3. Global and Regional Features

Let’s take a brief look at a few major trends and milestones in feature metrics research. While this brief outline is not intended to be a precise, inclusive look at all key events and research, it describes some general trends in mainstream industry thinking and academic activity.

One of the most basic metrics is texture, which is the description of the surface of an image channel, such as color intensity, like an elevation map or terrain map. Texture can be expressed globally or within local regions. Texture can be expressed locally by statistical relationships among neighboring pixels in a region, and it can be expressed globally by summary relationships of pixel values within an image or region. For a sampling of the literature covering a wide range of texture methods, see references [13,59,60,310,16–20,312,313].

According to Gonzalez [4], there are three fundamental classes of texture in image analysis: statistical, structural, and spectral. Statistical measures include histograms, scatter plots, and SDMs. Structural techniques are more concerned with locating patterns or structural primitives in an image, such as parallel lines, regular patterns, and so on. These techniques are described in [11,1,5,8]. Spectral texture is derived from analysis of the frequency domain representation of the data. That is, a fast Fourier transform is used to create a frequency domain image of the data, which can then be analyzed using Fourier techniques.

Histograms reveal overall pixel value distributions but say nothing about spatial relationships. Scatter plots are essentially two-dimensional histograms, and do not reveal any spatial relationships. A good survey is found in reference[315].

Texture has been used to achieve several goals:

In computer vision, texture metrics are devised to describe the perceptual attributes of texture by using discrete methods. For instance, texture has been described perceptually with several properties, including:

If textures can be recognized, then image regions can be segmented based on texture and the corresponding regions can be measured using shape metrics such as area, perimeter, and centroid (as will be discussed in Chapter 6). Chapter 2 included a survey of segmentation methods, some of which are based on texture. Segmented texture regions can be recognized and compared for computer vision applications. Micro-textures of a local region, such as the LBP discussed in detail in Chapter 6, can be useful as a feature descriptor, and macro-textures can be used to describe a homogenous texture of a region such as a lake or field of grass, and therefore have natural applications to image segmentation. In summary, texture can be used to describe global image content, image region content, and local descriptor region content. The distinction between a feature descriptor and a texture metric may be small.

Sensor limitations combined with compute and memory capabilities of the past have limited the development of texture metrics to mainly 2D gray scale metrics. However, with the advances toward pervasive computational photography in every camera providing higher resolution images, higher frame rates, deeper pixels, depth imaging, more memory, and faster compute, we can expect that corresponding new advances in texture metrics will be made.

Here is a brief historical survey of texture metrics.

The topic of statistical methods is vast, and we can only refer the reader to selected literature as we go along. One useful and comprehensive resource is the online NIST National Institute of Science and Technology Engineering Statistics Handbook,¹ including examples and links to additional resources and tools.

Statistical methods may be drawn upon at any time to generate novel feature metrics. Any feature, such as pixel values or local region gradients, can be expressed statistically by any number of methods. Simple methods, such as the histogram shown in Figure 3-1, are invaluable. Basic statistics such as minimum, maximum, and average values can be seen easily in the histogram shown in Chapter 2 (Figure 2-22). We survey several applications of statistical methods to computer vision here.

Edges, lines, contours, or ridges are basic textural features [316,317]. A variety of simple metrics can be devised just by analyzing the edge structure of regions in an image. There are many edge metrics in the literature, and a few are illustrated here.

The edge metrics can be computed locally or globally on image regions as follows:

Several other methods can be used to compute edge statistics. The representative methods are shown here; see also Shapiro and Stockton [517] for a standard reference.

Cross-correlation [34] is a metric showing similarity between two signals with a time displacement between them. Auto-correlation is the cross-correlation of a signal with a time-displaced version of itself. In the literature on signal processing, cross-correlation is also referred to as a sliding inner product or sliding dot product. Typically, this method is used to search a large signal for a smaller pattern.

Using the Wiener-Khinchin theorem as a special case of the general cross-correlation theorem, cross-correlation can be written as simply the Fourier transform of the absolute square of the function f _v, as follows:

In computer vision, the feature used for correlation may be a 1D line of pixels or gradient magnitudes, a 2D pixel region, or a 3D voxel volume region. By comparing the features from the current image frame and the previous image frame using cross-correlation derivatives, we obtain a useful texture change correlation metric.

