A coupled-contour model for effective segmentation of cotton cross-sections in wide-field microscopic images

Abstract

Image analysis of a fiber cross-section can provide direct measurements for cotton maturity. Effective segmentation of fiber contours in a cross-sectional image is paramount for accurate fiber geometrical measurements. In a wide-field microscopic image, the adhesion, breakage, and ambiguity (low contrast or noise) of fiber contours make the segmentation rather challenging. This paper presents a new approach for contour segmentation that takes advantage of the shape features of the triple concentric contours, called the coupled-contour model (CCM), of a cross-section, and a CCM-based algorithm developed to locate, split, merge, and refine fiber contours based on the established rules concerning contour features. For a wide-field microscopic image (12 megapixels), this CCM-based algorithm could detect >500 fiber cross-sections with a recall rate of 93.53% and a precision rate of 98.13%, and reduced the errors in maturity measurements by 50%.

Keywords

cotton cross-section contour detection contour refinement algorithm performance

Maturity is one of the determinants in evaluating cotton quality because it can affect the tenacity, dyeability, and many other properties of cotton fibers.^1–3 Cross-sectional analysis of cotton fibers provides direct, accurate measurements for fiber fineness and maturity, which are often regarded as the reference data used to validate or calibrate other indirect measurements of these important cotton properties.^4–6 Much research has been conducted with image analysis technology to measure cotton maturity and other fiber parameters from its cross-section image.^7–13 The success of a cross-sectional method using image analysis largely relies on two techniques: fiber cross-sectioning and cross-section detection.¹ Cross-sectioning, including the fiber and grinding and cutting, is the most import step in obtaining analyzable images of fibers.^14,15 A well-established cross-sectioning protocol addressed technical issues in fiber embedding, medium polymerization, and microtome setting, and greatly improved the separability and contrast of individual fibers in the image captured on a microscope with transmitting light.^6,16–18

Cross-section detection is a computational process to extract fiber cross-section contours from an image for geometrical analysis. Results of cross-section detection influence the accuracy of cotton maturity measurements directly. Due to variations in shape and thickness of fiber cross-sections across a sliced sample, fiber borders in different regions of an image may exhibit different levels of contrast or sharpness. Some cross-sections may adhere to each other, while others may be scratched by a dull cutting blade. These image defects are the object recognition problems targeted by image segmentation algorithms, such as the weighted skeleton,³ background flooding,¹ dual-thresholding,¹⁹ and watershed,²⁰ and by contour extracting algorithms, such as snake models,²¹ the gradient-vector-flow model,^22,23 and level-set model.²⁴ In a modern light microscope, a high-resolution, wide-field digital camera is often equipped for fast image acquisition and high-volume measurements of fiber properties by increasing the field of view (FOV) of the microscope and the number of fiber cross-sections in one image frame. Figure 1 is an image captured through a 12-megapixel digital camera and a 20× objective lens on a light microscope (Olympus CH30). This image covers a view area approximately 10 times larger than traditional microscopic images.¹ However, the aforementioned problems in cross-section detection are intensified in a high-resolution, wide-field image because of its high variability in lighting and focus, making these algorithms less viable when processing an image containing cross-sections with vastly distinctive shapes and features.

Figure 1.

A high-resolution, wide-field microscopic image (4272 × 2848 pixels) of cotton cross-sections: (1) a cross-section with a small lumen (mature); (2) a cross-section with a large open lumen (immature); (3) a cross-section with a collapsed lumen; (4) a self-rolling cross-section; (5) a cross-section with impurities; (6) an overlapped cross-section; (7) multiple cross-sections adhered together; (8) a cross-section with connected fiber and lumen borders by a blade scratch; (9) a cross-section with touching fiber and lumen borders.

This paper presents a new approach for cotton cross-section segmentation by utilizing contour shape features to correct and refine defective contours of fiber and lumen borders. A cross-section was first modeled by a triple concentric contour set, which is called the coupled-contour model (CCM) of the cross-section. Based on the CCM, effective rules concerning contour features were established to (1) locate cross-section contours, (2) split adhering contours and merge broken contours, and (3) refine defective contours. In the experiment, the performances, such as recall, precision, and time efficiency, of this CCM-based contour segmentation algorithm were evaluated.

Method

Coupled-contour model of cotton cross-sections

As shown in Figure 2, a typical cotton cross-section is defined by its fiber border and lumen border, which are normally separated by the cellulosic wall. From the two separated borders, three contours can be generated: the fiber outer contour c_o, the fiber inner contour c_i, and the lumen contour c_l. Normally, c_o and c_i are two concentric parallel contours very close to each other. The distance between c_o and c_i is the width of the fiber border, which is influenced by the thickness of the cross-section and the local focusing status. Thus, a cross-section can be depicted by a triple contour set $(c_{o}, c_{i}, c_{l})$ , which is called the CCM of the cross-section.

Figure 2.

Triple contours of a cotton-section. c_o: fiber outer contour; c_i: fiber inner contour; c_l: lumen contour.

