leibniz 0.2.0

The package provides a differentiable vector graphics rasterization loss.
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263

//! Boundary length distributions.

use ::burn::tensor::{Int, Tensor, backend::Backend};

use crate::{
    base::geometry::{Command, Indices},
    burn::geometry::{Contour, Quadratic, contour},
};

const EPSILON: f32 = 1e-6;

/// Segment CDF with shape `[segments]`.
pub type Cdf<B> = Tensor<B, 1>;

/// Segment lengths with shape `[segments]`.
pub type Lengths<B> = Tensor<B, 1>;

/// Segment PMF with shape `[segments]`.
pub type Pmf<B> = Tensor<B, 1>;

/// Boundary segment indices with shape `[samples]`.
pub type SegmentIndices<B> = Tensor<B, 1, Int>;

/// Unit samples with shape `[samples]`.
pub type UnitSamples<B> = Tensor<B, 1>;

/// Boundary length distribution for one contour.
pub struct Distribution<B: Backend> {
    lengths: Lengths<B>,
    pmf: Pmf<B>,
    cdf: Cdf<B>,
}

/// Boundary sample records for one contour.
pub struct Records<B: Backend> {
    segment_count: usize,
    segment_indices: SegmentIndices<B>,
    t: UnitSamples<B>,
    segment_pmf: Pmf<B>,
}

impl<B: Backend> Distribution<B> {
    /// Sample boundary records.
    pub fn sample(&self, samples: UnitSamples<B>) -> Records<B> {
        let segment_count = self.segment_count();
        let samples = samples.clamp(0.0, 1.0);
        // Broadcasting the CDF once avoids rebuilding repeated segment grids.
        let cdf = self.cdf.clone().unsqueeze_dim::<2>(0);
        let sample_grid = samples.clone().unsqueeze_dim::<2>(1);
        let segment_indices = cdf
            .greater_equal(sample_grid)
            .int()
            .argmax(1)
            .squeeze_dim::<1>(1);
        let previous_cdf = previous_cdf(self.cdf.clone());
        // Gather instead of select: the backend select funnels the index
        // tensor through a per-element iterator into a vector before the
        // lookup, measuring several times slower than gather for
        // sample-sized lookups.
        let lower = previous_cdf.gather(0, segment_indices.clone());
        let pmf = self.pmf.clone().gather(0, segment_indices.clone());
        let t = (samples - lower) / pmf.clone().clamp_min(EPSILON);

        Records {
            segment_count,
            segment_indices,
            t,
            segment_pmf: pmf,
        }
    }

    /// Number of segments.
    pub fn segment_count(&self) -> usize {
        self.lengths.dims()[0]
    }
}

impl<B: Backend> Records<B> {
    /// Number of segments in the sampled contour.
    pub const fn segment_count(&self) -> usize {
        self.segment_count
    }

    /// Segment indices.
    pub fn segment_indices(&self) -> SegmentIndices<B> {
        self.segment_indices.clone()
    }

    /// Segment probabilities.
    pub fn segment_pmf(&self) -> Pmf<B> {
        self.segment_pmf.clone()
    }

    /// Segment-local curve parameters.
    pub fn t(&self) -> UnitSamples<B> {
        self.t.clone()
    }
}

/// Build a boundary length distribution for one contour.
///
/// # Panics
///
/// Panics if the contour is empty or does not have segment shape
/// `[segments, 2, 2]`.
pub fn distribution<B: Backend>(contour: Contour<B>) -> Distribution<B> {
    let [segment_count, point_count, coordinate_count] = contour.dims();

    assert!(
        segment_count > 0
            && point_count == 2
            && coordinate_count == 2
            && contour.commands().len() == segment_count,
        "contours must have shape [segments, 2, 2]"
    );

    let mut lengths = Vec::with_capacity(segment_count);
    let arguments = contour.arguments();

    for segment in 0..segment_count {
        let indices = Indices::new(segment, segment_count);
        lengths.push(match contour.command(segment) {
            Command::Linear => linear_length(arguments.clone(), indices),
            Command::Quadratic => {
                Quadratic::from_arguments(arguments.clone(), indices).approximate_length()
            }
        });
    }

    let lengths = Tensor::cat(lengths, 0);
    let total = lengths.clone().sum().clamp_min(EPSILON);
    let pmf = lengths.clone() / total;
    let cdf = pmf.clone().cumsum(0);