By comparing displaced versions of an image with itself, we obtain a set of either local or global auto-correlation texture metrics. Auto-correlation can be used to detect repeating patterns or textures in an image, and also to describe the texture in terms of fine or coarse, where coarse textures show the auto-correlation function dropping of more slowly than fine textures. See also the discussion of correlation in Chapter 6 and Figure 6-20.

Basis transforms, such as the FFT, decompose a signal into a set of basis vectors from which the signal can be synthesized or reconstructed. Viewing the set of basis vectors as a spectrum is a valuable method for understanding image texture and for creating a signature. Several basis spaces are discussed in this chapter, including Fourier, HAAR, wavelets, and Zernike.

Although computationally expensive and memory intensive, the Fast Fourier Transform (FFT) is often used to produce a frequency spectrum signature. The FFT spectrum is useful for a wide range of problems. The computations typically are limited to rectangular regions of fixed sizes, depending on the radix of the transform (see Bracewell[227]).

As shown in Figure 3-3, Fourier spectrum plots reveal definite image features useful for texture and statistical analysis of images. For example, Figure 3-10 shows an FFT spectrum of LBP pattern metrics. Note that the Fourier spectrum has many valuable attributes, such as rotational invariance, as shown in Figure 3-3, where a texture image is rotated 90 degrees and the corresponding FFT spectrums exhibit the same attributes, only rotated 90 degrees.

Wavelets [227] are similar to Fourier methods, and have become increasingly popular for texture analysis [311], discussed later in the section on basis spaces.

Note that the FFT spectrum as a texture metric or descriptor is rotational invariant, as shown in the bottom left image of Figure 3-3. FFT spectra can be taken over rectangular 2D regions. Also, 1D arrays such as annuli or Cartesian coordinates of the shape taken around the perimeter of an object shape can be used as input to an FFT and as an FFT descriptor shape metric.

Haralick[6] proposed a set of 2D texture metrics calculated from directional differences between adjacent pixels, referred to as co-occurrence matrices, or spatial dependency matrices (SDM), or gray level co-occurrence matrices (GLCM). A complete set of four (4) matrices are calculated by evaluating the difference between adjacent pixels in the x, y, diagonal x and diagonal y directions, as shown in Figure 3-4, and further illustrated with a 4x4 image and corresponding co-occurence tables shown in Figure 3-5.

One benefit of the SDM as a texture metric is that it is easy to calculate in a single pass over the image. The SDM is also fairly invariant to rotation, which is often a difficult robustness attribute to attain. Within a segmented region or around an interest point, the SDM plot can be a valuable texture metric all by itself, therefore useful for texture analysis, feature description, noise detection, and pattern matching.

For example, if a camera has digital-circuit readout noise, it will show up in the SDM for the x direction only if the lines are scanned out of the sensor one at a time in the x direction, so using the SDM information will enable intelligent sensor processing to remove the readout noise. However, it should be noted that SDM metrics are not always useful alone, and should be qualified with additional feature information. The SDM is primarily concerned with spatial relationships, with regard to spatial orientation and frequency of occurrence. So, it is primarily a statistical measure.

The SDM is calculated in four orientations, as shown in Figure 3-4. Since the SDM is only concerned with adjacent pairs of pixels, these four calculations cover all possible spatial orientations. SDMs could be extended beyond 2x2 regions by using forming kernels extending into 5x5, 7x7, 9x9, and other dimensions.

A spatial dependency matrix is basically a count of how many times a given pixel value occurs next to another pixel value. Figure 3-5 illustrates the concept. For example, assume we have an 8-bit image (0. 255). If an SDM shows that pixel value x frequently occurs adjacent to pixels within the range x+1 to x-1, then we would say that there is a “smooth” texture at that intensity. However, if pixel value x frequently occurs adjacent to pixels within the range x+70 to x-70, we would say that there is quite a bit of contrast at that intensity, if not noise.

A critical point in using SDMs is to be sensitive to the varied results achieved when sampling over small vs. large image areas. By sampling the SDM over a smaller area (say 64x64 pixels), details will be revealed in the SDMs that would otherwise be obscured. The larger the size of the sample image area, the more the SDM will be populated. And the more samples taken, the more likely that detail will be obscured in the SDM image plots. Actually, smaller areas (i.e., 64x64 pixels) are a good place to start when using SDMs, since smaller areas are faster to compute and will reveal a lot about local texture.