The triple contours in a normal CCM should be isolated and concentric. Figure 3 lists a few examples of defective cross-sections and their abnormal contours. Figure 3(a) shows a scratched cross-section with the connected fiber and lumen borders leading to wrong c_i and c_l; Figure 3(b) shows three adhered cross-sections that generate wrong c_o; Figure 3(c) shows a self-rolling cross-section producing wrong c_o, c_i, and $c_{l} .$ By analyzing the sizes, shapes, and locations of the triple contours in a CCM, it is possible to detect the defects and to correct wrong contours.

Figure 3.

Defective cross-sections and contours. (a) A scratched cross-section with correct c_o but wrong c_i and c_l. The green points on c_i are the convex points of the inner contour, which can be used to split c_i into a new inner contour and a lumen contour. (b) Adhered cross-sections with wrong $c_{wo}$ but correctc_i and c_l. The green points on c_o are the concave points of the outer contour, which can be used to split c_ointo multiple outer contours. (c) A cross-section with touching c_i and $c_{l} .$ (Color online only.)

Geometrical features of the CCM

In an image denoted by I, a contour, c (e.g., c_o, c_i, or c_l), is an eight-connected closed curve, which can be expressed by an ordered set of pixels, $c = {p_{0}, p_{1}, \dots, p_{N - 1}}$ ( $N$ is the number of pixels in c). c can also be regarded as a polygon with N vertices in the clockwise order: $p_{0}, p_{1}, \dots, p_{N - 1}$ . Let $n_{c}$ , $CCP (c)$ , and $CVP (c)$ denote a normal vector of the plane that polygon c resides on, the set of concave points (CCP), and the set of convex points (CVP) on c, respectively. Then

CCP (c) = {p_{k} | (0 \leq k \leq N - 1) \land ((p_{k - K_{s}} p_{k} \times p_{k} p_{k + K_{s}}) \cdot n_{c} > 0) \land (∠ p_{k - K_{s}} p_{k} p_{k + K_{s}} \leq T_{angle})}

(1)

CVP (c) = {p_{k} | (0 \leq k \leq N - 1) \land ((p_{k - K_{s}} p_{k} \times p_{k} p_{k + K_{s}}) \cdot n_{c} < 0) \land (∠ p_{k - K_{s}} p_{k} p_{k + K_{s}} \leq T_{angle})}

(2)

where

T_{angle}

is an angle threshold for judging concave or convex points, which was set to 45^o in this research, and K_s is a step length used to regulate the accuracy of calculating a concave or convex angle at point p_k, which was set to be 5.

CCP

and

CVP

are the candidate points where contours are split or merged.

Let $Length (c)$ and $Area (c)$ denote the functions of calculating the length (or perimeter) and the area bounded by contour c. An important shape factor of c – circularity – can be defined as

Circularity (c) = 4 π Area (c) (Length (c)) 2

(3)

The geometrical features of fiber cross-sections are defined by the length (perimeter), area, and circularity of triple contours. For a given cotton variety, these parameters vary in certain ranges, which depend on the maturity of individual fibers and the cross-sectioning locations along fiber axes. The limits on these parameters can be set to distinguish abnormal contours from regular cross-sections. Let $T_{L}^{min}$ and $T_{L}^{max}$ be the upper and lower length thresholds, $T_{A}^{min}$ and $T_{A}^{max}$ the upper and lower area thresholds, and $T_{C} (0 \leq T_{C} \leq 1)$ the circularity threshold. When a CCM is known, the criteria to determine if the CCM represents a regular fiber cross-section can be expressed as a function, $Q_{c} (\cdot)$

Q_{c} (c) = (T_{L}^{min} \leq Length (c) \leq T_{L}^{max}) \land (T_{A}^{min} \leq Area (c) \leq T_{A}^{max}) \land (Circularity (c) > T_{C})

(4)

where c is c_o or c_i in the CCM. If both

Q_{c} (c_{o})

and

Q_{c} (c_{i})

are true, the CCM is a candidate for a normal cross-section. Otherwise, it indicates a contour of a defective cross-section or a non-lint object.

The candidate CCM can be further checked with the relationships among the triple contours $(c_{o}, c_{i}, c_{l})$ by a function defined as follows

Q_{oi} (c_{o}, c_{i}, c_{l}) = (Region (c_{o}) \supset Region (c_{i}) \supset Region (c_{l})) \land (Length (c_{o}) \geq Length (c_{i}) \geq Length (c_{l})) \land Q_{c} (c_{o}) \land Q_{c} (c_{i}) \land ((Area (c_{o}) - Area (c_{i})) / Area (c_{o}) \leq T_{AR})

(5)

where

Region (\cdot)

is a function to count the image pixels bounded by a contour and

T_{AR}

is a threshold used to filter false cross-sections by checking the difference between the areas of the inner and outer contours.

