From 066bfdf3adc6164c20f6846c447d524fa0279999 Mon Sep 17 00:00:00 2001 From: aj Date: Wed, 27 Nov 2019 22:45:48 +0000 Subject: [PATCH] added theory background --- .gitignore | 1 + report/coursework.lyx | 469 ++++++++++++++++++++++++++++++++++++++++- report/coursework.lyx~ | 136 ------------ report/coursework.pdf | Bin 72276 -> 199414 bytes 4 files changed, 469 insertions(+), 137 deletions(-) delete mode 100644 report/coursework.lyx~ diff --git a/.gitignore b/.gitignore index 8093953..e1b0ebf 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,4 @@ dataset descriptors *~ +*# diff --git a/report/coursework.lyx b/report/coursework.lyx index 4394b5d..8fa1b62 100644 --- a/report/coursework.lyx +++ b/report/coursework.lyx @@ -121,13 +121,479 @@ LatexCommand tableofcontents \end_layout \begin_layout Section -Theory +Introduction +\end_layout + +\begin_layout Standard +An application of computer vision and visual media processing is that of + viusal search, the ability to quantitatively identify features of an image + such that other images can be compared and ranked based on similarity. +\end_layout + +\begin_layout Standard +These measured features can be arranged as a data structure or descriptor + and a visual search system can be composed of the extraction and comparison + of these descriptors. +\end_layout + +\begin_layout Subsection +Extraction +\end_layout + +\begin_layout Standard +When arranged as three 2D arrays of intensity for each colour channel, an + image can be manipulated and measured to identify features using colour + and shape information. + The methods for doing so have varying applicability and efficacy to a visual + search system, many also have variables which can be tuned to improve performan +ce. +\end_layout + +\begin_layout Subsection +Comparison +\end_layout + +\begin_layout Standard +Typically a descriptor is a single column vector of numbers calculated about + an image. + This vector allows an image descriptor to plotted as a point in a feature + space of the same dimensionality as the vector. + Images that are close together in this feature space will indicate that + they have similar descriptors. + Methods for calculating the distance will determine how images are ranked. +\end_layout + +\begin_layout Subsection +Applications +\end_layout + +\begin_layout Standard +Visual search is used in consumer products to generate powerful results + such as Google Lens and Google reverse image search. + It also has applicability as smaller features of products such as 'related + products' results. +\end_layout + +\begin_layout Section +Theoretical Implementations +\end_layout + +\begin_layout Subsection +Average Colour +\end_layout + +\begin_layout Standard +Average colour represents one of the most basic descriptors capable of being + calculated about an image, an array of three numbers for the average red + green and blue intensity values found in the image. + +\end_layout + +\begin_layout Standard +These three numbers hold no information about the distribution of colour + throughout the image and no information based on edge and shape information. + The lack of either hinders it's applicability to any real world problems. + The only advantage would be the speed of calculation. +\end_layout + +\begin_layout Subsection +Global Colour Histogram +\end_layout + +\begin_layout Standard +A global colour histogram extracts colour distribution information from + an image which can be used as a descriptor. +\end_layout + +\begin_layout Standard +Each pixel in an image can be plotted as a point in it's 3D colour space + with the axes being red, green and blue intensity values for each pixel. + Visually inspecting this colour space will provide information about colour + scattering found throughout the image. + As different resolutions of images will produce datasets of different sizes + in the feature space, a descriptor must be devised that transforms this + data into a resolution agnostic form which can be compared. +\end_layout + +\begin_layout Standard +Each axes is partitioned into +\begin_inset Formula $q$ +\end_inset + + divisions so that a histogram can be calculated for each colour channel. + Each channel's intensity value, +\begin_inset Formula $val$ +\end_inset + +, can be converted into an integer bin value using equation +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:integer-bin-calc" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, where floor strips a float value into an integer by truncating all values + past the