# A Spatial Median Filter for Noise Removal in Digital Images

James C. Church, Yixin Chen, and Stephen V. Rice Department of Computer and Information Science, University of Mississippi {jcchurch,ychen,rice}@cs.olemiss.edu

Abstract

In this paper, six different image ﬁltering algorithms are compared based on their ability to reconstruct noiseaffected images. The purpose of these algorithms is to remove noise from a signal that might occur through the transmission of an image. A new algorithm, the Spatial Median Filter, is introduced and compared with current image smoothing techniques. Experimental results demonstrate that the proposed algorithm is comparable to these techniques. A modiﬁcation to this algorithm is introduced to achieve more accurate reconstructions over other popular techniques.

Figure 1 demonstrates ﬁve common ﬁltering algorithms applied to an original image.

(a)

(b)

(c)

(d)

(e)

(f)

1. Smoothing Algorithms

The inexpensiveness and simplicity of point-andshoot cameras, combined with the speed at which budding photographers can send their photos over the Internet to be viewed by the world, makes digital photography a popular hobby. With each snap of a digital photograph, a signal is transmitted from a photon sensor to a memory chip embedded inside a camera. Transmission technology is prone to a degree of error, and noise is added to each photograph. Signiﬁcant work has been done in both hardware and software to improve the signal-to-noise ratio in digital photography. In software, a smoothing ﬁlter is used to remove noise from an image. Each pixel is represented by three scalar values representing the red, green, and blue chromatic intensities. At each pixel studied, a smoothing ﬁlter takes into account the surrounding pixels to derive a more accurate version of this pixel. By taking neighboring pixels into consideration, extreme “noisy” pixels can be replaced. However, outlier pixels may represent uncorrupted ﬁne details, which may be lost due to the smoothing process. This paper examines four common smoothing algorithms and introduces a new smoothing algorithm. These algorithms can be applied to one-dimensional as well as two-dimensional signals.

Figure 1. Examples of common ﬁltering approaches. (a) Original Image (b) Mean Filtering (c) Median Filtering (d) Root Signal of Median Filtering (e) Component wise Median Filtering (f) Vector Median Filtering. The simplest of these algorithms is the Mean Filter as deﬁned in (1). The Mean Filter is a linear ﬁlter which uses a mask over each pixel in the signal. Each of the components of the pixels which fall under the mask are averaged together to form a single pixel. This new pixel is then used to replace the pixel in the signal studied. The Mean Filter is poor at maintaining edges within the image. 1 N ∑ xi N i=1

MEANFILT ER(x1 , ..., xN ) =

(1)

The use of the median in signal processing was ﬁrst introduced by J. W. Tukey [1]. The Median Filter is performed by taking the magnitude of all of the vectors within a mask and sorting the magnitudes, as deﬁned in (2). The pixel with the median magnitude is then used to replace the pixel studied. The Simple Median Filter has an advantage over the Mean ﬁlter in that it relies on median of the data instead of the mean. A single noisy pixel present in the image can signiﬁcantly skew

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the mean of a set. The median of a set is more robust with respect to the presence of noise. MEDIANFILT ER(x1 , ..., xN ) = MEDIAN( x1 2 , ..., xN 2 ) (2)

When ﬁltering using the Simple Median Filter, an original pixel and the resulting ﬁltered pixel of the sample studied are sometimes the same pixel. A pixel that does not change due to ﬁltering is known as the root of the mask. It can be shown that after sufﬁcient iterations of median ﬁltering, every signal converges to a root signal [2]. The Component Median Filter, deﬁned in (3), also relies on the...

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