T_{AR}

is often set around 0.6 because the coupled outer and inner contours (c_o and c_i) of a regular cross-section should be close to each other. If

Q_{oi} (c_{o}, c_{i}, c_{l})

is true, then the CCM, that is,

(c_{o}, c_{i}, c_{l})

, represents a measurable cross-section. Otherwise, the CCM needs the modifications to correct the imperfect

(c_{o}, c_{i}, c_{l})

After the $(c_{o}, c_{i}, c_{l})$ of a cross-section is validated, the mid-contour, c_f, between the c_o and c_i, can be generated to represent the ultimate fiber contour, and the maturity, θ, can be calculated as follows¹

θ = 4 π A_{f} (P_{f}) 2 = 4 π (Area (c_{f}) - Area (c_{l})) (Length (c_{f})) 2

(6)

where A_f and P_f are the area and perimeter of the cellulose wall bounded by c_f and c_l. After the maturity of all cross-sections in an image are measured, the descriptive statistics of maturity data (θ), including mean (

M_{θ}

), standard deviation (

{SD}_{θ}

), skewness (

S_{θ}

), and kurtosis (

K_{θ}

), can be obtained to examine the features of the maturity distribution.²⁵

CCM-based contour detection algorithm

Locating outer and inner contours

For any pixel $I (i, j)$ in image I, its neighborhood, denoted as $neighbor (i, j)$ , can be defined by a square or circle centered at $I (i, j)$ . A dynamic threshold, represented by $T_{neighbor (i, j)}$ , can be calculated using the adaptive thresholding algorithm,¹ and then image I can be segmented into background regions and cross-sections regions. Contours of cross-section regions are extracted, and those contours that are not included in any other contours are outer contours. For an outer contour c_o, the average threshold, $T_{Region (c_{o})}$ , for all the pixels inside $Region (c_{o})$ is calculated as

T_{Region (c_{o})} = \sum_{I (i, j) \in Region (c_{o})} T_{neighbor (i, j)} M

(7)

where M is the number of pixels of

Region (c_{o})

. Let

H (c_{o})

denote the convex hull of c_o, and

Δ T

be the tolerance for pixel contrast. The image within

H (c_{o})

can be segmented by a threshold interval

[T_{Region (c_{o})} - Δ T, T_{Region (c_{o})} + Δ T]

After the segmentation, there may be multiple contours in c_o. Let $C_{2 Δ T}$ be the set of all the found contours, and $c_{o}'$ be the optimized contour of $c_{o} .$ Optimizing c_o can be performed as follows where $BBox (c)$ is the function calculating the bounding box of c. $BBox (c) ≅ BBox (c_{o})$ implies that the difference between the bounding boxes of c and c_o is smaller than a pre-specified value, which was set to 4 pixels in this paper. If C_c denotes a set of contours enclosed by c_o, then the set of optimized inner contours $C_{c_{o}'}^{i}$ within $c_{o}'$ can be detected by

C_{c_{o}'}^{i} = {c | (c \in C_{c} \cup C_{2 Δ T}) \land (Region (c_{o}') \supset Region (c)) \land (= c' \in C_{c} \cup C_{Δ T} (Region (c') \supset Region (c)))}

(9)

For any inner contour $c_{i} \in C_{c_{o}'}^{i}$ , lumen contours, denoted by $C_{c_{o}'}^{l}$ , can be also found by

C_{c_{o}'}^{l} = {c | (c \in C_{c}) \land (Region (c_{i}) \supset Region (c))}

(10)

Figure 4 shows two cotton cross-section images ((a) and (d)) and the outer and inner contours located by using the CCM model ((b) and (e)) and the distance regularized level set evolution model (DRLSE)¹⁴ ((c) and (f)). The two images contain three (a) and six (d) adhered cross-sections, respectively. Therefore, their outer contours, $c_{o}'$ or c_o (red lines), would be the contours of multiple cross-sections. Because the CCM includes an optimizing process, the $c_{o}'$ extracted by the CCM ((b) and (e)) is more accurate than the c_o by DRLSE ( $Δ t = 1$ , $μ = 0.2$ , $λ = 5$ , $α = 5$ ) ((c) and (f)). DRLSE needed 325 iterations or 8.580 s for processing image (a), and 370 iterations 17.019 s for image (c), as opposed to <0.065 s in the CCM. More importantly, DRLSE and other snake models cannot detect inner contours. Because of the thickness of the fiber borders, the inner contours of the connected cross-sections remain isolated. The CCM is able to detect individual inner contours c_i, as shown in (b) and (e) (green lines). The $c_{o}'$ of adhered cross-sections can be split using the procedures described in the Splitting outer contours of adhered cross-sections section.

Figure 4.

Adhering cotton cross-sections ((a) and (d)), and the contours extracted by the coupled-contour model ((b) and (e)) and by distance regularized level set evolution mode (DRLSE) ( $Δ t = 1$ , $μ = 0.2$ , $λ = 5$ , $α = 5$ ) ((c) and (f)). (Color online only.)

Merging and splitting inner contours

As shown in Figure 3, an inner contour may be broken into small contours (a), connected with other inner contours (b), or adhered to lumen contour (c). In these cases, inner contours need to be merged or split.

