09/22/2009

Blob Measurements or Statistics

Blob statistics are selected in groups by category.

These measurements are used alone, or combined into form factors to describe blob size and shape.

## Blob statistics.

Each blob statistic has a long descriptive name, and a short nickname.  Lispix gives the nickname in most tabular output.

Area

The area of a blob is the pixel count inside the boundary.  Lispix stores the boundary between pixes, and displays it this way when the zoom is large enough.  Blobs do not have holes - the outline of a blob is single and simply connected.  Any holes in the object, which would appear as non-red areas inside an object, are filled in by the blobber before reporting blob statistics.  Holes are not counted in the area when initially blobbing objects - the number of pixels in the blob less holes must be equal to or greater than the Min Area.

### Perimeter

The perimeter is the length of the between-pixel outline, in pixel units.  A pixel has an edge length of 1, and an area of 1 (pixel).  This measurement may be greater than the perimeter value you might expect on visual examination, depending on the wigglyness of the outline.  The variation of this wigglyness with smoothing is the basis of the boundary fractal dimension statistics.  In general, the perimeter depends strongly on the resolution of the image, whereas the area, or maximum caliper diameter, for example, do not.  (Note - that the perimeter of Mathematical fractals is infinite.)

• perimeter                             prim      Length of the outline.
• perimeter percentage edge points      p%e       Fraction of perimeter points (pixels) on edge of image
• perimeter edge count                pe-cnt      Number of perimeter points on edge of image
• perimeter centroid x                p-cent x    Centroid calculated from perimeter
• perimeter centroid y                p-cent y
• perimeter average 20 steps          prm av20    Length of smoothed perimeter - 20 adjacent perimeter points. averaged together.  All of the perimeter points averaged together collapse into the centroid above.

### Caliper Diameter

Caliper (or Feret) diameters are good, intuitive, orientation independent linear measures of size, which are affected very little by image resolution.  If one imagines rotating the object between the jaws of an imaginary caliper while keeping the both jaws in contact with the object, then we naturally have these four values

• maximum caliper diameter            mxcl d      Maximum distance between caliper jaws.  Measure of Length.
• maximum caliper diameter angle      mxcl ang    Angle at which this occurs - orientation of the diameter itself, clockwise from the X-axis (horizontal, or East).
• minimum caliper diameter            mncl d      Minimum distance between caliper jaws.  Measure of Width.
• minimum caliper diameter angle      mncl ang

Lispix calculates the caliper diameters from the convex hull, using a fast an exact method.  Another common method involves rotation of coordinate axes to find the maximum and minimum calpers (Russ 1995, p. 514), but for long, thin objects, the rotation increment must be small even for a good approximate measurement.

### Convex Hull

The convex hull, or "taught band outline" is useful for characterizing shape because it is not sensitive to image noise or resolution as is the perimeter, and it ignores concavities in the outline.  For example, the perimeter (length of the outline) and convex hull perimeter are the same for a sphere or rectangle, but very different for a star or banana shape.

• convex hull area                    CH area     area of convex hull or taut band outline.
• convex hull perimeter               CH prim     this perimeter is not sensitive to image noise or resolution.
• convex hull num points              CH n p      Number of sides in the convex hull.
• convex hull max edge                CH mx e     Length of maximum edge of the convex hull.
• convex hull num short edges         CH shrt     Number of edges shorter than 2*perimeter/n-sides.
• convex hull centroid x              CH cnt x
• convex hull centroid y              CH cnt y

### Inscribed Circle

The diameter of the maximum inscribed circle is a good width measurment of width.  It may be smaller than the minimum caliper diameter.  Its center is the best place to aim an electron beam for elemental analysis because it is the point in the object that is furthest from any point of the boundary.

• inscribed circle radius             i cir r     a realistic width parameter.
• inscribed circle x                  i cir x
• inscribed circle y                  i cir y

### Intensity Related Parameters

The intensity related parameters are different in kind from all of the other parameters that Lispix measures except for the Surface Fractal Dimensions.  The intensities of the pixels within the object are averaged or measured in some way.  All pixels within the boundary are counted, including any "holes".

• mean intensity                      int        Average intensity withn the boundary.
• intensity range                     int rng
• integrated intensity                sum-int
• centroid x                          centr x    This centroid is almost the same as p-cent
• centroid y                          centr y
• weighted centroid x                 Wctr x     Intensity weighted centroid.
• weighted centroid y                 Wctr y

### Bounding Rectangle

The bounding rectangle of an object is the rectangle you would make by dragging the mouse, that just touches the object on all four sides.  It is the smallest rectangle to enclose the object, for that specific orientation.  Except for perfectly round objects, the bounding rectangle is orientation dependent -- the only parameter that Lispix measures that is.

• bdr-x1                              bdr-x1     Coordinates - upper right & lower left corners.
• bdr-y1                              bdr-y1
• bdr-x2                              bdr-x2
• bdr-y2                              bdr-y2
• bdr width                           bdr width  Width and height.
• bdr height                          bdr height

### Boundary Fractal Dimensions

The boundary fractal dimension describes the roughness or tortuosity of the boundary.  There are a number of ways to measure the fractal dimension, and all of them give different results.  The method Lispix uses here is by Adler and reflects the change in measured boundary length vs. degree of averaging.  This method seems the most reliable for this purpose.  I have checked it extensively using computer generated fractals.