The Haralick metrics are shown in Figure 3-6.

The statistical characteristics of the SDM have been extended by several researchers to add more useful metrics [26], and SDMs have been applied to 3D volumetric data by a number of researchers with good results [25].

The Laws metrics [52] provide a structural approach to texture analysis, using a set of masking kernels to measure texture energy or variation within fixed sized local regions, similar to the 2x2 region SDM approach but using larger pixel areas to achieve different metrics.

The basic Laws algorithm involves classifying each pixel in the image into texture based on local energy, using a few basic steps:

Laws defines a set of nine separable kernels to produce a set of texture region energy metrics, and some of the kernels work better than others in practice. The kernels are composed via matrix multiplication from a set of four vector masks L5, E5, S5, and R5, described below. The kernels were originally defined as 5x5 masks, but 3x3 approximations have been used also, as shown below.

To create 2D masks, vectors Ln, En, Sn, and Rn (as shown above) are convolved together as separable pairs into kernels; a few examples are shown in Figure 3-9.

Note that Laws texture metrics have been extended into 3D for volumetric texture analysis.[51][52]

In contrast to the various structural and statistical methods of texture analysis, the LBP operator [18,58] computes the local texture around each region as an LBP binary code, or micro-texture, allowing simple micro-texture comparisons to segment regions based on like micro-texture. (See the very detailed discussion on LBP in Chapter 6 for details and references to the literature, and especially Figure 6-6.) The LBP operator [173] is quite versatile, easy to compute, consumes a low amount of memory, and can be used for texture analysis, interest points, and feature description. As a result, the LBP operator is discussed is several places in this book.

As shown in Figure 3-10, the uniform set of LBP operators, composed of a subset of the possible LBPs that are by themselves rotation invariant, can be binned into a histogram, and the corresponding bin values are run through an FFT as a 1D array to create an FFT spectrum, which yields a robust metric with strong rotational invariance.

A close cousin to dynamic texture research is the field of activity recognition (discussed in Chapter 6), where features are parts of moving objects that compose an activity—for example, features on arms and legs that are tracked frame to frame to determine the type of motion or activity, such as walking or running. One similarity between activity recognition and dynamic textures is that the features or textures change from frame to frame over time, so for both activity recognition and dynamic texture analysis, tracking features and textures often requires a spatio-temporal approach involving a data structure with a history buffer of past and current frames, which provides a volumetric representation to the data.

For example, VLBP and LBP-TOP (discussed in Chapter 6) provide methods for dynamic texture analysis by using the LBP constructed to operate over three dimensions in a volumetric structure, where the volume contains image frames n-2, n-1, and n stacked into the volume.

In this work, we classify image moments under the term polygon shape descriptors in the taxonomy (see Chapter 5). Details on several image moments used for 2D shape description will be covered in Chapter 6, under “Object Shape Metrics for Blobs and Objects.”

Common properties of moments in the context of 1D distributions and 2D images include:

Moments can be used to create feature descriptors that are invariant to several robustness criteria, such as scale, rotation, and affine variations. The taxonomy of robustness and invariance criteria is provided in Chapter 5. For 2D shape description, in 1961 Hu developed a theoretical set of seven 2D planar moments for character recognition work, derived using invariant algebra, that are invariant under scale, translation, and rotation [7]. Several researchers have extended Hu’s work. An excellent resource for this topic is Moments and Moment Invariants in Pattern Recognition, by Jan Flusser et al.[518].

Point metrics can be used for the following: (1) feature description, (2) analysis and visualization, (3) thresholding and segmentation, and (4) image processing via programmable LUT functions (discussed in Chapter 2). Point metrics are often overlooked. Using point metrics to understand the structure of the image data is one of the first necessary steps toward devising the image pre-processing pipeline to prepare images for feature analysis. Again, the place to start is by analysis of the histogram, as shown in Figures 3-1 and 3-11. The basic point metrics can be determined visually, such as minima, maxima, peaks, and valleys. False coloring of the histogram regions for data visualization is simple using color lookup tables to color the histogram regions in the images.

Here is a summary of statistical point metrics:

Global histograms treat the entire image. In many cases, image matching via global histograms is simple and effective, using a distance function such as SSD. As shown in Figure 3-12, histograms reveal quantitative information on pixel intensity, but not structural information. All the pixels in the region contribute to the histogram, with no respect to the distance from any specific point or feature. As discussed in Chapter 2, the histogram itself is the basis of histogram modification methods, allowing the shape of the histogram to be stretched, compressed, or clipped as needed, and then used as an inverse lookup table to rearrange the image pixel intensity levels.