The first case is to connect broken inner contours. For outer contour $c_{o}'$ , if $Q_{c} (c_{o}')$ is true (see equation (4)), and $| C_{c_{o}'}^{i} | > 1$ ( $| \cdot |$ is an operator to compute the cardinal number of a set), then contours in $C_{c_{o}'}^{i}$ could be broken inner contours and need to be merged. Let $C_{c_{o}'}^{i} = {c_{i}^{0}, c_{i}^{1}, \dots, c_{i}^{M}}$ , where $M = | C_{c_{o}'}^{i} |$ . For contour $c_{i}^{j}$ in $C_{c_{o}'}^{i}$

c_{i}^{j} = {p_{0}^{j}, p_{1}^{j}, \dots, p_{N_{j} - 1}^{j}} (0 \leq j \leq M, N_{j} = | c_{i}^{j} |)

(11)

Let the distance of $p_{k}^{j} (0 \leq k \leq N_{j} - 1)$ to outer contour $c_{o}'$ be $dis (p_{k}^{j}, c_{o}')$ . We have

dis (p_{k}^{j}, c_{o}') = {min}_{p \in c_{o}'} dis (p_{k}^{j}, p)

(12)

where

dis (p_{k}^{j}, p)

is the Euclidean distance from

p_{k}^{j}

to p (another way to calculate

dis (p_{k}^{j}, p)

will be discussed in detail in the Refining fiber contour section). The longest fragment of

c_{i}^{j}

that is close to

c_{o}'

can be found by the

Closer (c_{i}^{j}, c_{o}')

function as follows

Closer (c_{i}^{j}, c_{o}') = {p_{k}^{j}, p_{k + 1}^{j}, \dots, p_{k + L}^{j} | dis (p_{k + l}^{j}, c_{o}') \leq T_{dis} 0 \leq l \leq L L < N_{j} - 1}

(13)

where L is the number of points in the longest fragment and

T_{dis}

is the distance threshold that depicts the thickness of fiber border (set to 5 pixels in this paper). If

k + l > N_{j} - 1

, then

p_{k + l}^{j} = p_{k + l - N_{j}}^{j}

. A new contour,

c_{i}'

, can be formed by linking all fragments in

{Closer (c_{i}^{j}, c_{o}') | 0 \leq j \leq M}

in the direction of

c_{o}'

. Another contour,

c_{l}'

(Region (c_{i}') \supset Region (c_{l}'))

, can be formed by linking the remaining fragments of

c_{i}^{0}, c_{i}^{1}, \dots, c_{i}^{M}

in the direction of

c_{o}'

. If

Q_{c} (c_{i}')

is true,

c_{i}'

is selected as the inner contour. Furthermore, if

Q_{oi} (c_{o}', c_{i}', c_{l}')

is true (see equation (5)),

(c_{o}', c_{i}', c_{l}')

is deemed as a regular cross-section. Figure 5 shows two examples of generating new inner contours by merging contours. The green points in the figure are the convex points (CVP) of the inner contours determined by equation (2), and they are the points where contours split or merge.

Figure 5.

Merging inner contours to form new inner contour and lumen contour: (a) cross-section with inner contour noise; (b) contours extracted by adaptive local threshold segmentation where the green points are convex (corner) points; (c) new inner contours and lumen contours. (Color online only.)

The second case is to split the adhered inner contours and lumen contour. For outer contour $c_{o}'$ , if $Q_{c} (c_{o}')$ is true and $| C_{c_{o}'}^{i} | = | {c_{i}} | = 1$ , but $Q_{oi} (c_{i})$ is false, contour c_i could be a contour adhering inner contour and lumen contour, and c_i needs to be split. Let $c_{i} = {p_{0}, p_{1}, \dots, p_{N_{j} - 1}} (N_{j} = | c_{i} |)$ . A pair of points in c_i, $(p_{i_{0}}, p_{j_{0} 0 oughbe}) (0 \leq i_{0} < j_{0} \leq N_{j} - 1)$ , where two contours are adhered to each other, needs to be located. So, c_i could be split into an inner contour and a lumen contour by $(p_{i_{0}}, p_{j_{0} 0 oughbe})$ . Usually, $p_{i 0}$ and $p_{j 0}$ are the two convex points of c_i, and could be found in $CCP (c_{i})$ , determined by equation (1). The two contours generated by splitting c_i with any two concave points $p_{i}, p_{j} (0 \leq i < j \leq N_{j} - 1)$ in $CCP (c_{i})$ are

c_{split}^{0} (p_{i}, p_{j}) = {p_{i + 1}, p_{i + 2}, \dots, p_{j}} \cup Line (p_{j}, p_{i})

(14)

and

where

Line (p_{j}, p_{i})

is a set of points on line from p_j to p_i. If

c_{split}^{0}

and

c_{split}^{1}

satisfy

Region (c_{split}^{0}) \cap Region (c_{split}^{1}) = Ø

(16)

then

p_{i} and p_{j}

can define a valid split, which is denoted as

{split}_{c_{i}} (p_{i}, p_{j}) = {c_{split}^{0}, c_{split}^{1}}

(17)

The larger one of $c_{split}^{0} and c_{split}^{1}$ is the inner contour (c_i) and the other is the lumen contour (c_l). Then, $(p_{i_{0}}, p_{j_{0} 0 oughbe})$ could be found by solving the expression below

