (m, n), and ad is that of the noise. Noise suppression is more effective for larger values of Q.

### 5.2 Change Enhancement by Image Subtraction

Image subtraction is generally performed between two images that have significant similarities between them. The purpose of image subtraction is to enhance the differences between two images (1). Images that are not captured under the same or very similar conditions may need to be registered [17]. This may be the case if the images have been acquired at different times or under different settings. The output image may have a very small dynamic range and may need to be rescaled to the available display range. Given two images f1(m, n) and f2(m, n), the rescaled output image g(m, n) is obtained with b(m, n) =/1(m, n) — n)

ones is used to avoid ringing. The commonly used Butterworth low-pass and high-pass filters are defined respectively as

D(M, V) = VM2 + V2, which can be compared to a threshold DT to determine if (m, v) is high or low. The simplest approach to image smoothing is the ideal low-pass filter Wl(m, v), defined to be 1 when D(m, v) < DT and 0 otherwise. Similarly, the ideal high-pass filter WH(m, v) can be defined to be 1 when D(m, v) > DT and 0 otherwise. However, these filters are not typically used in practice, because images that they produce generally have spurious structures that appear as intensity ripples, known as ringing [5]. The inverse Fourier transform of the rectangular window Wl(m, v) or WH(m, v) has oscillations, and its convolution with the spatial-domain image produces the ringing. Because ringing is associated with the abrupt 1 to 0 discontinuity of the ideal filters, a filter that imparts a smooth transition between the desired frequencies and the attenuated

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