Boundary points are averaged using a worm of given length (number of points) that crawls around the perimeter.  The averaged boundary point is the averaged coordinates for the worm.  The smoothed boundary consists of all the averaged points.  The more averaging, the more points in the worm, the greater the equivalent step size, the smoother the boundary, the smaller the measured perimeter.  The boundaries of Mathematical fractals looks identical regardless of size scale or magnification.  Of course this is not true for real objects, or for images displayed on the screen because there is no detail beyond some magnification.  For fractals, then, on the one hand, the perimeter grows indefinately as the step size shrinks.  Think of the step size as the span of calipers used to "walk" around the boundary.  For real objects on the other hand, the behaviour of the boundary length (perimeter) with step size is in itself a function of the step size.  This means that the fractal dimension, Db, that one measures by this method is not necessarily constant for changing step sizes.  The more "fractal like" the object is, the less Db will change with step size.  In any case, Lispix reports an averaged Db for seven lograrithmically spaced step sizes.  The Db values indicate fuzziness, tortuosity or "legginess" of an object, for small, medium and large step sizes respectively.  (Adjacent step sizes tend to correlate, i.e. to not be independent.)

Lispix reports fractal statistics for seven step length ranges.  The number following the parameter name or nickname denotes the range, shown as "x" below.

• Boundary Frac D-X                   BF D X     Fractal dimenson of boundary, for step size range X (1 to 7)
• Boundary av StepLen X               BavSL X    Average step size for this range.
• Boundary Count X                    BN X       Number of data points for this range (number of averaged slope measurements).

### Surface Fractal Dimension

The Surface Fractal Dimension measurement is based on a fractal measure of the intensity profile across an object.  Alex Pentland, 1984 and 1986 showed that the fractal dimension of a surface corresponds to the fractal dimension of the brightness of the image of the surface.  This was for Lambertian illumination, which may not be the case exactly for electron micrographs.  The surface brightness texture of particles is sometimes a better discriminator of shape than the the boundary, because the boundary, being an outline of projecting, rather than sectioning, hides cracks.

In any case, the fractal dimension of the brightness should be a useful descriptive parameter for scanning electron micrographs.  Dubue, 1989, shows a method for determing the fractional dimension of an intensity profile.  Lispix uses this method, averaging the results, for consecutive scan lines that are either perpendicular or parallel to the maximum caliper diameter of the object.  These are called the "Surface perpendicular Fractal Dimension" or the "Surface Parallel Fractal Dimension", ie. SrF D, and SlF D, respectively.  The numbers after the names, ranging from 1 to 5, denote the step size ranges, which are the same as used for Db (boundary fractal dimension measures) except for lack of the two largest step ranges which are not useful at this point because of limited image sizes.  There can be far more points going around the perimeter of a "rough" object, than going across a diameter.

The surface fractal dimension for any step length range, is the average of a number of intensity profiles.  The intensity profiles are all parallel to each other, 2 pixels apart, and run either parallel or perpendicular to the maximum caliper diameter.  Intensity profiles that are too short because they occur near the edge of the particle, are ommitted.  Most of the surface of the particle is covered twice in this way - by scan lines in two representative and perpendicular directions.  Horizontal and vertical directions are not used here, because the results would be dependent on the orientation of the particle in the image.

Here are the Surface Fractal Dimension parameters, for step length ranges, X, where X = 1, 2, 3, 4, and 5.

• Surface perpendicular Frac D-X      SrF D X
• Surface perpendicular D Range X     SrD Rg X
• Surface perpendicular av StepLen X  SravSL X
• Surface perpendicular Count X       SrN X
• Surface parallel Frac D-X           SlF D X
• Surface parallel D Range X          SlD Rg X
• Surface parallel av StepLen X       SlavSL X
• Surface parallel Count X            SlN X

### Boundary and Surface Fractal Dimension for "Flat Regions"

Lispix also reports the flattest region of the derivitive or slope of the Richardson plot - the step size range for which the fractal dimension measurement is most constant.  These parameters are less useful than the ones above, because the step size range for the "flat region" varies from particle to particle, making comparison difficult.  Although it might be interesting from a theoretical point of view to know where the flat region of the Richardson Plot is, I am thinking of discontinuing these parameters.  I suspect that the averaged fractal dimension measures over given step size ranges will be useful whether or not they represent a flat region.

• Boundary flat step min              Bf s1     Minimum step size for the region.
• Boundary flat step max              Bf s2     Maximum step size for the region.
• Boundary flat D ave                 Bf Dav    Average Db for the "flat" region.
• Boundary flat D range               Bf Drg    Range of Db for the region.
• Boundary flat Num points            Bf Npt
• Surface perpendicular flat step min Srf s1
• Surface perpendicular flat step max Srf s2
• Surface perpendicular flat D ave    Srf Dav
• Surface perpendicular flat D range  Srf Drg
• Surface perpendicular flat Num points     Srf Npt
• Surface parallel flat step min      Slf s1
• Surface parallel flat step max      Slf s2
• Surface parallel flat D ave         Slf Dav
• Surface parallel flat D range       Slf Drg
• Surface parallel flat Num points    Slf Npt