(p_{i_{0}}, p_{j_{0} 0 oughbe}) = {argmax}_{p_{i}, p_{j} \in CCP (c_{i}) p_{i} \neq p_{j}} Area (Bigger ({Split}_{c_{i}} (p_{i}, p_{j})))

(18)

where Bigger( ) is a function to select a bigger contour in

split (p_{i}, p_{j})

. Let

Split (p_{i 0}, p_{j 0}) = {c_{i}', c_{l}'}

be a valid split defined by

(p_{i 0}, p_{j 0})

. If

Q_{c} (c_{i}')

is true,

c_{i}'

could be the real inner contour. Furthermore, if

Q_{oi} (c_{o}', c_{i}', c_{l}')

is true,

(c_{o}', c_{i}', c_{l}')

is the detected cross-section to be measured. Figure 6 shows two examples of splitting a contour into a new inner contour and a lumen contour for cross-sections. The splitting on one contour starts at two close convex points (green points).

Figure 6.

Split inner contours to form a new inner contour and lumen contour: (a) cross-section with inner contour noise; (b) contours extracted by adaptive local threshold segmentation where the green points are convex points; (c) new inner contours and lumen contours. (Color online only.)

Splitting outer contours of adhered cross-sections

As shown in Figures 4(b) and 4(e), adhered cross-sections generated one outer contour, $c_{o}'$ , and multiple inner contours, c_i. $c_{o}'$ , must be split into several contours by detecting concave points to make each outer contour match with only one inner contour.

For outer contour $c_{o}'$ and $CCP (c_{o}')$ ={ $p_{0}, \dots, p_{N_{CCP} - 1}$ }( $N_{CCP} isthenumber$ of concave points in $CCP (c_{o}')$ ), a pair of concave points, $(p_{i}, p_{j}) (0 \leq i, j \leq N_{CCP} - 1, i \neq j)$ , can be used to create a valid split, $split (p_{i}, p_{j})$ , if it satisfies equations (14)–(17), and the following expression

\forall (c \in C_{c_{o}'}^{i}) (Line (p_{i}, p_{j}) \cap c = Ø)

(19)

If there are multiple valid splits are found, all valid splits are denoted by a set, $C_{split} (c_{o}')$

C_{split} (c_{o}') = {split (p_{i_{0}}, p_{j_{0}}), \dots, split (p_{i_{m}}, p_{j_{m}}), \dots, \times split (p_{i_{N_{v} - 1}}, p_{j_{N_{v} - 1}})}

(20)

where N_v is the number of valid splits. This splitting procedure takes multiple iterations until no new cross-section can be found by splitting the outer contour. In each iteration, contour c (

c = c_{o}'

in the first iteration) is split by valid splits in

C_{split} (c_{o}')

as shown below.

Valid splits in $C_{split} (c_{o}')$ are selected one by one and used to split c into two contours until one valid split, $Split (p_{i_{m}}, p_{j_{m}}) = {c_{split}^{0} (p_{i_{m}}, p_{j_{m}}), c_{split}^{1} (p_{i_{m}}, p_{j_{m}})} (0 \leq m \leq N_{v} - 1)$ , is found such that $c_{split}^{0} (p_{i_{m}}, p_{j_{m}})$ or $c_{split}^{1} (p_{i_{m}}, p_{j_{m}})$ include only one inner contour c_i.

If $c_{split}^{0} (p_{i_{m}}, p_{j_{m}})$ is a contour that includes only one inner contour c_i, let c_l be lumen contour found by (10). If $Q_{oi} (c_{split}^{0} (p_{i_{m}}, p_{j_{m}}), c_{i}, c_{l})$ is true, $(c_{split}^{0} (p_{i_{m}}, p_{j_{m}}), c_{i}, c_{l})$ is one detected cross-section to be measured. Let $c = c_{split}^{1} (p_{i_{m}}, p_{j_{m}})$ and go to the next iteration.

If $c_{split}^{1} (p_{i_{m}}, p_{j_{m}})$ is a contour that includes only one inner contour c_i, let c_l be lumen contour found by (10). If $Q_{oi} (c_{split}^{1} (p_{i_{m}}, p_{j_{m}}), c_{i}, c_{l})$ is true, $(c_{split}^{1} (p_{i_{m}}, p_{j_{m}}), c_{i}, c_{l})$ is one detected cross-section to be measured. Let $c = c_{split}^{0} (p_{i_{m}}, p_{j_{m}})$ and go to the next iteration.

Otherwise, iteration stops.

Figure 7 shows examples of splitting outer contour $c_{o}'$ of the three adhering cotton cross-sections in Figure 4(a). A set of concave points of $c_{o}'$ is $CCP (c_{o}') = {p_{0}, \dots, p_{5}}$ and is shown in black points in Figure 7(b). There are two valid splits: $split (p_{1}, p_{5})$ and $split (p_{2}, p_{4})$ , as shown by the yellow lines in Figure 7(d).

Figure 7.

Split outer contour of the three adhered cross-sections in Figure 4(b): (a) contours extracted; (b) concave points of outer contours; (c) valid splits (yellow lines): $split (p_{1}, p_{5})$ , $split (p_{2}, p_{4})$ ; (d) split outer contour by $split (p_{1}, p_{5})$ ; (e) split outer contour by $split (p_{2}, p_{4})$ , and three adhered cross-sections are split successfully. (Color online only.)