Local histograms of pixel intensity values can be used as attributes of a feature descriptor, and also used as the basis for remapping pixel values from one histogram shape to another, as discussed in Chapter 2, by reshaping the histogram and reprocessing the image accordingly. Chapter 6 discusses a range of feature descriptors such as SIFT, SURF, and LBP which make use of feature histograms to bin attributes such as gradient magnitude and direction.

The scatter diagram can be used to visualize the relationship or similarity between two image datasets for image analysis, pattern recognition, and feature description. Pixel intensity from two images or image regions can be compared in the scatter plot to visualize how well the values correspond. Scatter diagrams can be used for feature and pattern matching under limited translation invariance, but they are less useful for affine, scale, or rotation invariance. Figure 3-13 shows an example using a scatter diagram to look for a pattern in an image, the target pattern is compared at different offsets, the smaller the offset, the better the correspondence. In general, tighter sets of peak features indicate a strong structural or pattern correspondence; more spreading of the data indicates weaker correspondence. The farther away the pattern offset moves, the lower the correspondence.

Note that by analyzing the peak features compared to the low-frequency features, correspondence can be visualized. Figure 3-14 shows scatter diagrams from two separate images. The lack of peaks along the axis and the presence of spreading in the data show low structural or pattern correspondence.

The scatter plot can be made, pixel by pixel, from two images, where pixel pairs form the Cartesian coordinate for scatter plotting using the pixel intensity of image 1 is used as the x coordinate, and the pixel intensities of image 2 as the y coordinate, then the count of pixel pair correspondence is binned in the scatter plot. The bin count for each coordinate can be false colored for visualization. Figure 3-15 provides some code for illustration purposes.

For feature detection, as shown in Figure 3-12, the scatter plot may reveal enough correspondence at coarse translation steps to reduce the need for image pyramids in some feature detection and pattern matching applications. For example, the step size of the pattern search and compare could be optimized by striding or skipping pixels, searching the image at 8 or 16 pixel intervals, rather than at every pixel, reducing feature detection time. In addition, the scatter plot data could first be thresholded to a binary image, masked to show just the peak values, converted into a bit vector, and measured for correspondence using HAMMING distance for increased performance.

Multi-resolution histograms [10] have been used for texture analysis [54], and also for feature recognition [55]. The PHOG descriptor described in Chapter 6 makes use of multi-scale histograms of feature spectra—in this case, gradient information. Note that the multi-resolution histogram provides scale invariance for feature description. For texture analysis [54], multi-resolution histograms are constructed using an image pyramid, and then a histogram is created for each pyramid level and concatenated together [10], which is referred to as a multi-resolution histogram. This histogram has the desirable properties of algorithm simplicity, fast computation, low memory requirements, noise tolerance, and high reliability across spatial and rotational variations. See Figure 3-16. A variation on the pyramid is used in the method of Zhao and Pietikainen [15], employing a multi-dimensional pyramid image set from a volume.

Steps involved in creating and using multi-resolution histograms are as follows:

For some applications, computing the histogram using radial samples originating at the shape centroid can be valuable [136][137]. To do this, a line is cast from the centroid to the perimeter of the shape, and pixel values are recorded along each line and then binned into histograms. See Figure 3-17.

The perimeter or shape of an object can be the basis of a shape histogram, which includes the pixel values of each point on the perimeter of the object binned into the histogram. Besides recording the actual pixel values along the perimeter, the chain code histogram (CCH) that will be discussed in Chapter 6 shows the direction of the perimeter at connected edge point coordinates. Taken together, the CCH and contour histograms provide useful shape information.

The Fourier family of transforms was covered in detail in Chapter 2, in the context of image pre-processing and filtering. However, the Fourier frequency components can also be used for feature description. Using the forward Fourier transform, an image is transformed into frequency components, which can be selectively used to describe the transformed pixel region, commonly done for image coding and compression, and for feature description.

The Fourier descriptor provides several invariance attributes, such as rotation and scale. Any array of values can be fed to an FFT to generate a descriptor—for example, a histogram. A common application is illustrated in Figure 3-20, describing the circularity of a shape and finding the major and minor axis as the extrema frequency deviation from the sine wave. A related application is finding the endpoints of a flat line segment on the perimeter by fitting FFT magnitude’s of the harmonic series as polar coordinates against a straight line in Cartesian space.

