Object detection:
from one label to a set of objects

Assume a mutually exclusive cat/dog/other classification task.

The input type has not changed, but the requested description now includes two classes.

These illustrative scores sum to one. Selecting the largest score returns one image-level label.

Independent class scores can describe co-occurrence. They do not allocate one record per instance.

The distinction is about representation, not accuracy.

Class and geometry define the target record. Add an inferred score only after establishing that record. A score is not automatically a calibrated probability that the complete detection is correct.

Use this one format in the main derivation. Width and height span the complete box. Other conventions and letterboxing remain in the reference route.

The feature extractor is shared. One box head with four outputs describes only one designated instance.

Annotation policy defines the target instances. Empty images have an empty target set.

Rows can be permuted without changing the set. The class must stay paired with its own box.

For a fixed input shape, the head emits a fixed-capacity tensor. We need a representation and selection rule for a variable set.

K is a capacity, chosen before observing the particular scene. Empty slots need a no-object target.

Every finite candidate design has a capacity limit.

The target set does not identify a first dog. This is an assignment problem; spatial addresses are one possible response.

Retain the feature map instead of collapsing it immediately to one global vector.

The grid indexes output records. Shared head weights operate across locations.

This diagram will recur as later operations are introduced. Faded stages are still unresolved.

This is a generic teaching abstraction. It is not the exact parameterization of every detector.

Here S=4 and C=3: cat, dog and other. The toy head emits 16 records with 8 values each. The class term uses a softmax over the foreground classes for positive candidates.

A cell is not an independent image crop. The dog box deliberately crosses cell boundaries.

The normalized box is a constructed teaching target. The generated dog photograph identifies the object; it is not a measured dataset annotation. The same values, target record and prediction recur throughout the numerical trace.

The normalized box is a constructed teaching target. The generated dog photograph identifies the object; it is not a measured dataset annotation. The same values, target record and prediction recur throughout the numerical trace. Row and column are zero based.

The normalized box is a constructed teaching target. The generated dog photograph identifies the object; it is not a measured dataset annotation. The same values, target record and prediction recur throughout the numerical trace. The offsets are fractions of one cell. Width and height are fractions of the full image.

The normalized box is a constructed teaching target. The generated dog photograph identifies the object; it is not a measured dataset annotation. The same values, target record and prediction recur throughout the numerical trace.

These are fixed illustrative predictions, not measured detector output. Every loss and decoding calculation uses exactly these values from the shared JSON.

The simplified centre-cell policy chooses exactly one positive. Modern detectors use model-specific rules, often with several positives per object.

This toy policy has positives and background, with no ignore band. Other policies can exclude ambiguous candidates. Do not silently treat ignore as background.

Binary cross-entropy for this positive target reduces to minus log p. Natural logarithms are used.

Categorical cross-entropy selects the probability of the annotated class. This term supervises the assigned positive, not the background candidate.

We sum absolute differences, rather than averaging. Geometry combines two local centre offsets and two image-normalized sizes in this deliberately simple pedagogical loss. It is not attributed to a YOLO version.

The total is computed from unrounded component values; displayed components are rounded to three decimals. This is the positive-candidate loss, not the complete batch loss.

Background does not learn the dog box from the adjacent positive cell.

This separate 8×8 example motivates imbalance; it does not change the 4×4 numerical trace. Easy negative terms can dominate an unweighted sum.

The record and losses are a generic dense-detector teaching abstraction. Original YOLOv1 confidence, anchor-free heads, distributional regression and DETR are not identical parameterizations.

Use the predicted offsets, not the target offsets. The model has not changed its record; decoding only interprets its parameterization.

This product is the declared score for the toy factorization. Other heads use other definitions. It does not add another learned field to the prediction.

We have followed one dog candidate through target creation, supervision and decoding. The parcel scene now isolates selection among several candidates.

The parcel photograph and A-E values are constructed teaching data. Candidate IDs, coordinates, scores and colours stay fixed in every slide and the lab. Colours identify candidates, not correctness.

Filtering tests the score, not spatial overlap. E remains faintly visible so the identity and geometry do not disappear from the explanation.

All areas are in normalized image units and come from the displayed parcel boxes. B lies inside A in this example. IoU near 1 means strong overlap; near 0 means little overlap.

Scores, geometry and ID colours do not change. Green KEEP and red suppression text encode state separately. NMS is greedy prediction-to-prediction comparison, not truth matching.

The reported count is the number of retained predictions, not a guarantee of the true count.

The example deliberately aligns the two boxes so only class grouping changes. Same-class crowded objects can still suppress each other.

This is the dense-detector answer to the opening question.

The operands and purpose differ, even when all three use boxes and overlap. Training assignment is not NMS or evaluation matching.