decimal point. +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +bin\:val=floor\left(q\cdotp\frac{val}{256}\right)\label{eq:integer-bin-calc} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +This allows each pixel to now be represented as a 3D point of three 'binned' + values, a full RGB colour space has been reduced to three colour histrograms, + one for each channel. + In order to arrange this as a descriptor each point should be further reduced + to a single number so that a global histogram can be formed of these values. + This is done by taking decimal bin integers and concatenating them into + a single number in base +\begin_inset Formula $q$ +\end_inset + +. + For an RGB colour space, each pixel can be augmented as shown in equation + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:base-conversion" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\begin_inset Formula +\begin{equation} +pixel\:bin=red\:bin\cdotp q^{2}+green\:bin\cdotp q^{1}+blue\:bin\cdotp q^{0}\label{eq:base-conversion} +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Calculating a histogram of each pixel's bin value will function as a descriptor + for the image once normalised by count. + This normalisation will remove the effect of changing resolutions of image. +\end_layout + +\begin_layout Standard +Each descriptor plots an image as a point in a +\begin_inset Formula $q^{3}$ +\end_inset + +-dimensional feature space where similiarity can be computed using a suitable + distance measure (L1 norm for example). +\end_layout + +\begin_layout Subsubsection +Efficacy +\end_layout + +\begin_layout Standard +The advantage of global colour histogram over the average RGB descriptor + is that amounts of colours are now represented in the descriptor. + Clusters of similar colours representing objects or backgrounds will be + captured and can be compared. +\end_layout + +\begin_layout Standard +A global histogram, however, holds no spatial colour information, this is + lost by plotting the pixels in their colour space. + +\end_layout + +\begin_layout Standard +This suggests that performing a pixel shuffling operation on the image will + not affect the extracted descriptor which has implications on the adequacy + of the methodology for a visual search system. +\end_layout + +\begin_layout Subsection +Spatial Colour +\end_layout + +\begin_layout Standard +Spatial techniques involve calculating descriptors tht are discriminative + between colour and shape information in different regions of the image. + This is done by dividing the image into a grid of cells and then calculating + individual 'sub-descriptors' which are concatenated into the global image + descriptor. +\end_layout + +\begin_layout Standard +These sub-descriptors can be calculated using any approprate method however + a main consideration should be the dimensionality of the final descriptor. + This can be calculted using the following equation, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +D_{total}=W\cdotp H\cdotp D_{sub-descriptor} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Where +\begin_inset Formula $W$ +\end_inset + + and +\begin_inset Formula $H$ +\end_inset + + refer to the number of columns and rows of the determined grid respectively. +\end_layout + +\begin_layout Standard +It would be feasible to calculate a colour histogram however this already + generates a desciptor of +\begin_inset Formula $q^{3}$ +\end_inset + + dimensionality, where +\begin_inset Formula $q$ +\end_inset + + is the number of divisions. +\end_layout + +\begin_layout Standard +For example using a +\begin_inset Formula $q$ +\end_inset + + value of 4 and a spatial grid of 6 x 4 would produce a descriptor in 1536 + dimenions, while a +\begin_inset Formula $q$ +\end_inset + + of 6 with a a grid of 10 x 6 is 12,960 dimensional. +\end_layout + +\begin_layout Standard +This is an extremely high value and will increase the time taken to calculate + and compare descriptors. +\end_layout + +\begin_layout Standard +For a spatial colour descriptor the average RGB values for each cell can + be used as these sub descriptors will be three dimensional reducing the + total value. +\end_layout + +\begin_layout Subsection +Spatial Texture +\end_layout + +\begin_layout Standard +Spatial texture replaces the colour sub-desciptor from before with a descriptor + that reflects the texture found in the image as described by the edges + that can be detected. + +\end_layout + +\begin_layout Subsubsection +Edge Detection +\end_layout + +\begin_layout Standard +Edges can be detected in an image by finding areas where neighbouring pixels + have significantly different intensities. +\end_layout + +\begin_layout Standard +Mathematically this can be seen as taking the first derivative of the image + by convolving it with a Sobel filter. + The Sobel filters are a pair of 3x3 kernels, one for each axes (see figure + +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:3x3-Sobel-filter" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +), which approximates the gradient of the intensity of an image. +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Formula $S_{x}=\begin{bmatrix}-1 & 0 & +1\\ +-2 & 0 & +2\\ +-1 & 0 & +1 +\end{bmatrix}$ +\end_inset + + +\begin_inset space \qquad{} +\end_inset + + +\begin_inset Formula $S_{y}=\begin{bmatrix}+1 & +2 & +1\\ +0 & 0 & 0\\ +-1 & -2 & -1 +\end{bmatrix}$ +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +3x3 Sobel filter kernels for +\begin_inset Formula $x$ +\end_inset + + and +\begin_inset Formula $y$ +\end_inset + + axes +\begin_inset CommandInset label +LatexCommand label +name "fig:3x3-Sobel-filter" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The results of convolving each filter with the image are two images that + express the intensity of edges in that axes. + +\end_layout + +\begin_layout Standard +From here a composite edge magnitude image of the two can be calculated + as shown, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +G_{composite}=\sqrt{G_{x}^{2}+G_{y}^{2}} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +With the angles of the edges calculated as follows, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\Theta=\arctan\left(\frac{G_{y}}{G_{x}}\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Subsubsection +Application +\end_layout + +\begin_layout Standard +To create a descriptor, both the angle and magnitude information will be + used, the descriptor itself will reflect information about the angle of + the edges found. +\end_layout + +\begin_layout Standard +First the image grid cells will be thresholded using the magnitude values. + Magnitude values can be seen to represent the confidence with which edges + can be found and so here a decision is effectively being made as to what + are and are not edges, this value can be tuned to best match the applcation. +\end_layout + +\begin_layout Standard +Once a thresholded edge maginute image has been found, a normalised histogram + will be calculated for the angles of these edges. + This histograms of each grid cell will act as the descriptor when concatenated + into a vector of dimensionality, +\begin_inset Formula $D$ +\end_inset + +, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +D_{total}=W\cdotp H\cdotp q +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Where +\begin_inset Formula $q$ +\end_inset + + refers to the number of edge histogram bins. +\end_layout + +\begin_layout Section +Test Methods \end_layout \begin_layout Section Results \end_layout +\begin_layout Section +Discussion +\end_layout + \begin_layout Section Conclusions \end_layout @@ -140,6 +606,7 @@ Conclusions \end_layout \begin_layout Standard +\start_of_appendix \begin_inset CommandInset bibtex LatexCommand bibtex btprint "btPrintCited" diff --git a/report/coursework.lyx~ b/report/coursework.lyx~ deleted file mode 100644 index d7d84b8..0000000 --- a/report/coursework.lyx~ +++ /dev/null @@ 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For more info see http://www.lyx.org/ -\lyxformat 544 -\begin_document -\begin_header -\save_transient_properties true -\origin unavailable -\textclass article -\use_default_options true -\maintain_unincluded_children false -\language english -\language_package default -\inputencoding auto -\fontencoding global -\font_roman "default" "default" -\font_sans "default" "default" -\font_typewriter "default" "default" -\font_math "auto" "auto" -\font_default_family default -\use_non_tex_fonts false -\font_sc false -\font_osf false -\font_sf_scale 100 100 -\font_tt_scale 100 100 -\use_microtype false -\use_dash_ligatures true -\graphics default -\default_output_format default -\output_sync 0 -\bibtex_command default -\index_command default -\paperfontsize default -\spacing single -\use_hyperref false -\papersize default -\use_geometry true -\use_package amsmath 1 -\use_package amssymb 1 -\use_package cancel 1 -\use_package esint 1 -\use_package mathdots 1 -\use_package mathtools 1 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