    Distribution { lengths, pmf, cdf }
}

pub fn unit_samples<B: Backend>(count: usize, device: &B::Device) -> UnitSamples<B> {
    (Tensor::<B, 1, Int>::arange(0..count as i64, device).float() + 0.5) / count as f32
}

fn linear_length<B: Backend>(arguments: contour::Arguments<B>, indices: Indices) -> Tensor<B, 1> {
    let start = point(arguments.clone(), indices.start());
    let end = point(arguments, indices.end());

    (end - start).powi_scalar(2).sum().sqrt()
}

fn point<B: Backend>(arguments: contour::Arguments<B>, index: [usize; 2]) -> Tensor<B, 1> {
    arguments
        .slice([index[0]..index[0] + 1, index[1]..index[1] + 1, 0..2])
        .squeeze_dim::<2>(0)
        .squeeze_dim::<1>(0)
}

fn previous_cdf<B: Backend>(cdf: Cdf<B>) -> Cdf<B> {
    let segment_count = cdf.dims()[0];
    let zeros = Tensor::<B, 1>::zeros([1], &cdf.device());

    if segment_count == 1 {
        zeros
    } else {
        Tensor::cat(vec![zeros, cdf.slice_dim(0, 0..segment_count - 1)], 0)
    }
}

#[cfg(test)]
mod tests {
    use ::burn::tensor::{Tensor, TensorData, backend};

    use super::super::tests::{
        assert_floats, assert_ints, samples, segments, small_triangle, triangle,
    };
    use super::{Cdf, Distribution, EPSILON, Lengths, Pmf, distribution};
    use crate::{
        base::geometry::Command,
        burn::{geometry::Contour, tests::Backend},
    };

    #[test]
    fn builds_contour_length_distribution() {
        let contours = contours::<Backend, _>([triangle(), small_triangle()]);

        assert_eq!(contours.distributions.len(), 2);
        assert!(!contours.distributions.is_empty());
        assert_floats(contours.lengths.clone(), [12.0, 6.0]);
        assert_floats(contours.pmf.clone(), [2.0 / 3.0, 1.0 / 3.0]);
        assert_floats(contours.cdf.clone(), [2.0 / 3.0, 1.0]);
        assert_floats(contours.distributions[1].lengths.clone(), [1.5, 2.0, 2.5]);
    }

    #[test]
    fn builds_quadratic_length_distribution() {
        let distribution = distribution::<Backend>(triangle());

        assert_floats(distribution.lengths.clone(), [3.0, 4.0, 5.0]);
        assert_floats(distribution.pmf.clone(), [0.25, 1.0 / 3.0, 5.0 / 12.0]);
        assert_floats(distribution.cdf.clone(), [0.25, 7.0 / 12.0, 1.0]);
    }

    #[test]
    #[should_panic]
    fn rejects_empty_contour_distribution() {
        let _ = distribution::<Backend>(segments([]));
    }

    #[test]
    #[should_panic]
    fn rejects_invalid_segment_shape_distribution() {
        let invalid = Tensor::<Backend, 3>::from_data(
            TensorData::new(vec![0.0; 6], [1, 3, 2]),
            &Default::default(),
        );

        let _ = distribution::<Backend>(Contour::new(vec![Command::Quadratic], invalid));
    }

    #[test]
    fn samples_boundary_records() {
        let distribution = distribution::<Backend>(triangle());
        let records = distribution.sample(samples([0.125, 5.0 / 12.0, 19.0 / 24.0]));

        assert_eq!(records.segment_count(), 3);
        assert_ints(records.segment_indices(), [0, 1, 2]);
        assert_floats(records.t(), [0.5, 0.5, 0.5]);
        assert_floats(records.segment_pmf(), [0.25, 1.0 / 3.0, 5.0 / 12.0]);
    }

    struct Contours<B: backend::Backend> {
        distributions: Vec<Distribution<B>>,
        lengths: Lengths<B>,
        pmf: Pmf<B>,
        cdf: Cdf<B>,
    }

    fn contours<B, I>(contours: I) -> Contours<B>
    where
        B: backend::Backend,
        I: IntoIterator<Item = Contour<B>>,
    {
        let distributions = contours.into_iter().map(distribution).collect::<Vec<_>>();

        assert!(!distributions.is_empty(), "contours must not be empty");

        let lengths = Tensor::cat(
            distributions
                .iter()
                .map(|distribution| distribution.lengths.clone().sum())
                .collect(),
            0,
        );
        let total = lengths.clone().sum().clamp_min(EPSILON);
        let pmf = lengths.clone() / total;
        let cdf = pmf.clone().cumsum(0);

        Contours {
            distributions,
            lengths,
            pmf,
            cdf,
        }
    }
}