In this simplified rule, process predictions in descending score order and choose the greatest-IoU available same-class truth. Official evaluators also handle crowd, ignore, area and detection limits.

The example has two matched predictions, one false positive, and no missed truth.

AP is not precision at one arbitrary deployment threshold. Apply the benchmark protocol, including its detection limits. An aggressive prefilter can truncate the ranking and reduce measured recall.

Each prediction appears with its point; future labels are hidden. The two true positives refer to distinct targets. Axes remain fixed in all builds.

The green envelope uses the best precision at or beyond each recall. This toy continuous area differs from protocol-specific sampled AP such as COCO 101-point averaging.

Constructed per-class AP values use the same protocol. Their mean is not pooled class-agnostic AP.

COCO additionally specifies detection caps, area ranges, recall sampling and crowd/ignore handling. Evaluation thresholds do not rerun NMS.

Queries are learned vectors, not text prompts or fixed class labels. Their count still limits output capacity.

Distinguish K, the number of queries, from K in the attention notation for keys. In original DETR, the decoder combines self-attention among queries with cross-attention to encoded image features.

Do not assume the shown pairing follows from row order. This is an illustrative lowest-cost assignment; a model computes costs from class scores and box geometry.

Matched queries learn class and geometry. Unmatched queries learn no-object. The mechanism and purpose precede the name. Original DETR trains a set predictor with this matching objective.

This compares the teaching dense model with original DETR. Neither is universally superior, and both retain finite capacity and can make errors.

Annotations assign supervision while training the fixed-capacity head. At inference, decoding and selection return a scene-dependent subset.

The offline lab embeds the same canonical JSON and pure numerical functions as this lecture. No detector or network download is needed. The optional class-ambiguity scenario changes a label only, not candidate geometry or score.

Both routes use fixed capacity to describe a variable scene. The next representation question is what to predict when a box is insufficient.

The format must accompany the data. Centre coordinates are the means of corresponding corners. Width and height are corner differences. Do not confuse full sizes with half sizes.

Continuous coordinates are used. Centre = top-left + half-size. Divide horizontal values by image width and vertical values by image height.

Scale the coordinates, then add the padding offsets. The example uses continuous geometry and symmetric vertical padding. Crops and flips also transform targets.

Check image dimensions, class IDs, dtypes and coordinate format. Transform the image and target together. Many torchvision detection models reserve class 0 for background.

This round trip uses the target offsets. The main inference example uses different predicted offsets. Width and height remain normalized to the image. Other architectures use different box parameterizations.

Original YOLOv1 shares class probabilities between the two box predictors in a cell. Its confidence combines object presence and localization quality: Pr(object) × IoU. This is not the independent objectness term in our toy record.

The predictors specialize in geometry through training. Two independent objects with centres in the same cell are still difficult to represent. These original predictors are not predefined anchor templates.

Anchor-based heads can emit several records at one location. Assignment is model-specific and can use IoU and heuristics. Anchor-free detectors do not require these reference boxes.

Illustrative code, not a complete detector. Define all dimensions consistently. Separate branches and extra regression channels are common.

Feature Pyramid Networks combine coarse semantic features with higher-resolution maps through a top-down pathway and lateral connections. Image resizing alone cannot guarantee small-object performance.

Natural logarithms. A high probability is good for a positive and poor for background. Use numerically stable losses accepting logits, rather than taking logs manually.

The choice depends on parameterization and failure modes. Ordinary IoU offers limited guidance for disjoint boxes. GIoU adds enclosing-region geometry. There is no universal ranking of losses.

Clamp intersection width and height at zero for non-overlap. Validate corner ordering before computing areas.

batched_nms compares only matching group indices. Do not mix images under the same class IDs without image-specific groups. Decoding and score filtering are still separate operations.

Apply the declared benchmark protocol. One erroneous prediction and one missed target can occur at the same time.

The envelope is 1 up to recall 0.5 and 2/3 from 0.5 to 1. COCO samples 101 recall levels; do not label this exact continuous toy area as COCO AP.

Both complete lists have the same precision and recall. Putting both correct predictions first improves AP. This controlled example assumes the true positives match distinct targets.

The model includes proposals, decoding, internal score selection and NMS. This keep line adds a user cutoff. Inspect the API before applying another NMS stage.

This is a structural distinction, not a universal speed or accuracy ranking. Faster R-CNN is a two-stage example. Architecture, resolution, hardware and implementation determine practical performance.

Photographs are generated teaching illustrations reused from the existing lecture. Their overlays, predictions and scores are constructed examples, not measured detector outputs. Original provenance remains in lecture9/figures/PROVENANCE.md.