CCM-based contour refinement algorithm

Refining fiber contour

For a detected cross-section, the outer and inner contours can be refined if they contain abnormal fragments caused by impurities or other noise. The refinement is based on analyzing the curvature of the inner and outer contour, and the distance between them. Figure 8(a) gives one cross-section example whose outer contour was bounded with a small object, and Figure 8(b) shows its detected contours, $(c_{o}', c_{i}, c_{l})$ , with a small window highlighting the erroneous section on $c_{o}'$ , and Figure 8(c) is a close-up view of the small window in Figure 8(b). Assume that $c_{o}'$ has M points, that is, $c_{o}' = {P_{0}, P_{1}, \dots, P_{M - 1}}$ , and c_i has K points, that is, $c_{i} = {p_{0}, p_{1}, \dots, p_{K - 1}}$ . For any point on c_i, $p_{k} \in c_{i} (0 \leq k \leq K - 1)$ , the curvature of c_i at p_k can be assessed by $a_{p_{k}} = ∠ p_{k - K_{s}} p_{k} p_{k + K_{s}}$ (K_s is set to 5 according to (1) and (2)), and the curvature change from p_k to $p_{k + 1}$ can be measured by differential angle $a_{p_{k}}' = a_{p_{(k + 1)}} - a_{p_{k}}$ . The tangent vector of c_i at point p_k is denoted by $v_{k}^{t} ⇀$ that the angle between $v_{k}^{t} ⇀$ and $p_{k - 5} p_{k}$ is equal to the angle between $v_{k}^{t} ⇀$ and $p_{k} p_{k + 5} ⇀$ . The normal vector of c_i at point p_k is denoted by $v_{k}^{n} ⇀$ , which passes through p_k and is vertical to $v_{k}^{t} ⇀$ . It is possible that $v_{k}^{n} ⇀$ will intersect $c_{o}'$ at a point $p_{m} (0 \leq m \leq M - 1)$ . The distance between p_k and p_m, $dis (p_{k}, p_{m})$ , can be calculated and denoted by $d_{p_{k}}^{⊥}$ .

Figure 8.

Outer and inner contour optimization: (a) image of one cross-section; (b) contours detected, the outer contour $c_{o}'$ is red, the inner contour c_i is green, and the lumen contour c_l is blue; (c) calculate the distance from p_k (point on the inner contour) to the outer contour. (Color online only.)

After the differential angles, ${a_{p_{0}}', a_{p_{1}}', \dots, a_{p_{k}}', \dots, a_{p_{K - 1}}'}$ , and the distances, ${d_{p_{0}}^{⊥}, d_{p_{1}}^{⊥}, \dots, d_{p_{k}}^{⊥}, \dots, d_{p_{K - 1}}^{⊥}}$ , for all the points on c_i are calculated in the aforementioned way, they can be used to detect an abnormal change on c_i. Figures 9(a) and (b) show the differential angle and distance curves (in green) of c_i by plotting ${d_{p_{k}}^{⊥}}$ and ${a_{p_{k}}'}$ against index k (abscissa). The two curves ( $d_{p_{k}}^{⊥}$ and $a_{p_{k}}'$ ) for outer contour $c_{o}'$ (in red) can also be generated in the same way, and shown in the same figure for comparison. From the distance curves ( $d_{p_{k}}^{⊥}$ in Figure 9(a)), a sharp peak occurs on both $c_{o}'$ and c_i, indicating an anomalous segment corresponding to an impurity or noise on the contour (a dashed circle). Because $c_{o}'$ and c_i contain different numbers of points (M > K), the peak positions on the two curves shifted.

Figure 9.

The distance curve (a) and differential angle curves (b) of outer contour $c_{o}'$ and inner contour c_i. The abscissa is point index k. (Color online only.)

The abnormal segments on the $c_{o}'$ and c_i distance curves can be detected by distance threshold, $T_{dis}$ , introduced in equation (13), and denoted as

F_{a} (c_{o}') = {P_{m_{1}}, \dots, P_{m_{2}}} (0 \leq m_{1} < m_{2} \leq M - 1)

(21)

and

F_{a} (c_{i}) = {p_{k_{1}}, \dots, p_{k_{2}}} (0 \leq k_{1} < k_{2} \leq K - 1)

(22)

Their corresponding fragments on the $a_{p_{k}}'$ curves (Figure 8(b)) are denoted as

{a_{m_{1}}^{o}, a_{m_{1} + 1}^{o}, \dots, a_{m_{2}}^{o}}, {a_{k_{1}}^{i}, a_{k_{1} + 1}^{i}, \dots, a_{k_{2}}^{i}}

(23)

and

{a_{m_{1}}^{o'}, a_{m_{1} + 1}^{o'}, \dots, a_{m_{2}}^{o'}}, {a_{k_{1}}^{i'}, a_{k_{1} + 1}^{i'}, \dots, a_{k_{2}}^{i'}}