In Figure 3-20, a complex wave is plotted as a dark gray circle unrolled around a sine wave function or a perfect circle. Note that the Fourier transform of the lengths of each point around the complex function yields an approximation of a periodic wave, and the Fourier descriptor of the shape of the complex wave is visible. Another example illustrating Fourier descriptors is shown in Figure 6-29.

The Hadamard transform [4,9] uses a series of square waves with the value of +1 or -1, which is ideal for digital signal processing. It is amenable to optimizations, since only signed addition is needed to sum the basis vectors, making this transform much faster than sinusoidal basis transforms. The basis vectors for the harmonic Hadamard series and corresponding transform can be generated by sampling Walsh functions, which make up an orthonormal basis set; thus, the combined method is commonly referred to as the Walsh-Hadamaard transform; see Figure 3-21.

The HAAR transform [4,9] is similar to the Fourier transform, except that the basis vectors are HAAR features resembling square waves, and similar to wavelets. HAAR features, owing to their orthogonal rectangular shapes, are suitable for detecting vertical and horizontal images features that have near- constant gray level. Any structural discontinuities in the data, such as edges and local texture, cannot be resolved very well by the HAAR features; see Figures 3-21 and 6-22.

The Slant transform [284], as illustrated in Figure 3-21, was originally developed for television signal encoding, and was later applied to general image coding [283,4]. The Slant transform is analogous to the Fourier transform, except that the basis functions are a series of slant, sawtooth, or triangle waves. The slant basis vector is suitable for applications where image brightness changes linearly over the length of the function. The slant transform is amenable to discrete optimizations in digital systems. Although the primary applications have been image coding and image compression, the slant transform is amenable to feature description. It is closely related to the Karhunen-Loeve transform and the Slant-Hadamaard transform [512].

Fritz Zernike, 1953 Nobel Prize winner, devised Zernike polynomials during his quest to develop the phase contrast microscope, while studying the optical properties and spectra of diffraction gratings. The Zernike polynomials [272–274] have been widely used for optical analysis and modeling of the human visual system, and for assistance in medical procedures such as laser surgery. They provide an accurate model of optical wave aberrations expressed as a set of basis polynomials, illustrated in Figure 3-22.

Zernike polynomials are analogous to steerable filters [388], which also contain oriented basis sets of filter shapes used to identify oriented features and take moments to create descriptors. The Zernike model uses radial coordinates and circular regions, rather than rectangular patches as used in many other feature description methods.

Zernike methods are widely used in optometry to model human eye aberrations. Zernike moments are also used for image watermarking[278] and image coding and reconstruction [279,281]. The Zernike features provide scale and rotational invariance, in part due to the radial coordinate symmetry and increasing level of detail possible within the higher-order polynomials. Zernike moments are used in computer vision applications by comparing the Zernike basis features against circular patches in target images [276,277].

Fast methods to compute the Zernike polynomials and moments exist [275,280,282], which exploit the symmetry of the basis functions around the x and y axes to reduce computations, and also to exploit recursion.

Steerable filters are loosely considered as basis functions here, and can be used for both filtering or feature description. Conceptually similar to Zernike polynomials, steerable filters [388,400] are composed by synthesizing steered or oriented linearly combinations of chosen basis functions, such as quadrature pairs of Gaussian filters and oriented versions of each function, in a simple transform.

Many types of filter functions can be used as the basis for steerable filters [389,390]. The filter transform is created by combining together the basis functions in a filter bank, as shown in Figure 3-23. Gain is selected for each function, and all filters in the bank are summed, then adaptively applied to the image. Pyramid sets of basis functions can be created to operate over scale. Applications include convolving oriented steerable filters with target image regions to determine filter response strength, orientation and phase. Other applications include filtering images based on orientation of features, contour detection, and feature description.

For feature description, there are several methods that could work—for example, convolving each steerable basis function with an image patch. The highest one or two filter responses or moments from all the steerable filters can then be chosen as the set-ordinal feature descriptor, or all the filter responses can be used as a feature descriptor. As an optimization, an interest point can first be determined in the patch, and the orientation of the interest point can be used to select the one or two steerable filters closest to the orientation of the interest point; then the closest steerable filers are used as the basis to compute the descriptor.

