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Image processing and Texture analysis Dr. S. Selinger Dennis Gabor College Budapest - Hungary Jump to first page Image processing Fundamentals Jump to first page Introduction Modern digital technology has made it possible to manipulate multi-dimensional signals with systems that range from simple digital circuits to advanced parallel computers. An image defined in the "real world" is considered to be a function of two real variables, for example, a(x,y) with a as the amplitude (e.g. brightness) of the image at the real coordinate position (x,y). An image may be considered to contain sub-images sometimes referred to as regions-of-interest, ROIs, or simply regions. Jump to first page Introduction (2) An image is digitized to convert it to a form which can be stored in a computer's memory or on some form of storage media such as a hard disk or CD-ROM. This digitization procedure can be done by a scanner, or by a video camera connected to a frame grabber board in a computer. Once the image has been digitized, it can be operated upon by various image processing operations. Jump to first page Introduction (3) We will restrict ourselves to two-dimensional (2D) image processing although most of the concepts and techniques that are to be described can be extended easily to three or more dimensions. An image defined in the "real world" is considered to be a function of two real variables, for example, a(x,y) with a as the amplitude (e.g. brightness) of the image at the real coordinate position (x,y). An image may be considered to contain sub- images sometimes referred to as regions-of-interest, ROI’s, or simply regions. Jump to first page Introduction (4) Image processing operations can be roughly divided into three major categories: Image Compression, Image Enhancement Restoration and Measurement Extraction. Jump to first page Introduction (5) Image Compression involves reducing the amount of memory needed to store a digital image. Image defects which could be caused by the digitization process or by faults in the imaging set-up (for example, bad lighting) can be corrected using Image Enhancement techniques. Once the image is in good condition, the Measurement Extraction operations can be used to obtain useful information from the image. Jump to first page Image Processing image in - image out Image Analysis image in - measurements out Image Understanding image in - high-level description out Jump to first page Digital Image Definitions A digital image a[m,n] described in a 2D discrete space is derived from an analog image a(x,y) in a 2D continuous space through a sampling process that is frequently referred to as digitization. The 2D continuous image a(x,y) is divided into N rows and M columns. The intersection of a row and a column is termed a pixel. The value assigned to the integer coordinates [m,n] with {m = 0,1,2,...,M-1} and {n = 0,1,2,...,N-1} is a[m,n] . Jump to first page Digital Image Definitions (Common Values) There are standard values for the various parameters encountered in digital image processing. Parameter Symbol Typical values Rows N 256,512,525,625,1024,1035 Columns M 256,512,768,1024,1320 Gray Levels L 2,64,256,1024,4096,16384 M = N = 2K where {K = 8,9,10}. The number of distinct gray levels is usually a power of 2, that is, L = 2B where B is the number of bits in the binary representation of the brightness levels. Jump to first page Digitization of a continuous image The image has been divided into N = 16 rows and M = 16 columns. The value assigned to every pixel is the average brightness in the pixel rounded to the nearest integer value. The process of representing the amplitude of the 2D signal at a given coordinate as an integer value with L different gray levels is usually referred to as amplitude quantization or simply quantization. Jump to first page Noise Real world signals usually contain departures from the ideal signal. Such departures are referred to as noise. Noise arises as a result of unmodelled or unmodellable processes going on in the production and capture of the real signal. Noise can generally be grouped into two classes: - independent noise. - noise which is dependent on the image data. Jump to first page Characteristics of Image Operations Types of operations The types of operations to transform an input image a[m,n] into an output image b[m,n] can be classified into three categories: point: the output value at a specific coordinate is dependent only on the input value at that same coordinate, local: the output value is dependent on the input values in the neighborhood of that same coordinate, global: the output value is dependent on all the values in the input image. Jump to first page Characteristics of Image Operations Types of neighborhoods The images can be sampled to the various neighborhoods Rectangular sampling: images are sampled by laying a rectangular grid over an image Hexagonal sampling: rectangular sampling hexagonal sampling 4-connected 8-connected 6-connected Jump to first page Algorithms Operations that are fundamental to digital image processing can be divided into four categories: operations based on the image histogram, on simple mathematics, on convolution, on mathematical morphology. Jump to first page Tools Certain tools are central to the processing of digital images. These include mathematical tools such as: convolution, Fourier analysis, statistical descriptions and manipulative tools such as: chain codes and run codes. Jump to first page Histogram-based Operations The histogram of an image represents the relative frequency of occurrence of the various gray levels in the image. A black and white image is represented by a two- dimensional array of numbers (or matrix), x(m,n), with values ranging from 0 to 255. A black pixel assumes the value x(m,n)=0 while a white pixel assumes the value x(m,n)=255. All other levels in between are assigned linearly to contrasts between the black and white extremes. The vertical lines in the histogram indicate how many pixels in an image assume a particular gray level. Jump to first page Histogram-based Operations (2) In the histogram of the white and black image we see that its pixels assume gray levels that are concentrated in a interval The histogram displays the overall range of the image, allowing you to determine the distribution of light and dark tones. Jump to first page Histogram-based Operations (3) From left to right the histogram displays the darkest to lightest pixel values. The vertical lines each represent one of 256 possible tones. The higher the line, the more of that particular tone exists in the image. Mean: the average brightness level for all image pixels. Std Dev: (Short for Standard Deviation) this value represents the difference in brightness of the average pixel from the Mean. Median: the value found in the exact center of the existing tonal range. Pixels: the total number of image pixels. Level: the brightness value of an individual point on the Histogram, position the cursor over it and click. Percentile: the percentage of existing image pixels possessing a value darker than the selected value. Jump to first page Convolution Convolution is a simple mathematical operation which is