(24)

From the $a_{p_{k}}'$ curves, it can be seen that only the $c_{o}'$ segment ( ${a_{m_{1}}^{o'}, \dots, a_{m_{2}}^{o'}}$ ) has larger variations, and the c_i segment ( ${a_{k_{1}}^{i'}, \dots, a_{k_{2}}^{i'}}$ ) retains low values, that is

\sum_{k = k_{1}}^{k_{2}} a_{k}^{i'} k_{2} - k_{1} + 1 < \sum_{m = m_{1}}^{m_{2}} a_{k}^{o'} m_{2} - m_{1} + 1

(25)

This proves that the abnormal noise occurs on outer contour $c_{o}'$ , and $F_{a} (c_{o}')$ needs to be rectified to be similar to $F_{a} (c_{i})$ . Let $F_{a}^{r} (c_{o}')$ represent the rectified abnormal segment, and $p_{k}^{r} (k_{1} \leq k \leq k_{2})$ be a point on line $Line (p_{k}, p_{m})$ and near p_k, that is

F_{a}^{r} (c_{o}') = {p_{k_{1}}^{r}, p_{k_{1} + 1}^{r}, \dots, p_{k_{2}}^{r}}

(26)

$p_{k}^{r}$ satisfies

{argmin}_{p_{k}^{r} \in Line (p_{k}, p_{m})} dis (p_{k}, p_{k}^{r}) = d_{normal} (c_{i})

(27)

where

d_{normal} (c_{i})

is the average distance of the abnormal segment between

c_{o}'

and c_i

d_{normal} (c_{i}) = \sum_{i = 0}^{k_{1}} d_{p_{i}}^{⊥} + \sum_{i = k_{2}}^{K - 1} d_{p_{i}}^{⊥} (K - 1) - (k_{2} - k_{1} + 1)

(28)

$p_{k}^{r}$ is then selected as a rectification point to replace p_m on $c_{o}'$ . If equation (26) is false, the abnormal noise occurs on inner contour c_i, and $F_{a}^{r} (c_{i}) = {p_{m_{1}}^{r}, \dots, p_{m_{2}}^{r}}$ can be calculated in the same way. Figure 10(a) shows the rectified outer contour segment of the cross-section in Figure 8(a). The new segment follows the shape of the inner contour and has the regular distance to the inner contour. After the mutual rectification with $c_{o}'$ and c_i, the middle curve between $c_{o}'$ and c_i can be used as the final contour, c_f, of the fiber (see Figure 10(b)).

Figure 10.

Contour rectification: (a) rectified outer contour $c_{o}'$ with inner contour c_i; (b) final fiber contour c_f (black), which is the mid contour between $c_{o}'$ and c_i.

Refining lumen contour

Figure 11 displays the cross-sections of three different fibers. When a cotton fiber is fully mature, its inner space is filled with cellulose (Figure 11(a)). An immature fiber has a large lumen that may be collapsed totally (Figure 11(b)) or partially (Figure 11(c)) during the processing or packaging. In all these cases, the detection of lumen can be easily missed (Figure 11(b)), or the contours of noisy regions near the lumen (Figure 11(a)) and discontinued lumen contours (Figure 11(c)) may be generated. To locate a lumen that is too small or incomplete in a cross-section, the skeleton of the cross-section provides useful information.

Figure 11.

Fiber cross-sections with wrong lumen detection: (a) mature fiber with a small lumen and noisy regions; (b) immature fiber with a collapsed lumen; (c) immature fiber with discontinued lumens.

After the fiber contour, c_f, is obtained with the detected c_o and c_i, the skeleton, $c_{f}^{skeleton}$ , of the cross-section indicated by c_f can be extracted in an iterative process¹ (the second column in Figure 11). Let $Region (c_{f}^{skeleton})$ be the image region defined by $c_{f}^{skeleton}$ , and $C_{c_{f}}^{l}$ be the set of contours initially detected within c_f. If the detected lumen contour is $C_{c_{f}}^{l} = Ø$ , $c_{f}^{skeleton}$ can be taken as a closed lumen contour of c_f (Figure 11(b)). Otherwise, $C_{c_{f}}^{l}$ are filters with the following operations to form a new set of lumen contours denoted by $C_{c_{f}}^{l'}$

C_{c_{f}}^{l'} = {c | (c \in C_{c_{f}}^{l}) \land (Region (c_{f}^{skeleton}) \cap Region (c) \neq Ø)}

(29)

The region defined by $C_{c_{f}}^{l}$ can be found as

Then, the contour of $Region (C_{c_{f}}^{l})$ can be extracted to be used as the refined lumen contour of c_f (the third column in Figure 11).