The Karhunen-Loeve transform (KLT)[4,9] was devised to describe a continuous random process as a series expansion, as opposed to the Fourier method of describing periodic signals. Hotelling later devised a discrete equivalent of the KLT using principal components. “KLT” is the most common name referring to both methods.

The basis functions are dependent on the eigenvectors of the underlying image, and computing eigenvectors is a compute-intensive process with no established fast transform known. The KLT is not separable to optimize over image blocks, so the KLT is typically used for PCA on small datasets such as feature vectors used in pattern classification, clustering, and matching.

Wavelets, as the name suggests, are short waves or wave-lets [334]. Think of a wavelet as a short-duration pulse such as a seismic tremor, starting and ending at zero, rather than a continuous or resonating wave. Wavelets are convolved with a given signal, such as an image, to find similarity and statistical moments. Wavelets can therefore be implemented like convolution kernels in the spatial domain. See Figure 3-24.

Wavelet analysis is a vast field [291,292] with many applications and useful resources available, including libraries of wavelet families and analysis software packages [289]. Fast wavelet transforms (FWTs) exist in common signal and image processing libraries. Several variants of the wavelet transform include:

Wavelets are designed to meet various goals, and are crafted for specific applications; there is no single wavelet function or basis. For example, a set of wavelets can be designed to represent the musical scale, where each note (such as middle C) is defined as having a duration of an eighth note wavelet pulse, and then each wavelet in the set is convolved across a signal to locate the corresponding notes in the musical scale.

When designing wavelets, the mother wavelet is the basis of the wavelet family, and then daughter wavelets are derived using translation, scaling, or compression of the mother wavelet. Ideally, a set of wavelets are overlapping and complementary so as to decompose data with no gaps and be mathematically reversible.

Wavelets are used in transforms as a set of nonlinear basis functions, where each basis function can be designed as needed to optimally match a desired feature in the input function. So, unlike transforms which use a uniform set of basis functions—as the Fourier transform uses sine and cosine functions—wavelets use a dynamic set of basis functions that are complex and nonuniform in nature. See Figure 3-25.

Wavelets have been used as the basis for scale and rotation invariant feature description [288], image segmentation [285,286], shape description [287], and obviously image and signal filtering of all the expected varieties, denoising, image compression, and image coding. A set of application-specific wavelets could be devised for feature description.

The Hough transform [228–230] and the Radon transform [299] are related, and the results are equivalent, in the opinion of many;[295][300] see Figure 3-26. The Radon transform is an integral transform, while the Hough transform is a discrete method, therefore much faster. The Hough method is widely used in image processing, and can be accelerated using a GPU [298] with data parallel methods. The Radon algorithm is slightly more accurate and perhaps more mathematically sound, and is often associated with x-ray tomography applied to reconstruction from x-ray projections. We focus primarily on the Hough transform, since it is widely available in image processing libraries.

Key applications for the Hough and Radon transforms are shape detection and shape description of lines, circles, and parametric curves. The main advantages include:

The disadvantages include:

The Hough transform is primarily a global or regional descriptor and operates over larger areas. It was originally devised to detect lines, and has been subsequently generalized to detect parametric shapes [301], such as curves and circles. However, adding more parameterization to the feature requires more memory and compute. Hough features can be used to mark region boundaries described by regular parametric curves and lines. The Hough transform is attractive for some applications, since it can tolerate gaps in the lines or curves and is not strongly affected by noise or some occlusion, but morphology and edge detection via other methods is often sufficient, so the Hough transform has limited applications.

The input to the Hough transform is a gradient magnitude image, which has been thresholded, leaving the dominant gradient information. The gradient magnitude is used to build a map revealing all the parameterized features in the image—for example, lines at a given orientation or circles with a given diameter. For example, to detect lines, we map each gradient point in the pixel space into the Hough parameter space, parameterized as a single point (d,θ) corresponding to all lines with orientation angle θ at distance d from the origin. Curve and circle parameterization uses different variables [301]. The parameter space is quantized into cells or accumulator bins, and each accumulator is updated by summing the number of gradient lines passing through the same Hough points. The accumulator method is modified for detecting parametric curves and circles. Thresholding the accumulator space and re-projecting only the highest accumulator values as overlays back onto the image is useful to highlight features.