fundamental to many common image processing operators. Convolution provides a way of `multiplying together' two arrays of numbers (generally of different sizes, but of the same dimensionality), to produce a third array of numbers of the same dimensionality. This can be used in image processing to implement operators whose output pixel values are simple linear combinations of certain input pixel values. Jump to first page Convolution (2) In an image processing context, one of the input arrays is normally just a gray level image. The second array is usually much smaller, and is also two dimensional (although it may be just a single pixel thick), and is known as the kernel. The convolution is performed by sliding the kernel over the image, generally starting at the top left corner, so as to move the kernel through all the positions where the kernel fits entirely within the boundaries of the image. Jump to first page Convolution (3) An example image and kernel that we will use to illustrate convolution: the value of the bottom right pixel in the output image will be given by: Jump to first page Convolution (4) If the image has M rows and N columns, and the kernel has m rows and n columns, then the size of the output image will have M-m+1 rows, and N-n+1 columns. Mathematically we can write the convolution as: (where i runs from 1 to M-m+1 and j runs from 1 to N-n+1) Jump to first page Fourier Transforms The Fourier Transform to decompose an image into its sine and cosine components. The Fourier transform produces another representation of a signal, specifically a representation as a weighted sum of complex exponential. Euler's formula: The output of the transformation represents the image in the Fourier or frequency domain, while the input image is the spatial domain equivalent. The Fourier Transform is used: image analysis, image filtering, image reconstruction and image compression. Jump to first page Fourier Transforms (2) Given an image a and its Fourier transform A, then the forward transform goes from the spatial domain (either continuous or discrete) to the frequency domain which is always continuous. Forward - The inverse Fourier transform goes from the frequency domain back to the spatial domain. Inverse - Jump to first page Digital Filters In image processing filters are mainly used to suppress: - high frequencies in the image (i.e. smoothing the image) - low frequencies (i.e. enhancing or detecting edges in the image) An image can be filtered either in the frequency or in the spatial domain. Jump to first page Linear Filtering Low Pass Filters ("smoothing”): - remove high spatial frequency noise from a digital image. Reconstruction filtering Enhancement filtering Moving Window Operations - affects one pixel of the image at a time, changing its value by some function of a "local" region of pixels ("covered" by the window). Neighborhood-averaging filters Median filters Mode filters Jump to first page Linear Filtering (2) Neighborhood-averaging filters : - replace the value of each pixel, a[i,j] say, by a weighted- average of the pixels in some neighborhood around it. (i.e.: a weighted sum of a[i+p,j+q], with p = -k to k, q = -k to k for some positive k; the weights are non-negative with the highest weight on the p = q = 0 term. If all the weights are equal then this is a mean filter is "linear”) Median filters: - replaces each pixel value by the median of its neighbors. (i.e. the value such that 50% of the values in the neighborhood are above, and 50% are below. ) Mode filters: - each pixel value is replaced by its most common neighbor. This is a particularly useful filter for classification procedures where each pixel corresponds to an object which must be placed into a class Jump to first page Smoothing Operations Linear Filters: - Uniform filter - Triangular filter - Gaussian filter Non-Linear Filters: - Median filter - Kuwahara filter Jump to first page Thresholding In many vision applications, it is useful to be able to separate out the regions of the image corresponding to objects in which we are interested (ROI), from the regions of the image that correspond to background. Thresholding often provides an easy and convenient way to perform this segmentation on the basis of the different intensities or colors in the foreground and background regions of an image. Jump to first page Thresholding (2) The segmentation is determined by a single parameter known as the intensity threshold. In a single pass, each pixel in the image is compared with this threshold. If the pixel's intensity is higher than the threshold, the pixel is set to, say, white, in the output. If it is less than the threshold, it is set to black. Jump to first page Edge Detectors Edges are places in the image with strong intensity contrast. Edges often occur at image locations representing object boundaries. Edge detection is extensively used in image segmentation when we want to divide the image into areas corresponding to different objects. Edges consist of mainly high frequencies. For the edge detection we can applying a highpass frequency filter in the Fourier domain or convoluting the image with an appropriate kernel in the spatial domain. Jump to first page Laplacian Edge Detection Build of morphing algorithm which operates on features automatically extracted from target images: View the following signal, with an edge as highlighted below : the gradient of this signal : Jump to first page Tutorial Images This is a demo showing different image processing techniques ORIGINAL image (photo "Robin Jeffers at Ton House" (1927) by Edward Weston) Jump to first page Tutorial Images SAMPLING RATE Here is the image with every 3rd pixel sampled, and the intermediate pixels filled in with the sampled values. Note the blocky appearance of the new image. Jump to first page Tutorial Images QUANTIZATION Here is the image with only 5 grayscale shades; the original has 184 shades. Note how much detail is retained with only 5 shades! Jump to first page Tutorial Images LOW PASS FILTERING Here is the image filtered with the following kernel: 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 this filter is a 3-by-3 mean filter (notice how it smooths the texture of the image while blurring out the edges) Jump to first page Tutorial Images LOW PASS FILTERING II Here is the image filtered with the following 5-by-5 mean kernel: 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 1/25 Notice: the difference between the images from the two filters Jump to first page Tutorial Images EDGE DETECTION Here is the image filtered with the following kernel: 0 -1 0 -1 4 -1 0 -1 0 This filter is a 2-dimensional Laplacian (actually the negative of the Laplacian) Notice: how it brings out the edges in the image. Jump to first page Tutorial Images EDGE DETECTION II Top right: Here is the image filtered with the following kernel: 0 -1 0 000 0 -1 0 -1 4 -1 + 0 1 0 = -1 5 -1 0 -1 0 000 0 -1 0 This is the Laplacian filter with the original image added back in Notice: how it brings out the edges in the image while maintaining the underlying grayscale information. Jump to first page

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posted: | 8/14/2012 |

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