Results and discussion

The above CCM-based algorithm was implemented in Microsoft Visual Studio 2010 and OpenCV 2.4.10 under Windows 7. All tests were executed on a Dell computer (Intel duo core CPU E8400 at 2.99 GHz with 4 GB RAM). The images of 15 different cotton varieties were collected for the software performance evaluations. As a reference method, a previously developed fiber image analysis software (FIAS) and the manual editing tools^1,25 were used to detect as many valid cross-sections as possible in these images, and the measurements were used as the ground truth when evaluating the recall rate and the precision of the algorithm.²⁶ The reference method included the automatic cross-section detections with our software and the post-operator corrections of miss- or wrongly detected cross-sections on the screen. Figure 12(a) displays an example image of cotton cross-sections with a selected region of interest (ROI) for a clearer demonstration with larger contours. Figures 12(b)–(e) present the extracted contours with noise (b), the clean background with highlighted defective contours (circles) (c), the corrected contours (d), and the refined lumen contours (e). Figure 12(f) shows the maturity (θ) distributions obtained by using the CCM-based algorithm and the reference method. The two distributions are highly correlated (R²= 0.984).

Figure 12.

An example image of fiber cross-sections with a selected region of interest (a), locating contours (b), noise filtering with defective contours being highlighted by circles (c), corrections of outer and inner contours (d), refinement of lumen contours (e), maturity (θ) distributions from the coupled-contour model-based algorithm, and the reference (f).

The recall and precision rates of the CCM-based algorithm and the FIAS algorithm¹ were calculated for the 15 cottons that are shown in Figure 13. Averagely, the CCM had 93.53% recall rate and 98.13% precision, which are higher than the FIAS’s 90.24% recall rate and 90.96% precision. For all the cottons analyzed, the CCM produced higher precisions than recall rates, which is especially true for cottons 10 and 11. These two cotton varieties have a high content of dead fibers, which are likely to adhere to each other.

Figure 13.

Recall rate and precision of the coupled-contour model (CCM)-based algorithm and fiber image analysis software (FIAS).

Table 1 lists the mean θ values of the 15 cottons measured by the CCM, the FIAS, and the reference method (i.e., the FIAS plus manual editing). From each cotton image, both the CCM and the FIAS generated higher mean θ values than the reference method. This is because many immature fibers with lower θ values were likely to be missed out in the automatic cross-section detections of the CCM and the FIAS due to their complex shapes, but were salvaged by repairing the cross-section borders manually in the reference method.²⁵ However, the θ values of all the cottons measured by the CCM were closer to the reference values, indicating that the CCM was more robust in detecting immature fibers than the FIAS.

Table 1.

Comparisons of the θ values of the 15 cotton samples

Cotton	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
FIAS	0.646	0.651	0.653	0.669	0.687	0.688	0.640	0.627	0.618	0.556	0.559	0.640	0.651	0.599	0.637
CCM	0.639	0.645	0.641	0.659	0.681	0.678	0.632	0.618	0.611	0.545	0.544	0.631	0.645	0.589	0.626
Reference	0.629	0.637	0.633	0.652	0.674	0.671	0.623	0.610	0.602	0.532	0.533	0.624	0.637	0.579	0.618

FIAS: fiber image analysis software; CCM: coupled-contour model.

The errors of θ values in the CCM and the FIAS were measured by the relative differences between their θ values (θ_ccm and θ_fias) and the reference values (θ_ref), that is, (θ_ccm – θ_ref)/θ_ref. Figure 14(a) shows the θ errors of the CCM and the FIAS over different cottons, and Figure 14(b) shows the θ errors of the two methods over different reference values. The θ errors of the CCM were consistently lower than those of the FIAS, as shown in Figure 14(a). The performance of the CCM was also less influenced by the level of cotton maturity (θ_ref), as shown in Figure 14(b). The FIAS appeared to be more sensitive to lower θ values. These results imply that the CCM can detect contours of cross-sections more accurately than the FIAS, especially for cotton with low maturity.

Figure 14.

Comparisons of θ errors of the coupled-contour model (CCM) and fiber image analysis software (FIAS): (a) θ errors over the 15 cottons; (b) θ errors over the reference θ values.

The time efficiency of the CCM-based algorithm was analyzed by counting time consumption in the main steps of the algorithm, including steps of “locating contours” (LC), “merging inner contours” (MIC), “splitting outer contours” (SOC), and “refining contours” (RC). Figure 15 shows the consumed time in the major steps when processing the 15 cottons. Since the cross-section images of these cottons were associated with the levels of their maturity, each cotton needed different amounts of time in different processing steps. Among these steps, locating and splitting contours (LC and SOC) took the major portions of the entire time, with MIC being the least or negligible. In particular, cottons 10 and 11 required significantly more time in SOC than other cottons, because these two cottons contained a large number of dead fibers, which are more difficult to split than mature fibers.

Figure 15.

Time consumption in major steps of the coupled-contour model-based algorithm.

Conclusions

This paper presented a new algorithm to process wide-field microscopic images of cotton cross-sections for maturity measurement. The paper utilized a triple concentric contour set, called the CCM, to represent a regular cotton cross-section and to tackle common problems associated with contour segmentation, such as contour adherence, breakage, and ambiguity. Based on the CCM, the algorithm established specific rules or functions for locating, splitting, merging, and refining fiber contours by taking advantage of their shared shape features. The experiments on the images of 15 different cottons demonstrated that this CCM-based algorithm could achieve a recall rate of 93.53% and a precision rate of 98.13%, and reduce the maturity calculation errors by 50%.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

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